{"id":46,"date":"2023-09-17T10:54:42","date_gmt":"2023-09-17T10:54:42","guid":{"rendered":"https:\/\/mathority.org\/pt\/representacao-de-funcoes\/"},"modified":"2023-09-17T10:54:42","modified_gmt":"2023-09-17T10:54:42","slug":"representacao-de-funcoes","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/representacao-de-funcoes\/","title":{"rendered":"Representa\u00e7\u00e3o de fun\u00e7\u00e3o"},"content":{"rendered":"<p class=\"has-text-align-left\">Neste artigo veremos <strong>como representar qualquer tipo de fun\u00e7\u00e3o em um gr\u00e1fico.<\/strong> Al\u00e9m disso, voc\u00ea encontrar\u00e1 exerc\u00edcios passo a passo resolvidos sobre a representa\u00e7\u00e3o de fun\u00e7\u00f5es em um gr\u00e1fico. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-representar-una-funcion-en-una-grafica\"><\/span> Como representar uma fun\u00e7\u00e3o em um gr\u00e1fico<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Para representar uma fun\u00e7\u00e3o em um gr\u00e1fico, as seguintes etapas devem ser executadas: <\/p>\n<div style=\"background-color:#FFF3E0; padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 10px; border-radius:30px;\">\n<ol style=\"color:#64B5F6; font-weight: bold;\">\n<li style=\"margin-bottom:16px\"> <span style=\"color:#000000;font-weight: normal;\">Encontre o <strong>dom\u00ednio<\/strong> da fun\u00e7\u00e3o.<\/span><\/li>\n<li style=\"margin-bottom:16px;\"> <span style=\"color:#000000;font-weight: normal;\">Calcule os <strong>pontos de corte<\/strong> da fun\u00e7\u00e3o com os eixos cartesianos.<\/span><\/li>\n<li style=\"margin-bottom:16px;\"> <span style=\"color:#000000;font-weight: normal;\">Calcule as <strong>ass\u00edntotas<\/strong> da fun\u00e7\u00e3o.<\/span><\/li>\n<li style=\"margin-bottom:16px;\"> <span style=\"color:#000000;font-weight: normal;\">Estude a monotonicidade da fun\u00e7\u00e3o e encontre seus <strong>extremos relativos<\/strong> .<\/span><\/li>\n<li style=\"margin-bottom:16px;\"> <span style=\"color:#000000;font-weight: normal;\">Estude a curvatura da fun\u00e7\u00e3o e encontre seus <strong>pontos de inflex\u00e3o<\/strong> .<\/span><\/li>\n<li> <span style=\"color:#000000;font-weight: normal;\"><strong>Trace<\/strong> os pontos de corte, ass\u00edntotas, extremos relativos e pontos de inflex\u00e3o e, em seguida, represente graficamente a fun\u00e7\u00e3o.<\/span> <\/li>\n<\/ol>\n<\/div>\n<h2 class=\"estil_titol_H2 wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-la-representacion-de-una-funcion\"><\/span> Exemplo de representa\u00e7\u00e3o de uma fun\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Para que voc\u00ea possa ver como uma fun\u00e7\u00e3o \u00e9 representada graficamente, resolveremos passo a passo o seguinte exerc\u00edcio:<\/p>\n<ul>\n<li> Trace a seguinte fun\u00e7\u00e3o racional em um gr\u00e1fico:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eb173dfd702785865be0051c9bcb7738_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x^2}{x-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"101\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> A primeira coisa a fazer \u00e9 <strong>calcular o dom\u00ednio da fun\u00e7\u00e3o<\/strong> . Esta \u00e9 uma fun\u00e7\u00e3o racional, ent\u00e3o precisamos igualar o denominador a zero para ver quais n\u00fameros n\u00e3o pertencem ao dom\u00ednio da fun\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2a57ca6c48b6f646aeb64eb7f05e4840_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x-1=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"73\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o, quando x for 1, o denominador ser\u00e1 0 e portanto a fun\u00e7\u00e3o n\u00e3o existir\u00e1. O dom\u00ednio da fun\u00e7\u00e3o, portanto, consiste em todos os n\u00fameros reais, exceto x=1.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-66d11e82f81cd2425ea2e6641e374baf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom } f= \\mathbb{R}-\\{1 \\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"138\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Para encontrar o <strong>ponto de intersec\u00e7\u00e3o com o eixo X<\/strong> , devemos resolver a equa\u00e7\u00e3o<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bbb52c33bfaff434771f0e4ddd4cf677_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)= 0.\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" style=\"vertical-align: -5px;\"><\/p>\n<p> Como a fun\u00e7\u00e3o sempre tem valor 0 no eixo X:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0bce6c022ed0fc63f4659af75888f96c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"67\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-598e2ac6e2410e5d89ee067071c1d280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^2}{x-1} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"73\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> O termo<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d6ca0f3c84745bcbcccc5f4ebf219891_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x -1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"40\" style=\"vertical-align: 0px;\"><\/p>\n<p> Isso envolve dividir todo o lado esquerdo, para que possamos multiplic\u00e1-lo por todo o lado direito:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f086c381edce77135440070151e8ce65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2 = 0 \\cdot (x-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"117\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-97bb5e9fb1f811f609395daafea9e9c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2 = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-69deb06de751e80bf5f09f379ee2bc53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>O ponto de intersec\u00e7\u00e3o com o eixo OX \u00e9 portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2c790019bd70403eba876c59c82c0f9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> E para encontrar o <strong>ponto de intersec\u00e7\u00e3o com o eixo Y<\/strong> , calculamos<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d449eebd1f011aebdf90931f3a66a3b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<p> Como x \u00e9 sempre 0 no eixo Y:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7617c5eab838a7e451fef14b9ccce246_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=\\cfrac{0^2}{0-1} = \\cfrac{0}{-1} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"179\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Portanto, o ponto de corte com o eixo OY \u00e9:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2c790019bd70403eba876c59c82c0f9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Neste caso, quando a fun\u00e7\u00e3o passa pela origem da coordenada, o ponto de intersec\u00e7\u00e3o com o eixo X coincide com o ponto de intersec\u00e7\u00e3o com o eixo Y.<\/p>\n<p> Depois de conhecermos o dom\u00ednio e os pontos de corte, precisamos <strong>calcular as ass\u00edntotas da fun\u00e7\u00e3o<\/strong> .<\/p>\n<p> Para ver se a fun\u00e7\u00e3o possui ass\u00edntotas verticais, precisamos calcular o limite da fun\u00e7\u00e3o em pontos que n\u00e3o pertencem ao dom\u00ednio (neste caso x=1). E se o resultado for infinito, \u00e9 uma ass\u00edntota vertical. Ainda:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d1933bd0c8df1e0e994ff71304ce3627_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 1} \\ \\cfrac{x^2}{x-1} = \\cfrac{1^2}{1-1} = \\cfrac{1}{0} = \\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"220\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Como o limite da fun\u00e7\u00e3o quando x tende a 1 d\u00e1 infinito, x=1 \u00e9 uma ass\u00edntota vertical: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/represente-fonctions-vertical-asymptote.webp\" alt=\"representar fun\u00e7\u00f5es, ass\u00edntota vertical\" class=\"wp-image-2626\" width=\"550\" height=\"604\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Uma vez calculada a ass\u00edntota vertical, \u00e9 necess\u00e1rio calcular os limites laterais da fun\u00e7\u00e3o em rela\u00e7\u00e3o a ela. Como n\u00e3o sabemos se a fun\u00e7\u00e3o tender\u00e1 a -\u221e ou +\u221e \u00e0 medida que se aproxima de x=1 pela esquerda, e n\u00e3o sabemos quando se aproxima de x=1 pela direita.<\/p>\n<p> Assim, procedemos ao c\u00e1lculo do limite lateral esquerdo da fun\u00e7\u00e3o em x=1:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a2a03f93135e37cc8d84b375dfc5b40e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 1^{-}} \\cfrac{x^2}{x-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"84\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Para calcular numericamente um limite lateral em um ponto, devemos substituir um n\u00famero na fun\u00e7\u00e3o que esteja muito pr\u00f3ximo do ponto. Nesse caso, queremos um n\u00famero bem pr\u00f3ximo de 1 \u00e0 esquerda, como 0,9. Portanto, substitu\u00edmos o ponto 0,9 na fun\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9828df62f758cce355c916237e82b766_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{0,9^2}{0,9-1}=\\cfrac{0,81}{-0,1}=-81\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"176\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Os limites laterais de uma ass\u00edntota s\u00f3 podem fornecer +\u221e ou -\u221e. E como substituindo um n\u00famero muito pr\u00f3ximo de 1 \u00e0 esquerda na fun\u00e7\u00e3o obtivemos um resultado negativo, o limite \u00e0 esquerda \u00e9 -\u221e:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e6a013ed3f883f6822c94fb274360b7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 1^{-}} \\cfrac{x^2}{x-1} = \\bm{-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"139\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Agora fazemos o mesmo procedimento com o limite do lado direito:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1aef4f24697f29600025e161323e07dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 1^{+}} \\cfrac{x^2}{x-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"84\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Substitu\u00edmos um n\u00famero muito pr\u00f3ximo de 1 \u00e0 direita na fun\u00e7\u00e3o. Por exemplo, ponto 1.1:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-93573af45a33dd1b799499d4568686f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{1,1^2}{1,1-1}=\\cfrac{1,21}{0,1}=+12,1\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"187\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Neste caso, o resultado do limite lateral \u00e9 um n\u00famero positivo. O limite \u00e0 direita \u00e9, portanto, +\u221e:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8800ea3138e573b9285eef458f08fa91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 1^{+}} \\cfrac{x^2}{x-1} = \\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"138\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Concluindo, em x = 1 a fun\u00e7\u00e3o tende para menos infinito \u00e0 esquerda e mais infinito \u00e0 direita: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/fonctions-graphiques-asymptote-verticale.webp\" alt=\"fun\u00e7\u00f5es gr\u00e1ficas, ass\u00edntota vertical\" class=\"wp-image-2627\" width=\"550\" height=\"602\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Por outro lado, a ass\u00edntota horizontal da fun\u00e7\u00e3o ser\u00e1 o resultado do limite infinito da fun\u00e7\u00e3o. Ainda: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-df700cb3b219bb9b3dff71de849ac381_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\ \\cfrac{x^2}{x-1} = \\cfrac{+\\infty}{+\\infty } =+\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"217\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div style=\"background-color:#FFFDE7; padding-top: 23px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px; border: 2.5px dashed #FFB74D; border-radius:20px;\">\n<p> <strong>Lembre-se<\/strong> de como calcular os limites infinitos de fun\u00e7\u00f5es racionais:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2c969e4b99985b44006e57d554ff0247_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to \\pm \\infty}}\\frac{a_nx^r+a_{n-1}x^{r-1}+a_{n-2}x^{r-2}+\\dots}{b_nx^s+b_{n-1}x^{s-1}+b_{n-2}x^{s-2}+\\dots}=\\left\\{ \\begin{array}{lcl} 0 &amp; \\text{si} &amp; r<s \\\\[3ex]=&quot;&quot; \\cfrac{a_n}{b_n}=&quot;&quot; &amp;=&quot;&quot; \\text{si}=&quot;&quot; r=&quot;s&quot; \\\\[5ex]=&quot;&quot; \\pm=&quot;&quot; \\infty=&quot;&quot;>s \\end{array}\\right.&#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;139&#8243; width=&#8221;767&#8243; style=&#8221;vertical-align: 0px;&#8221;><\/p>\n<\/p>\n<\/div>\n<p> O limite infinito da fun\u00e7\u00e3o nos deu +\u221e, ent\u00e3o a fun\u00e7\u00e3o n\u00e3o tem ass\u00edntota horizontal.<\/p>\n<p> Agora calculamos a ass\u00edntota obl\u00edqua. As ass\u00edntotas obl\u00edquas t\u00eam a forma<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8e4adcc4368f6296906b6231bf17a6a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=mx+n\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: -4px;\"><\/p>\n<p> . E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00c9 calculado com a seguinte f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9dcdd1bef23f97f1397a19964de98fa6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to +\\infty} f(x):x\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"147\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2be9bdf7dce3b2e7d79079b5528fe177_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to +\\infty} \\cfrac{x^2}{x-1}:x\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"156\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> O x \u00e9 como se tivesse 1 como denominador:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2f799c07d0e78080d55bc31bc5278446_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to +\\infty} \\cfrac{x^2}{x-1}:\\cfrac{x}{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"158\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> \u00c9 uma divis\u00e3o de fra\u00e7\u00f5es, ent\u00e3o multiplicamos elas transversalmente:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-66422a93cd9b73474054b6370ecbdc76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to +\\infty} \\cfrac{x^2 \\cdot 1 }{(x-1) \\cdot x}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"168\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-529870b7f0ef84afc97448cbc7855056_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to +\\infty} \\cfrac{x^2 }{x^2-x}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"140\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> E calculamos o limite:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e82cefa1bb58cc251d600848a5a0a57d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to +\\infty} \\cfrac{x^2 }{x^2-x} =  \\cfrac{+\\infty}{+\\infty } = \\cfrac{1}{1} = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"271\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Ent\u00e3o m = 1. Agora calculamos<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> com a seguinte f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-81d8f8b6af95602f96372b8abe4af497_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to +\\infty} \\bigl[f(x)-mx\\bigr]\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"174\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-10dfa8fdcfbf0c978e02374654a66b7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to +\\infty} \\left[\\cfrac{x^2}{x-1}-1x\\right] = \\cfrac{+\\infty}{+\\infty} -(+\\infty) = +\\infty - \\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"413\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p> Mas obtemos a indetermina\u00e7\u00e3o infinito menos infinito, ent\u00e3o temos que reduzir os termos a um denominador comum. Para fazer isso, multiplicamos e dividimos o termo x pelo denominador da fra\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-70026c2aed1bb58a120f8c18423d9ef5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to +\\infty}\\left[\\cfrac{x^2}{x-1}-x\\right]  = \\lim_{x \\to +\\infty} \\left[\\cfrac{x^2}{x-1}-\\cfrac{x\\cdot (x-1)}{x-1} \\right] = \\lim_{x \\to +\\infty} \\left[\\cfrac{x^2}{x-1}-\\cfrac{x^2-x}{x-1}\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"109\" width=\"582\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p> Agora que os dois termos t\u00eam o mesmo denominador, podemos agrup\u00e1-los:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7702287a02af6d8e3dddaa3f0c6eb1b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to +\\infty} \\left[\\cfrac{x^2-(x^2-x)}{x-1}  \\right] =\\lim_{x \\to +\\infty} \\left[\\cfrac{x}{x-1} \\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"340\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p> E finalmente resolvemos o limite:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-feb5faa9dc5d3b68d3273ad4d75d2bb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n =\\lim_{x \\to +\\infty} \\left[\\cfrac{x}{x-1} \\right] = \\cfrac{+\\infty}{+\\infty} = \\cfrac{1}{1} = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"278\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p> Portanto, n = 1. A ass\u00edntota obl\u00edqua \u00e9, portanto: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6fbe1cc5f3362ddbd80ed0b29c0bb4ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = mx+n\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dd133b92b6c5b350ce4383147df52e3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = 1x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dcfc739c1fd18f6fd834ff3e59b009e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y = x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Uma vez calculada a ass\u00edntota obl\u00edqua, representamo-la no mesmo gr\u00e1fico fazendo uma tabela de valores:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-73556269fc16e4cae71ddfde0ff51632_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6e696990e6cea37c0267d01c4553240f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c|c} x &amp; y \\\\ \\hline 0 &amp; 1 \\\\ 1 &amp; 2 \\\\ 2 &amp; 3 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"90\" width=\"52\" style=\"vertical-align: -40px;\"><\/p>\n<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/representent-fonctions-oblique-asymptote.webp\" alt=\"representar fun\u00e7\u00f5es, ass\u00edntota obl\u00edqua\" class=\"wp-image-2632\" width=\"501\" height=\"551\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Agora que conhecemos todas as ass\u00edntotas da fun\u00e7\u00e3o, precisamos analisar a <strong>monotonicidade da fun\u00e7\u00e3o<\/strong> . Ou seja, precisamos estudar em quais intervalos a fun\u00e7\u00e3o aumenta e em quais intervalos ela diminui. Portanto, calculamos a primeira derivada da fun\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-370529412c7f94fbe43e8d844520a185_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x^2}{x-1} \\ \\longrightarrow \\ f'(x)= \\cfrac{2x\\cdot (x-1) - x^2 \\cdot 1}{(x-1)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"364\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eafd7df834025eab179670d70e631871_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= \\cfrac{2x^2-2x - x^2}{(x-1)^2}  = \\cfrac{x^2-2x}{(x-1)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"260\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> E agora definimos a derivada igual a 0 e resolvemos a equa\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36700780d306ccf4975387990b1949fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6f17b3ce9d9690b68738698290f1b33f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^2-2x}{(x-1)^2}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"95\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> O termo<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cc1f4cc53676f0eb98290b3478031fef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(x-1\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -5px;\"><\/p>\n<p> Isso envolve dividir todo o lado esquerdo, para que possamos multiplic\u00e1-lo por todo o lado direito:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3d8bb0359e60db0b26d9bfce1b349e9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-2x=0\\cdot \\left(x-1\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"165\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-62138ee9fb8dc604ee836f1703379032_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-2x=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Extra\u00edmos o fator comum para resolver a equa\u00e7\u00e3o quadr\u00e1tica:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b243129a0d8853ec8716beb6d2d5c504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x(x-2)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Para que a multiplica\u00e7\u00e3o seja igual a 0, um dos dois elementos da multiplica\u00e7\u00e3o deve ser zero. Portanto, definimos cada fator igual a 0 e obtemos ambas as solu\u00e7\u00f5es da equa\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-55127e675ce8f7742db17d565c2ae507_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x\\cdot(x-2) =0   \\longrightarrow  \\begin{cases} \\bm{x=0} \\\\[2ex] x-2=0 \\ \\longrightarrow \\ \\bm{x= 2} \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"329\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Representamos agora na reta num\u00e9rica todos os pontos cr\u00edticos encontrados, ou seja, os pontos que n\u00e3o pertencem ao dom\u00ednio (x=1) e aqueles que cancelam a derivada (x=0 e x=2): <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/ligne-numerique-0-1-2.webp\" alt=\"\" class=\"wp-image-2443\" width=\"390\" height=\"77\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> E avaliamos o sinal da derivada em cada intervalo, para saber se a fun\u00e7\u00e3o aumenta ou diminui. Portanto, pegamos um ponto em cada intervalo (nunca os pontos cr\u00edticos) e observamos qual sinal a derivada tem nesse ponto: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8c77f1f797549bb4663fca07fcea2302_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=\\cfrac{x^2-2x}{\\left(x-1\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"128\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-171fa182722405650545d6e7fe14d5b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1) = \\cfrac{(-1)^2-2(-1)}{\\left((-1)-1\\right)^2} =\\cfrac{+3}{+4} = +0,75 \\  \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"369\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3daba13acad48408dfadae7c683d62d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(0,5) = \\cfrac{0,5^2-2\\cdot 0,5}{\\left(0,5-1\\right)^2} = \\cfrac{-0,75}{+0,25} = -3  \\  \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"363\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bd76e394ebfc759caaedca3a6ff66762_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1,5) = \\cfrac{1,5^2-2\\cdot 1,5}{\\left(1,5-1\\right)^2} = \\cfrac{-0,75}{+0,25} = -3  \\  \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"363\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e8d15f1093f1455f39042681fd9ab133_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(3) = \\cfrac{3^2-2\\cdot 3}{\\left(3-1\\right)^2} =\\cfrac{+3}{+4} = +0,75 \\  \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"313\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/ligne-numerique-0-1-2-positif-negatif-positif.webp\" alt=\"\" class=\"wp-image-2444\" width=\"390\" height=\"136\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Se a derivada for positiva, significa que a fun\u00e7\u00e3o est\u00e1 aumentando, e se a derivada for negativa, significa que a fun\u00e7\u00e3o est\u00e1 diminuindo. Portanto, os intervalos de crescimento e decl\u00ednio s\u00e3o:<\/p>\n<p class=\"has-text-align-center\"> <strong>Crescimento:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-11ebeca24ba262661dd73042a326110c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty, 0)\\cup (2,+\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"142\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Diminuir:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-206ab3f38b17a58b25209bf269265919_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,1)\\cup (1,2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Al\u00e9m disso, em x=0 a fun\u00e7\u00e3o vai de crescente para decrescente, ent\u00e3o x=0 \u00e9 um m\u00e1ximo relativo da fun\u00e7\u00e3o. E em x=2, a fun\u00e7\u00e3o vai de decrescente para crescente, ent\u00e3o x=2 \u00e9 um m\u00ednimo relativo da fun\u00e7\u00e3o.<\/p>\n<p> Finalmente, substitu\u00edmos os extremos encontrados na fun\u00e7\u00e3o original para encontrar a coordenada Y dos pontos:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d8bb02550f4c83abce02040f9e9ab495_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=\\cfrac{0^2}{0-1} = \\cfrac{0}{-1} = 0 \\ \\longrightarrow \\ (0,0)\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"268\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-74333ede5561c728c68899d68b31ee62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2)=\\cfrac{2^2}{2-1} = \\cfrac{4}{1} = 4 \\ \\longrightarrow \\ (2,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"254\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Os extremos relativos da fun\u00e7\u00e3o s\u00e3o, portanto:<\/p>\n<p class=\"has-text-align-center\"> <strong>M\u00e1ximo no ponto<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2c790019bd70403eba876c59c82c0f9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>M\u00ednimo para apontar<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a59b564601b4cd9f2bc149baa80c44a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(2,4)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Representamos o m\u00e1ximo e o m\u00ednimo no gr\u00e1fico: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/representent-les-fonctions-maximum-et-minimum.webp\" alt=\"representam as fun\u00e7\u00f5es m\u00e1xima e m\u00ednima\" class=\"wp-image-2633\" width=\"548\" height=\"604\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Por fim, basta <strong>estudar a curvatura da fun\u00e7\u00e3o<\/strong> , ou seja, estudar os intervalos de concavidade e convexidade da fun\u00e7\u00e3o. Para fazer isso, calculamos sua segunda derivada: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-06628d8d896e24d462c90c9d6a47fdfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=\\cfrac{x^2-2x}{(x-1)^2} \\ \\longrightarrow \\ f''(x)= \\cfrac{(2x-2)\\cdot (x-1)^2- (x^2-2x)\\cdot 2(x-1)\\cdot 1}{\\left(\\left(x-1\\right)^2\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"575\" style=\"vertical-align: -33px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c854f6cd07cc084869bbb880365c1d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{(2x-2)\\cdot (x-1)^2- (x^2-2x)\\cdot 2(x-1)}{(x-1)^4}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"377\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-15214503897d52ddd5c4f2865cc8a3b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{(2x-2)\\cdot (x-1)^{\\cancel{2}}- (x^2-2x)\\cdot 2\\cancel{(x-1)}}{(x-1)^{\\cancelto{3}{4}}} = \\cfrac{(2x-2)\\cdot (x-1)- (x^2-2x)\\cdot 2}{(x-1)^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"93\" width=\"582\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c72f4714446d0cb718dd19637001f822_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{2x^2-2x-2x+2- (2x^2-4x)}{(x-1)^3}  =\\cfrac{2x^2-2x-2x+2- 2x^2+4x}{(x-1)^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"563\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c3f75e315c71b8f2e66c226c897c6585_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x) =\\cfrac{2}{(x-1)^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"131\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> E agora igualamos a segunda derivada a zero e resolvemos a equa\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-981d85a257dd56afdb3fc7eb53d5eadf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-06d4145cca14c05a89eeea18d0cb9bf0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{2}{(x-1)^3} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"95\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-15603629285ab47e50ccf28d5ab28607_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2=0\\cdot \\left(x-1\\right)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"116\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e6f185609f51e8f3ae79e0e459644dc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> 2 nunca ser\u00e1 igual a 0, ent\u00e3o a equa\u00e7\u00e3o<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-981d85a257dd56afdb3fc7eb53d5eadf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<p> N\u00e3o h\u00e1 solu\u00e7\u00e3o.<\/p>\n<p> Representamos agora na reta num\u00e9rica todos os pontos cr\u00edticos encontrados, ou seja, os pontos que n\u00e3o pertencem ao dom\u00ednio (x=1) e aqueles que cancelam a segunda derivada (neste caso n\u00e3o h\u00e1 nenhum): <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/nombre-ligne-1.webp\" alt=\"\" class=\"wp-image-2564\" width=\"226\" height=\"88\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> E avaliamos o sinal da derivada em cada intervalo, para saber se a fun\u00e7\u00e3o \u00e9 convexa ou c\u00f4ncava. Portanto, pegamos um ponto em cada intervalo (nunca os pontos singulares) e observamos qual sinal a derivada tem neste ponto: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c3f75e315c71b8f2e66c226c897c6585_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x) =\\cfrac{2}{(x-1)^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"131\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d986c4a85ebad90dba014c4958a2d6fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(0) =\\cfrac{2}{(0-1)^3} = \\cfrac{2}{-1}=-2 \\  \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"275\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b42e050afcc2170bb221768109f1f839_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(2) =\\cfrac{2}{(2-1)^3} = \\cfrac{2}{1}=2 \\  \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"247\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/droite-numerique-1-concave-convexe.webp\" alt=\"\" class=\"wp-image-2565\" width=\"227\" height=\"151\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> E finalmente deduzimos os intervalos de concavidade e convexidade da fun\u00e7\u00e3o. Se a segunda derivada for positiva, significa que a fun\u00e7\u00e3o \u00e9 convexa.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> , e se a segunda derivada for negativa isso significa que a fun\u00e7\u00e3o \u00e9 c\u00f4ncava<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Os intervalos de concavidade e convexidade s\u00e3o, portanto:<\/p>\n<p class=\"has-text-align-center\"> <strong>Convexo<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-87d6843b66a0ebea6c769017a30a8d75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(1,+\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"61\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>C\u00f4ncavo<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36b85798b30f125fea3702a0671c77ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty,1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"60\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> No entanto, embora haja uma mudan\u00e7a na curvatura em x=1, n\u00e3o \u00e9 um ponto de inflex\u00e3o. Porque x=1 n\u00e3o pertence ao dom\u00ednio da fun\u00e7\u00e3o.<\/p>\n<p> Assim podemos terminar de representar a fun\u00e7\u00e3o usando tudo o que calculamos: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/representation-des-fonctions.webp\" alt=\"representa\u00e7\u00e3o de fun\u00e7\u00f5es\" class=\"wp-image-2634\" width=\"550\" height=\"608\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> A fun\u00e7\u00e3o representada no gr\u00e1fico fica assim: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/representation-graphique-de-la-fonction-rationnelle.webp\" alt=\"representa\u00e7\u00e3o gr\u00e1fica da fun\u00e7\u00e3o racional\" class=\"wp-image-2635\" width=\"546\" height=\"598\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-representacion-de-funciones\"><\/span> Exerc\u00edcios resolvidos para representar fun\u00e7\u00f5es<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Fa\u00e7a um gr\u00e1fico da seguinte fun\u00e7\u00e3o polinomial: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4fbbd639713355d58e743cb927faeee0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=x^3-3x^2+4\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"155\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> A primeira coisa a fazer \u00e9 calcular o dom\u00ednio de defini\u00e7\u00e3o da fun\u00e7\u00e3o. Esta \u00e9 uma fun\u00e7\u00e3o polinomial, portanto o dom\u00ednio consiste apenas em n\u00fameros reais:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4f565027fd5d2a4381e3a23d183c9f76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom } f= \\mathbb{R}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"90\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Para encontrar o ponto de intersec\u00e7\u00e3o com o eixo X, resolvemos <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bbb52c33bfaff434771f0e4ddd4cf677_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)= 0.\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0bce6c022ed0fc63f4659af75888f96c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"67\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-32372d448fb254237a89bd11fa071711_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^3-3x^2+4=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"129\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Esta \u00e9 uma equa\u00e7\u00e3o de grau maior que 2. Portanto, fatoramos a equa\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f40f79838a9f4eb8ef8092860e41bfe2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x+1)(x^2-4x+4)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"189\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o x=-1 \u00e9 uma solu\u00e7\u00e3o. E calculamos as outras solu\u00e7\u00f5es resolvendo a equa\u00e7\u00e3o quadr\u00e1tica resultante:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e79a2a2f6650c4095c0dca52188c40c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}x &amp; =\\cfrac{-b \\pm \\sqrt{b^2-4ac}}{2a} =\\cfrac{-(-4) \\pm \\sqrt{(-4)^2-4\\cdot 1 \\cdot 4}}{2\\cdot 1} \\\\[2ex] &amp;=\\cfrac{+4 \\pm \\sqrt{16-16}}{2} =\\cfrac{4 \\pm \\sqrt{0}}{2} = \\cfrac{4 }{2 } = 2\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"105\" width=\"406\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Os pontos de intersec\u00e7\u00e3o com o eixo X s\u00e3o, portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d9a1ceadc45948f3a942ccb21109ccf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-1,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9109f132c97f810054198982440ac8c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(2,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E para encontrar o ponto de intersec\u00e7\u00e3o com o eixo Y, calculamos<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d449eebd1f011aebdf90931f3a66a3b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<p> Como x \u00e9 sempre 0 no eixo Y:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2a6de1788b17877aa807a31eae47e8a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=0^3-3\\cdot0^2+4 = 4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"196\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O ponto de intersec\u00e7\u00e3o com o eixo Y \u00e9, portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d0aa7bcc7acd9b70190168abbe3d05d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,4)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Para ver se a fun\u00e7\u00e3o possui ass\u00edntotas verticais, precisamos calcular o limite da fun\u00e7\u00e3o em pontos que n\u00e3o pertencem ao dom\u00ednio. Neste caso, o dom\u00ednio inclui todos os n\u00fameros reais. A fun\u00e7\u00e3o, portanto, n\u00e3o tem ass\u00edntota vertical.<\/p>\n<p class=\"has-text-align-left\"> Por outro lado, a ass\u00edntota horizontal da fun\u00e7\u00e3o ser\u00e1 o resultado do limite infinito da fun\u00e7\u00e3o. Ainda:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ea1063bd9a29fc7ccfd470464efcd868_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\ x^3-3x^2+4 =(+\\infty)^3 = +\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"283\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O limite infinito da fun\u00e7\u00e3o nos deu +\u221e, ent\u00e3o a fun\u00e7\u00e3o n\u00e3o tem ass\u00edntota horizontal.<\/p>\n<p class=\"has-text-align-left\"> Agora calculamos a ass\u00edntota obl\u00edqua. As ass\u00edntotas obl\u00edquas t\u00eam a forma<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ad313410fc976bc53709807aa8aed8e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=mx+n.\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"95\" style=\"vertical-align: -4px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00c9 calculado com a seguinte f\u00f3rmula: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-df30ba17002e63ee33654a94955bbac9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to +\\infty} f(x):x = \\lim_{x \\to +\\infty} \\left( x^3-3x^2+4\\right): x =\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"376\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7b5d944c651bb86db8c00770059ebea7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle = \\lim_{x \\to +\\infty} \\cfrac{x^3-3x^2+4}{x} = \\cfrac{+\\infty}{+\\infty} = +\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"286\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O limite nos deu +\u221e, ent\u00e3o a fun\u00e7\u00e3o tamb\u00e9m n\u00e3o possui ass\u00edntota obl\u00edqua.<\/p>\n<p class=\"has-text-align-left\"> Para estudar a monotonicidade da fun\u00e7\u00e3o, devemos primeiro calcular a sua derivada:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f9453115e5750d874ed1f96014f8481b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)= x^3-3x^2+4 \\ \\longrightarrow \\ f'(x)= 3x^2-6x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"335\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora definimos a derivada igual a 0 e resolvemos a equa\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4890b9dfeb634c4d7a349351be73b5d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f360aca5f393d3a9c7e882e09f37fa7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x^2-6x=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"100\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7c36141e8c1b73c1c73f9fea0dc115cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x(3x-6)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"106\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d23e2b378508baca9f51117fc8767e90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x\\cdot(3x-6) =0 \\longrightarrow \\begin{cases} \\bm{x=0} \\\\[2ex] 3x-6=0 \\ \\longrightarrow \\ x= \\cfrac{6}{3} = 2 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"381\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Representamos agora na reta num\u00e9rica todos os pontos singulares obtidos, ou seja, os pontos que n\u00e3o pertencem ao dom\u00ednio (neste caso, todos pertencem) e aqueles que cancelam a derivada (x=0 e x =2) : <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/ligne-numerique-0-2.webp\" alt=\"\" class=\"wp-image-2638\" width=\"285\" height=\"75\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> E avaliamos o sinal da derivada em cada intervalo, para saber se a fun\u00e7\u00e3o aumenta ou diminui. Portanto, pegamos um ponto em cada intervalo (nunca os pontos singulares) e observamos qual sinal a derivada tem neste ponto: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0412923f1191d192d18528b63a0e57ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1)=3(-1)^2-6(-1)= 3+6 = 9\\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"343\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e29cd090a269505f244837bcbd11be75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1)=3\\cdot 1^2-6\\cdot 1= 3-6 = -3\\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"314\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b1e444ac85ebaf441a325b83eb48714d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(3)=3\\cdot 3^2-6\\cdot 3= 27-18 = 9\\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"317\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/ligne-numerique-0-2-monotonie.webp\" alt=\"\" class=\"wp-image-2639\" width=\"286\" height=\"135\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Se a derivada for positiva, significa que a fun\u00e7\u00e3o est\u00e1 aumentando, e se a derivada for negativa, significa que a fun\u00e7\u00e3o est\u00e1 diminuindo. Portanto, os intervalos de crescimento e decl\u00ednio s\u00e3o:<\/p>\n<p class=\"has-text-align-center\"> <strong>Crescimento:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b16ae35401c1d15b6d08334338f92172_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty,0)\\cup (2,+\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"142\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Diminuir:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6928e231184fd28bd944d9531a322d74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A fun\u00e7\u00e3o vai de crescente a decrescente em x=0, ent\u00e3o x=0 \u00e9 o m\u00e1ximo da fun\u00e7\u00e3o. E a fun\u00e7\u00e3o vai de decrescente para crescente em x=2, ent\u00e3o x=2 \u00e9 o m\u00ednimo da fun\u00e7\u00e3o.<\/p>\n<p class=\"has-text-align-left\"> Finalmente, substitu\u00edmos os extremos encontrados na fun\u00e7\u00e3o original para encontrar as coordenadas Y dos pontos: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-27cd3181606a77a6e28a89ac0e82f545_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=0^3-3\\cdot 0^2+4 = 4 \\ \\longrightarrow \\ (0,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"285\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a261755ca09d967ba2a2b3cbc84c1d8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2)=2^3-3\\cdot 2^2+4 = 8-3 \\cdot 4 +4 = 0 \\ \\longrightarrow \\ (2,0)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"400\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Os extremos relativos da fun\u00e7\u00e3o s\u00e3o, portanto:<\/p>\n<p class=\"has-text-align-center\"> <strong>M\u00e1ximo no ponto<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d0aa7bcc7acd9b70190168abbe3d05d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,4)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>M\u00ednimo para apontar<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9109f132c97f810054198982440ac8c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(2,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Para estudar a curvatura da fun\u00e7\u00e3o, calculamos sua segunda derivada:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aec39b1d4c0168da2b76e56643593a08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= 3x^2-6x \\ \\longrightarrow \\ f''(x)= 6x-6\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"296\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora definimos a segunda derivada igual a 0 e resolvemos a equa\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f618f4961c18c45be60fc496ad4896e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-905af0a687269e0a0775640143c2ea2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6x-6=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c9c29e31dbe2952c3f947c7999aaea32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6x=6\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1b7492f8f450a70fe2916618ad1021b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= \\cfrac{6}{6} = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"76\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Representamos na reta todos os pontos singulares encontrados, ou seja, os pontos que n\u00e3o pertencem ao dom\u00ednio (neste caso todos pertencem) e aqueles que cancelam a derivada (x=1): <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/nombre-ligne-1.webp\" alt=\"\" class=\"wp-image-2564\" width=\"201\" height=\"78\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> E agora avaliamos o sinal da segunda derivada em cada intervalo, para saber se a fun\u00e7\u00e3o \u00e9 c\u00f4ncava ou convexa. Portanto, pegamos um ponto em cada intervalo (nunca os pontos singulares) e observamos qual sinal a segunda derivada tem neste ponto: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e82925d4c4cfc1e0d51b97ea177d8508_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(0)=6\\cdot 0-6= -6 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"225\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d532f146f034f820e7a7a72860a8bc54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(2)=6\\cdot 2-6= 12-6= 6 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"283\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/droite-numerique-1-concave-convexe.webp\" alt=\"\" class=\"wp-image-2565\" width=\"206\" height=\"138\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Se a segunda derivada for positiva, significa que a fun\u00e7\u00e3o \u00e9 convexa.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> , e se a segunda derivada for negativa isso significa que a fun\u00e7\u00e3o \u00e9 c\u00f4ncava<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Os intervalos de concavidade e convexidade s\u00e3o, portanto:<\/p>\n<p class=\"has-text-align-center\"> <strong>Convexo<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-87d6843b66a0ebea6c769017a30a8d75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(1,+\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"61\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>C\u00f4ncavo<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36b85798b30f125fea3702a0671c77ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty,1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"60\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Al\u00e9m disso, a fun\u00e7\u00e3o muda de c\u00f4ncava para convexa em x=1, ent\u00e3o x=1 \u00e9 um ponto de inflex\u00e3o da fun\u00e7\u00e3o.<\/p>\n<p class=\"has-text-align-left\"> Finalmente, substitu\u00edmos os pontos de inflex\u00e3o encontrados na fun\u00e7\u00e3o original para encontrar a coordenada Y dos pontos:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ad643a4b4579b1f3a7484efda7d37f51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1)=1^3-3\\cdot 1^2+ 4= 1 -3 +4 =2 \\ \\longrightarrow \\ (1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"379\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Os pontos de viragem da fun\u00e7\u00e3o s\u00e3o, portanto:<\/p>\n<p class=\"has-text-align-center\"> <strong>Pontos de viragem:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-47f62d161c48f3aa8d8c81141352f01c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(1,2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Finalmente, com base em todas as informa\u00e7\u00f5es que calculamos, representamos graficamente a fun\u00e7\u00e3o: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/representation-graphique-fonction-polynomiale.webp\" alt=\"representa\u00e7\u00e3o gr\u00e1fica da fun\u00e7\u00e3o polinomial\" class=\"wp-image-2640\" width=\"419\" height=\"464\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 2<\/h3>\n<p> Fa\u00e7a um gr\u00e1fico da seguinte fun\u00e7\u00e3o racional: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1e778797ead4e88a87997bf75163584e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=\\frac{x^2+2}{x^2-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"109\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para encontrar o dom\u00ednio da fun\u00e7\u00e3o, igualamos o denominador. traga a fra\u00e7\u00e3o para zero e resolva a equa\u00e7\u00e3o resultante: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8818c0eafadfe429ce54f48546e7c06c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-1= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"81\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-959000af33497314f9a59a9bed2a19c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-003a71cb0ec797dfcd0cca915b03a795_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{x^2}=\\sqrt{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"79\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b2e5d1349000e44cc1988f98254e0389_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\pm 1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8b555e0164cc0f3755093da248489375_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom } f= \\mathbb{R}-\\{-1, +1 \\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"182\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Segundo, determinamos os limites da fun\u00e7\u00e3o com o eixo x igual \u00e0 express\u00e3o alg\u00e9brica da fun\u00e7\u00e3o. a\u00e7o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0bce6c022ed0fc63f4659af75888f96c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"67\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f4fd6078af8da3dbdaadfef085a088a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^2+2}{x^2-1}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"81\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-396d8c2a41a4ee6344c8a7ac4824d785_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+2=0\\cdot (x^2-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"155\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5655aa321be25373afe5d61e410eb5cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"81\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-23644022351ee7016d72eda1f084c02d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"63\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-07d269376ad563a1d6aa25499b471829_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\sqrt{-2} \\quad \\color{red}\\bm{\\times}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"127\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> N\u00e3o existe raiz quadrada de um n\u00famero negativo. Portanto, a fun\u00e7\u00e3o n\u00e3o intercepta o eixo X.<\/p>\n<p class=\"has-text-align-left\"> E para encontrar o ponto de intersec\u00e7\u00e3o com o eixo do computador, avaliamos a fun\u00e7\u00e3o em x=0.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f88af2bf1737fdf34e3aed57447206b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=\\cfrac{0^2+2}{0^2-1}= \\cfrac{2}{-1} = -2\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"200\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O ponto de intersec\u00e7\u00e3o com o eixo Y \u00e9, portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b1761f5972412ad38102968850bf6220_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,-2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Para ver se a fun\u00e7\u00e3o possui ass\u00edntotas verticais, precisamos calcular o limite da fun\u00e7\u00e3o em pontos que n\u00e3o pertencem ao dom\u00ednio (neste caso x=-1 e x=+1). E se o resultado for infinito, \u00e9 uma ass\u00edntota vertical. Ainda:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9213a1057c2accea4cd10e01b44e0a0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -1} \\cfrac{x^2+2}{x^2-1} = \\cfrac{(-1)^2+2}{(-1)^2-1} =\\cfrac{1+2}{1-1}= \\cfrac{3}{0} = \\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"333\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Como o limite da fun\u00e7\u00e3o quando x se aproxima de -1 d\u00e1 o infinito, x=-1 \u00e9 uma ass\u00edntota vertical.<\/p>\n<p class=\"has-text-align-left\"> Calculamos os limites laterais da ass\u00edntota x=-1 substituindo um n\u00famero muito pr\u00f3ximo a ela na fun\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3734e234658371143046355d62d2b15c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(-1,1)=\\cfrac{(-1,1)^2+2}{(-1,1)^2-1} =+15,29 \\longrightarrow \\lim_{x \\to -1^{-}} \\ \\cfrac{x^2+2}{x^2-1} = +\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"463\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-97f3b163c528f25bd56635fd28639ecb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(-0,9)=\\cfrac{(-0,9)^2+2}{(-0,9)^2-1} =-14,79 \\longrightarrow \\lim_{x \\to -1^{+}} \\ \\cfrac{x^2+2}{x^2-1} = -\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"463\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora vamos ver se x=+1 \u00e9 uma ass\u00edntota vertical:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-19d299b6cd11eea52abec214e2d5dbf5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +1} \\cfrac{x^2+2}{x^2-1} = \\cfrac{1^2+2}{1^2-1} =\\cfrac{1+2}{1-1}= \\cfrac{3}{0} = \\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"305\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Como o limite da fun\u00e7\u00e3o quando x se aproxima de +1 d\u00e1 infinito, x=+1 \u00e9 uma ass\u00edntota vertical.<\/p>\n<p class=\"has-text-align-left\"> Calculamos os limites laterais da ass\u00edntota x=1 substituindo um n\u00famero muito pr\u00f3ximo dela na fun\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1e721a23e7ead77eb1974f5f4ccb781e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(0,9)=\\cfrac{(0,9)^2+2}{(0,9)^2-1} =-14,79 \\longrightarrow \\lim_{x \\to +1^{-}} \\ \\cfrac{x^2+2}{x^2-1} = -\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"435\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e5caabec3166aad7e5c7d6a8d25b2388_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(1,1)=\\cfrac{(1,1)^2+2}{(1,1)^2-1} =+15,29 \\longrightarrow \\lim_{x \\to +1^{+}} \\ \\cfrac{x^2+2}{x^2-1} = +\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"435\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Por outro lado, a ass\u00edntota horizontal da fun\u00e7\u00e3o ser\u00e1 o resultado do limite infinito da fun\u00e7\u00e3o. Ainda:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7aa229147046645987101dfc3b4cfec1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\ \\cfrac{x^2+2}{x^2-1} = \\cfrac{+\\infty}{+\\infty } =\\cfrac{1}{1} = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"236\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O limite infinito da fun\u00e7\u00e3o nos deu 1, ent\u00e3o a fun\u00e7\u00e3o tem uma ass\u00edntota horizontal em y=1.<\/p>\n<p class=\"has-text-align-left\"> Como a fun\u00e7\u00e3o possui uma ass\u00edntota horizontal, ela n\u00e3o ter\u00e1 uma ass\u00edntota obl\u00edqua.<\/p>\n<p class=\"has-text-align-left\"> Diferenciamos a fun\u00e7\u00e3o e depois estudamos os intervalos de crescimento e diminui\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5641380d832933540291b6dbcb8e151a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x^2+2}{x^2-1}  \\ \\longrightarrow \\ f'(x)= \\cfrac{2x \\cdot (x^2-1) -(x^2+2) \\cdot 2x}{\\left(x^2-1 \\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"433\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c2348d02afd4187badfc0b04346dc34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= \\cfrac{2x^3-2x - (2x^3+4x) }{\\left(x^2-1 \\right)^2} = \\cfrac{-6x}{\\left(x^2-1 \\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"330\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora definimos a derivada igual a 0 e resolvemos a equa\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4890b9dfeb634c4d7a349351be73b5d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b60ec42d7180d4f5c76674c6a8915e4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{-6x}{\\left(x^2-1 \\right)^2}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"102\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e8c7d7a359a454599e1662d8f4c64859_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-6x=0\\cdot\\left(x^2-1 \\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"149\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-430a0f35bef91d5e64798d69fc210c41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-6x= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"64\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fd71758c91c04f304eae7e7fd164eb1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\cfrac{0}{-6} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"91\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Representamos na reta todos os pontos cr\u00edticos calculados, que s\u00e3o os pontos que n\u00e3o pertencem ao dom\u00ednio (x=-1 e x=+1) e aqueles que cancelam a derivada (x=0): <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/number-line-1-0-1.webp\" alt=\"\" class=\"wp-image-2643\" width=\"397\" height=\"77\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> E avaliamos o sinal da derivada em cada intervalo, para saber se a fun\u00e7\u00e3o aumenta ou diminui. Portanto, pegamos um ponto em cada intervalo (nunca os pontos singulares) e observamos qual sinal a derivada tem neste ponto: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8110ed8ec9e600c9ec17fa5ffa3c088f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-2)= \\cfrac{-6(-2)}{\\left((-2)^2-1 \\right)^2} = \\cfrac{12}{9} =1,33 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"48\" width=\"327\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b982f336007ee0be2dcee5d52be0105_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-0,5)= \\cfrac{-6(-0,5)}{\\left((-0,5)^2-1 \\right)^2} = \\cfrac{3}{0,56} =5,33 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"48\" width=\"377\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-82b12d15e99f6b63fe4d10bfb77b32e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(0,5)= \\cfrac{-6\\cdot 0,5}{\\left(0,5^2-1 \\right)^2} = \\cfrac{-3}{0,56} =-5,33 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"350\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7166a97e07562fc86f89483322af2efd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(2)= \\cfrac{-6\\cdot 2}{\\left(2^2-1 \\right)^2} = \\cfrac{-12}{9} =-1,33 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"321\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/number-line-1-0-1-croissant-decroissant.webp\" alt=\"\" class=\"wp-image-2645\" width=\"400\" height=\"141\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> A fun\u00e7\u00e3o aumenta onde a derivada \u00e9 positiva e a fun\u00e7\u00e3o diminui onde a fun\u00e7\u00e3o \u00e9 negativa:<\/p>\n<p class=\"has-text-align-center\"> <strong>Crescimento:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-14d2fb1f352a42d7f0e8b6c91776fe24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty,-1)\\cup (-1,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"147\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Diminuir:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9df00656c8f1b8a5b5660489739aecc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,1)\\cup (1,+\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"120\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A fun\u00e7\u00e3o vai de crescente a decrescente em x=0, ent\u00e3o x=0 \u00e9 um m\u00e1ximo local da fun\u00e7\u00e3o.<\/p>\n<p class=\"has-text-align-left\"> Substitu\u00edmos o extremo encontrado na fun\u00e7\u00e3o original para encontrar a coordenada Y do ponto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7a0031b1f78a601b773cab39f1f54d4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=\\cfrac{0^2+2}{0^2-1}= \\cfrac{2}{-1} = -2 \\ \\longrightarrow \\ (0,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"303\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Os extremos relativos da fun\u00e7\u00e3o s\u00e3o, portanto:<\/p>\n<p class=\"has-text-align-center\"> <strong>M\u00e1ximo no ponto<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b1761f5972412ad38102968850bf6220_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,-2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Para estudar a curvatura da fun\u00e7\u00e3o, calculamos sua segunda derivada:<\/p>\n<p class=\"ql-center-displayed-equation\" style=\"line-height: 280px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-802852beb818dd5a0dce2f30374f3a88_l3.png\" height=\"280\" width=\"687\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"f'(x)=\\cfrac{-6x}{\\left(x^2-1 \\right)^2}  \\ \\longrightarrow <span class=&quot;ql-right-eqno&quot;>   <\/span><span class=&quot;ql-left-eqno&quot;>   <\/span><img src=&quot;https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-273969cf60ee8cf3413ee2f8b1db7688_l3.png&quot; height=&quot;129&quot; width=&quot;476&quot; class=&quot;ql-img-displayed-equation quicklatex-auto-format&quot; alt=&quot;\\[f''(x)= \\cfrac{-6 \\cdot \\left(x^2-1 \\right)^2 - (-6x) \\cdot 2(x^2-1) \\cdot 2x}{ \\left(\\left(x^2-1 \\right)^2\\right)^2}$$ f''(x)= \\cfrac{-6 \\left(x^2-1 \\right)^2 -(-6x)\\cdot 4x(x^2-1)}{\\left(x^2 -1\\right)^4} =\\]&quot; title=&quot;Rendered by QuickLaTeX.com&quot;\/> \\cfrac{-6 \\left(x^2-1 \\right)^2 + 24x^2(x^2-1)}{\\left(x^2 -1\\right)^4}&#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221;><\/p>\n<\/p>\n<p class=\"has-text-align-left\">Todos os termos t\u00eam<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2050d569319abb9789111cc5f49b21cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x^2-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"60\" style=\"vertical-align: -5px;\"><\/p>\n<p> , podemos, portanto, simplificar a fra\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-675ece9a985c5d3a11397a0fc84d7b5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{-6 \\left(x^2-1 \\right)^{\\cancel{2}} + 24x^2\\cancel{(x^2-1)}}{\\left(x^2-1 \\right)^\\cancelto{3}{4}}  =\\cfrac{-6 \\left(x^2-1 \\right) + 24x^2}{\\left(x^2 -1\\right)^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"52\" width=\"475\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3dba9e99bfb4fcfd547c2d2edaafb4b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{-6x^2+6  + 24x^2}{\\left(x^2 -1\\right)^3} =\\cfrac{18x^2+6}{\\left(x^2 -1\\right)^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"300\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora definimos a segunda derivada igual a 0 e resolvemos a equa\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f618f4961c18c45be60fc496ad4896e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f373044cf16a3d9191973dbd6a9a21f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{18x^2+6}{\\left(x^2 -1\\right)^3}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"102\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c431872e1278c9771b294701b56bf632_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"18x^2+6=0\\cdot \\left(x^2 -1\\right)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"182\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\">\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f760897a45aa19dc1c2d5cbd6864de12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"18x^2+6= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"98\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9ffbd0e0d0183993e003026edffe4f0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"18x^2=-6\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"81\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2753f9ca0b7d0a2054856746d7ba3c26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=\\cfrac{-6}{18}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"74\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b4cd1f1648c3b49fe6be0a39600890b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=-0,33\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"90\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c3ab76c31e735be0675b2a6c095c213e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\sqrt{-0,33} \\quad \\color{red}\\bm{\\times}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"153\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> N\u00e3o existe raiz quadrada de um n\u00famero negativo. Ent\u00e3o n\u00e3o h\u00e1 ponto que corresponda<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-981d85a257dd56afdb3fc7eb53d5eadf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora representamos na reta todos os pontos singulares encontrados, ou seja, os pontos que n\u00e3o pertencem ao dom\u00ednio (x=-1 e x=+1) e aqueles que cancelam a segunda derivada (neste caso n\u00e3o h\u00e1 qualquer): <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/nombre-ligne-1-1.webp\" alt=\"\" class=\"wp-image-2596\" width=\"294\" height=\"75\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> E avaliamos o sinal da segunda derivada em cada intervalo, para saber se a fun\u00e7\u00e3o \u00e9 c\u00f4ncava ou convexa. Portanto, pegamos um ponto em cada intervalo (nunca os pontos singulares) e observamos qual sinal a segunda derivada tem neste ponto: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b84333a96a947c0a9c2ef605e8426f77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(-2)=\\cfrac{18(-2)^2+6}{\\left((-2)^2 -1\\right)^3}= \\cfrac{78}{27}  = 2,89 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"331\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2ec750b2e7e3b0b47e650695ad1f4259_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(0)=\\cfrac{18\\cdot 0^2+6}{\\left(0^2 -1\\right)^3}= \\cfrac{6}{-1}  = -6 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"292\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7f079ac6677ac2cfc283b9b4ff817a13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(2)=\\cfrac{18\\cdot 2^2+6}{\\left(2^2 -1\\right)^3}= \\cfrac{78}{27}  = 2,89 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"299\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/ligne-numerique-1-1-positif-negatif-positif.webp\" alt=\"\" class=\"wp-image-2597\" width=\"297\" height=\"132\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Se a segunda derivada for positiva, significa que a fun\u00e7\u00e3o \u00e9 convexa.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> , e se a segunda derivada for negativa isso significa que a fun\u00e7\u00e3o \u00e9 c\u00f4ncava<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Os intervalos de concavidade e convexidade s\u00e3o, portanto:<\/p>\n<p class=\"has-text-align-center\"> <strong>Convexo<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dc2d3b8519f698562d39a2807dc7a906_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty, -1) \\cup (1, +\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"156\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>C\u00f4ncavo<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3c8937d362a3ba07fa9068381afe74a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-1,1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> No entanto, embora haja uma mudan\u00e7a na curvatura em x=-1 e em x=1, estes n\u00e3o s\u00e3o pontos de inflex\u00e3o. Porque eles n\u00e3o pertencem ao dom\u00ednio da fun\u00e7\u00e3o.<\/p>\n<p class=\"has-text-align-left\"> E finalmente, representamos graficamente a fun\u00e7\u00e3o usando todos os c\u00e1lculos realizados: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/fonctions-graphiques-exercice-resolu.webp\" alt=\"exerc\u00edcio de fun\u00e7\u00f5es gr\u00e1ficas resolvido\" class=\"wp-image-2646\" width=\"436\" height=\"439\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 3<\/h3>\n<p> Trace a seguinte fun\u00e7\u00e3o racional em um gr\u00e1fico: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3ac9ccc5e8540cca38f599ed36507792_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=\\frac{x^3}{x^2-4}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"109\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Esta \u00e9 uma fun\u00e7\u00e3o racional, ent\u00e3o precisamos igualar o denominador a 0 para ver quais n\u00fameros n\u00e3o pertencem ao dom\u00ednio da fun\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a34dfe78d673534873a2013c16e1b353_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-4= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"81\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c269e23a1070b3e5556abece040af75a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=4\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c4c32359b264b28ac80f2606c09d5a2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{x^2}=\\sqrt{4}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"79\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c06a55e3acdd1e283973786926b27716_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\pm 2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e666f828709575f965b5120fbdda085e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom } f= \\mathbb{R}-\\{-2, +2 \\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"182\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Para encontrar o ponto de intersec\u00e7\u00e3o com o eixo X, resolvemos<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bbb52c33bfaff434771f0e4ddd4cf677_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)= 0.\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" style=\"vertical-align: -5px;\"><\/p>\n<p> Como a fun\u00e7\u00e3o sempre tem valor 0 no eixo X: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0bce6c022ed0fc63f4659af75888f96c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"67\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4fb6c030af3bbc674466d58da3704303_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^3}{x^2-4}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"81\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d913fc1b7622c590b24cb0bcff8f07a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^3=0\\cdot (x^2-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"125\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-085b61b70fdf3a335b744ed8bc4f06a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^3=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-432867a310687d633fea4e3e72197b03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\sqrt[3]{0}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"91\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O ponto de intersec\u00e7\u00e3o com o eixo X \u00e9, portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2c790019bd70403eba876c59c82c0f9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E para encontrar o ponto de intersec\u00e7\u00e3o com o eixo Y, calculamos<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d449eebd1f011aebdf90931f3a66a3b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<p> Como x \u00e9 sempre 0 no eixo Y:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-21000bf141667a94b3deec79162d963f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=\\cfrac{0^3}{0^2-4} = \\cfrac{0}{-4} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"187\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O ponto de intersec\u00e7\u00e3o com o eixo Y \u00e9, portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2c790019bd70403eba876c59c82c0f9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Neste caso, o ponto de intersec\u00e7\u00e3o com o eixo X coincide com o ponto de intersec\u00e7\u00e3o com o eixo Y, pois a fun\u00e7\u00e3o passa pela origem das coordenadas.<\/p>\n<p class=\"has-text-align-left\"> Para ver se a fun\u00e7\u00e3o possui ass\u00edntotas verticais, precisamos calcular o limite da fun\u00e7\u00e3o em pontos que n\u00e3o pertencem ao dom\u00ednio (neste caso x=-2 e x=+2). E se o resultado for infinito, \u00e9 uma ass\u00edntota vertical. Ainda:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-08adb9903ed9589da031a215aca7cf82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -2} \\cfrac{x^3}{x^2-4} = \\cfrac{(-2)^3}{(-2)^2-4} =\\cfrac{-8}{4-4}= \\cfrac{-8}{0} = \\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"354\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Como o limite da fun\u00e7\u00e3o quando x se aproxima de -2 d\u00e1 infinito, x=-2 \u00e9 uma ass\u00edntota vertical.<\/p>\n<p class=\"has-text-align-left\"> Calculamos os limites laterais da ass\u00edntota x=-2 substituindo um n\u00famero muito pr\u00f3ximo a ela na fun\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-91d1a99bc96ac49d3b83d2ba9459558a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(-2,1)=\\cfrac{(-2,1)^3}{(-2,1)^2-4} =-22,59 \\longrightarrow \\lim_{x \\to -2^{-}}  \\cfrac{x^3}{x^2-4} = -\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"457\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a430ef18ddb27cf657aa3e810c0a5e24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(-1,9)=\\cfrac{(-1,9)^3}{(-1,9)^2-4} =+17,59 \\longrightarrow \\lim_{x \\to -2^{+}}  \\cfrac{x^3}{x^2-4} = +\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"457\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora vamos ver se x=+2 \u00e9 uma ass\u00edntota vertical:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-38e9b49183f8680c3ee84f96454334d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +2} \\cfrac{x^3}{x^2-4} = \\cfrac{(2)^3}{(2)^2-4} =\\cfrac{8}{4-4}= \\cfrac{8}{0} = \\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"319\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Como o limite da fun\u00e7\u00e3o quando x se aproxima de +2 d\u00e1 infinito, x=+2 \u00e9 uma ass\u00edntota vertical.<\/p>\n<p class=\"has-text-align-left\"> Calculamos os limites laterais da ass\u00edntota x=2 substituindo um n\u00famero muito pr\u00f3ximo dela na fun\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-28d87cfb93cd7a3673f853003fcd158d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(1,9)=\\cfrac{1,9^3}{1,9^2-4} =-17,59 \\longrightarrow \\lim_{x \\to 2^{-}}  \\cfrac{x^3}{x^2-4} = -\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"405\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-be984dcc8e4149888e44944c44675d87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(2,1)=\\cfrac{2,1^3}{2,1^2-4} =22,59 \\longrightarrow \\lim_{x \\to 2^{+}}  \\cfrac{x^3}{x^2-4} = +\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"391\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Por outro lado, a ass\u00edntota horizontal da fun\u00e7\u00e3o ser\u00e1 o resultado do limite infinito da fun\u00e7\u00e3o. Ainda:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aa6c8c730e78a8f4ea067adbd5f487be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\ \\cfrac{x^3}{x^2-4} = \\cfrac{+\\infty}{+\\infty } =+\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"225\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O limite infinito da fun\u00e7\u00e3o nos deu +\u221e, ent\u00e3o a fun\u00e7\u00e3o n\u00e3o tem ass\u00edntota horizontal.<\/p>\n<p class=\"has-text-align-left\"> Agora calculamos a ass\u00edntota obl\u00edqua. As ass\u00edntotas obl\u00edquas t\u00eam a forma<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ad313410fc976bc53709807aa8aed8e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=mx+n.\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"95\" style=\"vertical-align: -4px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00c9 calculado com a seguinte f\u00f3rmula: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8bac34a30967593c9e1d8d1dc6ff816c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to +\\infty} f(x):x = \\lim_{x \\to +\\infty} \\cfrac{x^3}{x^2-4}: x =\\lim_{x \\to +\\infty} \\cfrac{x^3}{x^2-4}: \\frac{x}{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"446\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8f5c82ccc7edabf4fd047ea738ac8124_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m =\\lim_{x \\to +\\infty}\\cfrac{x^3\\cdot 1}{(x^2-4)\\cdot x} =\\lim_{x \\to +\\infty}\\cfrac{x^3}{x^3-4x}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"309\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d10814f4c377eb6df5e7ef2eb1ba1e1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m =\\lim_{x \\to +\\infty}\\cfrac{x^3}{x^3-4x} = \\frac{+\\infty}{+\\infty} = \\frac{1}{1} = \\bm{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"276\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Uma vez conhecida a inclina\u00e7\u00e3o da ass\u00edntota obl\u00edqua, determinamos a intercepta\u00e7\u00e3o usando a seguinte f\u00f3rmula: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-de4326a40acf34b64a28c9da8250bf00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to +\\infty} \\left[f(x)-mx\\right] = \\lim_{x \\to +\\infty} \\left[ \\cfrac{x^3}{x^2-4}-1x\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"355\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-13478ac6f6fac958ec8b2a714c28bc3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to +\\infty} \\left[ \\cfrac{x^3}{x^2-4}-x\\right] = \\cfrac{+\\infty}{+\\infty} - (+\\infty) = \\bm{+\\infty - \\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"412\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Mas obtemos a indetermina\u00e7\u00e3o \u221e \u2013 \u221e. \u00c9, portanto, necess\u00e1rio reduzir os termos a um denominador comum. Para fazer isso, multiplicamos e dividimos x pelo denominador da fra\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5e939b43a3405ba644d4b60bb4bacadb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to +\\infty} \\left[ \\cfrac{x^3}{x^2-4}-\\cfrac{x \\cdot (x^2-4)}{(x^2-4)}\\right] =\\lim_{x \\to +\\infty} \\left[ \\cfrac{x^3}{x^2-4}-\\cfrac{x^3-4x}{x^2-4}\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"484\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1ce9e619c697b7b25ec55b52ec531fd1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n =  \\lim_{x \\to +\\infty} \\cfrac{x^3- (x^3-4x)}{x^2-4} = \\lim_{x \\to +\\infty} \\cfrac{4x}{x^2-4}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"320\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6395fc2ac5efce4007680594b4e78fa9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n =\\lim_{x \\to +\\infty} \\cfrac{4x}{x^2-4} = \\cfrac{+\\infty}{\\infty} =\\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"231\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Resumindo, a ass\u00edntota obl\u00edqua \u00e9: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6fbe1cc5f3362ddbd80ed0b29c0bb4ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = mx+n\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b76adff686f2ca940f3054478fa10fc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = 1x + 0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fca2e14d8d10e98015169017e681c9e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y = x }\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Para estudar a monotonicidade da fun\u00e7\u00e3o, devemos primeiro calcular a sua derivada: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1397d0e8e73bd7b1d851411dee28daed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x^3}{x^2-4}  \\ \\longrightarrow \\ f'(x)= \\cfrac{3x^2 \\cdot (x^2-4) - x^3 \\cdot 2x }{\\left(x^2-4\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"396\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-495f08a881718b2734ef1db17b5f39ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= \\cfrac{3x^4-12x^2-2x^4}{\\left(x^2-4\\right)^2} = \\cfrac{x^4-12x^2}{\\left(x^2-4\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"298\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora definimos a derivada igual a 0 e resolvemos a equa\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4890b9dfeb634c4d7a349351be73b5d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1d8ad11a06814972752eadc64b861f62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^4-12x^2}{\\left(x^2-4\\right)^2}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"108\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bd6cbd9bc26086bfc990fdf2152e75f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^4-12x^2=0\\cdot \\left(x^2-4\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"192\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-05799ab35c660f110890406329cc4b25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^4-12x^2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"108\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-be662a07bcc839ad08a7e30c0538fc1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2(x^2-12)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"121\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dc1f64cdcd293da4fee1ef02fff9a588_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x^2\\cdot(x^2-12) =0 \\longrightarrow \\begin{cases} x^2 =0 \\ \\longrightarrow \\ \\bm{x=0} \\\\[2ex] x^2-12=0 \\ \\longrightarrow \\ x=\\sqrt{12} \\ \\longrightarrow \\ \\bm{x= \\pm 3,46} \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"527\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Representamos agora na reta todos os pontos singulares encontrados, ou seja, os pontos que n\u00e3o pertencem ao dom\u00ednio (x=-2 e x=+2) e aqueles que cancelam a derivada (x=0, x=- 3,46 e x = +3,46): <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/number-line-346-2-0.webp\" alt=\"\" class=\"wp-image-2653\" width=\"532\" height=\"70\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> E avaliamos o sinal da derivada em cada intervalo, para saber se a fun\u00e7\u00e3o aumenta ou diminui. Portanto, pegamos um ponto em cada intervalo (nunca os pontos singulares) e observamos qual sinal a derivada tem neste ponto: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2dc401187921384b6e083fd0d662e404_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-4)= \\cfrac{(-4)^4-12(-4)^2}{\\left((-4)^2-4\\right)^2} = \\cfrac{64}{144} = 0,44 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"367\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c2195913b32457357954d053d8f37b91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-3)= \\cfrac{(-3)^4-12(-3)^2}{\\left((-3)^2-4\\right)^2} = \\cfrac{-27}{25} = -1,08 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"394\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-840f32597f014205b3223516b1557773_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1)= \\cfrac{(-1)^4-12(-1)^2}{\\left((-1)^2-4\\right)^2} = \\cfrac{-11}{9} = -1,22 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"394\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88eb3af4c1a297fdbdeae6470edf7165_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1)= \\cfrac{1^4-12\\cdot1^2}{\\left(1^2-4\\right)^2} = \\cfrac{-11}{9} = -1,22 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"338\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2fce37cc9db62e965c31ae2ff6d6ad09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(3)= \\cfrac{3^4-12\\cdot 3^2}{\\left(3^2-4\\right)^2} = \\cfrac{-27}{25} = -1,08 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"338\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-688354135682d1cadec94bbdcac7cc46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(4)= \\cfrac{4^4-12\\cdot 4^2}{\\left(4^2-4\\right)^2} = \\cfrac{64}{144} = 0,44 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"311\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/ligne-numerique-346-monotonie.webp\" alt=\"\" class=\"wp-image-2654\" width=\"534\" height=\"127\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Se a derivada for positiva, significa que a fun\u00e7\u00e3o est\u00e1 aumentando, e se a derivada for negativa, significa que a fun\u00e7\u00e3o est\u00e1 diminuindo. Portanto, os intervalos de crescimento e decl\u00ednio s\u00e3o:<\/p>\n<p class=\"has-text-align-center\"> <strong>Crescimento:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fcc905c2730bc3771063bf7280f05002_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty,-3,46)\\cup (3,46,+\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"207\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Diminuir:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6812d1ef2a6df5de54448b0f42751758_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-3,46,-2)\\cup(-2,0)\\cup (0,2) \\cup (2,3,46)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"308\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A fun\u00e7\u00e3o vai de aumentar para diminuir em x=-3,46, ent\u00e3o x=-3,46 \u00e9 o m\u00e1ximo da fun\u00e7\u00e3o. E a fun\u00e7\u00e3o vai de decrescente para crescente em x=3,46, ent\u00e3o x=3,46 \u00e9 o m\u00ednimo da fun\u00e7\u00e3o.<\/p>\n<p class=\"has-text-align-left\"> Determinamos as coordenadas Y das extremidades relativas: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-673718f25b824c11e7777325974ffeb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(-3,46)=\\cfrac{(-3,46)^3}{(-3,46)^2-4} = \\cfrac{-41,42}{7,97}=-5,20 \\ \\longrightarrow \\ (-3,46,-5,20)\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"529\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bfebef64900ebf04ef1a0fa2f969e1f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(3,46)=\\cfrac{3,46^3}{3,46^2-4} = \\cfrac{41,42}{7,97}=5,20 \\ \\longrightarrow \\ (3,46,5,20)\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"424\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Os extremos relativos da fun\u00e7\u00e3o s\u00e3o, portanto:<\/p>\n<p class=\"has-text-align-center\"> <strong>M\u00e1ximo no ponto<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6670c6858270b2890ec3a8b85d68cc23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-3,46,-5,20)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"117\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>M\u00ednimo para apontar<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-58e02eb58912f164efba8d6b648e45bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(3,46,5,20)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"89\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Para estudar a curvatura da fun\u00e7\u00e3o, calculamos a segunda derivada da fun\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-50df15bb48cacf8f031b640994661e47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{\\left(4x^3-24x\\right)\\cdot \\left(x^2-4\\right)^2 - \\left(x^4-12x^2\\right)\\cdot 2\\left(x^2-4\\right)\\cdot 2x }{ \\left(\\left(x^2-4\\right)^2 \\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"483\" style=\"vertical-align: -33px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3b05b5f09c2adbfead593df2cdf2ad29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{\\left(4x^3-24x\\right)\\cdot \\left(x^2-4\\right)^2 - \\left(x^4-12x^2\\right)\\cdot 4x\\left(x^2-4\\right) }{\\left(x^2-4\\right)^4 }\" title=\"Rendered by QuickLaTeX.com\" height=\"52\" width=\"461\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-92e1aa280d06bf8b58045845d5e21f37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{\\left(4x^3-24x\\right)\\cdot \\left(x^2-4\\right)^{\\cancel{2}} - \\left(x^4-12x^2\\right)\\cdot 4x\\cancel{\\left(x^2-4\\right)} }{\\left(x^2-4\\right)^{\\cancelto{3}{4}} }\" title=\"Rendered by QuickLaTeX.com\" height=\"52\" width=\"458\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a912eff3359969b6ffbef96a3f16932d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{\\left(4x^3-24x\\right)\\cdot \\left(x^2-4\\right) - \\left(x^4-12x^2\\right)\\cdot 4x}{\\left(x^2-4\\right)^3 }\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"386\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8bc35bdd2b70bbac52fa0f24bbefa261_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{4x^5-16x^3-24x^3+96x - \\left(4x^5-48x^3\\right) }{\\left(x^2-4\\right)^3 }\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"381\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-045971f71cc11ced77ea0df9f2c514fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{4x^5-16x^3-24x^3+96x - 4x^5+48x^3 }{\\left(x^2-4\\right)^3 }\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"365\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3144a0aa00ee8ec427752f05f0fac40c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{8x^3+96x  }{\\left(x^2-4\\right)^3 }\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"145\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora definimos a segunda derivada igual a 0 e resolvemos a equa\u00e7\u00e3o: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f618f4961c18c45be60fc496ad4896e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e8ed519add27a4d51c75b49179e632ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{8x^3+96x  }{\\left(x^2-4\\right)^3 }=0\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"109\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b5c0d1e44accc3b68a67598f5c4d834c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"8x^3+96x =0\\cdot \\left(x^2-4\\right)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"193\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4c52a7448e4acc67488ef5747cc3bed9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"8x^3+96x =0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"109\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4f2884a1c9b8dabf6ea5323f2ac71b2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x(8x^2+96)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"123\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-31adba554b44aa92fd7227506440ccaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x\\cdot(8x^2+96) =0 \\longrightarrow \\begin{cases} \\bm{x =0} \\\\[2ex] 8x^2+96=0 \\ \\longrightarrow \\ x^2=\\cfrac{-96}{8}} = -12 \\ \\longrightarrow \\ x= \\sqrt{-12} \\ \\color{red}\\bm{\\times} \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"635\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-635e2ffa452a5a66a4bcacb0e111c5ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= \\sqrt{-12}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"81\" style=\"vertical-align: -3px;\"><\/p>\n<p> N\u00e3o h\u00e1 solu\u00e7\u00e3o, pois n\u00e3o existe raiz negativa de um n\u00famero real.<\/p>\n<p class=\"has-text-align-left\"> Representamos agora na reta todos os pontos singulares encontrados, ou seja, os pontos que n\u00e3o pertencem ao dom\u00ednio (x=-2 e x=+2) e aqueles que cancelam a segunda derivada (x=0): <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/droite-numerique-2-0-2.webp\" alt=\"\" class=\"wp-image-2399\" width=\"370\" height=\"72\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> E avaliamos o sinal da segunda derivada em cada intervalo, para saber se a fun\u00e7\u00e3o \u00e9 c\u00f4ncava ou convexa. Portanto, pegamos um ponto em cada intervalo (nunca os pontos singulares) e observamos qual sinal a segunda derivada tem neste ponto: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3b5618d1ab96a078d50507f45155595b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(-3)=\\cfrac{8(-3)^3+96(-3)  }{\\left((-3)^2-4\\right)^3 } = \\cfrac{-504}{125}=-4,03 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"408\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e2e868f1f815d4155a187c55b004cc13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(-1)=\\cfrac{8(-1)^3+96(-1)  }{\\left((-1)^2-4\\right)^3 } = \\cfrac{-104}{-27}=3,85 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"394\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-acc8341ccd6a1c8a3cd3d6a0ce888dba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(1)=\\cfrac{8\\cdot 1^3+96\\cdot 1  }{\\left(1^2-4\\right)^3 } = \\cfrac{104}{-27}=-3,85 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"348\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-54d98824f72954de12bc065471a610e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(3)=\\cfrac{8\\cdot 3^3+96\\cdot 3  }{\\left(3^2-4\\right)^3 } = \\cfrac{504}{125}=4,03 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"329\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/ligne-numerique-2-0-2-courbure.webp\" alt=\"\" class=\"wp-image-2568\" width=\"371\" height=\"124\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Se a segunda derivada for positiva, significa que a fun\u00e7\u00e3o \u00e9 convexa.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> , e se a segunda derivada for negativa isso significa que a fun\u00e7\u00e3o \u00e9 c\u00f4ncava<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Os intervalos de concavidade e convexidade s\u00e3o, portanto:<\/p>\n<p class=\"has-text-align-center\"> <strong>Convexo<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0cd739761b2dc845594c0a0696a240c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-2,0)\\cup (2,+\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>C\u00f4ncavo<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5e741ac026627200772655094f921f26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty,-2)\\cup (0,2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> No entanto, embora haja uma mudan\u00e7a na curvatura em x=-2 ex=+2, estes n\u00e3o s\u00e3o pontos de inflex\u00e3o. Porque x=-2 e x=+2 n\u00e3o pertencem ao dom\u00ednio da fun\u00e7\u00e3o. Por outro lado, em x=0 h\u00e1 uma mudan\u00e7a na curvatura (a fun\u00e7\u00e3o passa de convexa para c\u00f4ncava) e isso pertence \u00e0 fun\u00e7\u00e3o, ent\u00e3o x=0 \u00e9 um ponto de inflex\u00e3o.<\/p>\n<p class=\"has-text-align-left\"> Substitu\u00edmos os pontos de inflex\u00e3o encontrados na fun\u00e7\u00e3o original para encontrar a outra coordenada do ponto de inflex\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8a1a8a10485a749c631531b80ce05642_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0) = \\cfrac{0^3}{0^2-4}  = \\cfrac{0}{-4} =0\\ \\longrightarrow \\ (0,0)\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"276\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Os pontos de viragem da fun\u00e7\u00e3o s\u00e3o, portanto:<\/p>\n<p class=\"has-text-align-center\"> <strong>Pontos de viragem:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2c790019bd70403eba876c59c82c0f9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Finalmente, com base em todas as informa\u00e7\u00f5es que calculamos, representamos a fun\u00e7\u00e3o: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/representation-graphique-des-fonctions-exercices-resolus.webp\" alt=\"representa\u00e7\u00e3o gr\u00e1fica de fun\u00e7\u00f5es resolvidas, exerc\u00edcios\" class=\"wp-image-2655\" width=\"449\" height=\"518\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Coment\u00e1rio: Observe que a fun\u00e7\u00e3o cruza a ass\u00edntota obl\u00edqua no ponto<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-791f3561f68c75b943d5af446c9f988f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,0) .\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<p> Na verdade, as ass\u00edntotas obl\u00edquas determinam sobretudo o comportamento da fun\u00e7\u00e3o quando x tende para +\u221e e -\u221e, de fato, a fun\u00e7\u00e3o nunca cruza a ass\u00edntota obl\u00edqua \u00e0 direita do gr\u00e1fico (x\u2192+\u221e) e \u00e0 esquerda de o gr\u00e1fico (x\u2192-\u221e). Por\u00e9m, \u00e9 muito raro que a fun\u00e7\u00e3o cruze a ass\u00edntota obl\u00edqua no meio, \u00e9 um caso muito especial.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Neste artigo veremos como representar qualquer tipo de fun\u00e7\u00e3o em um gr\u00e1fico. Al\u00e9m disso, voc\u00ea encontrar\u00e1 exerc\u00edcios passo a passo resolvidos sobre a representa\u00e7\u00e3o de fun\u00e7\u00f5es em um gr\u00e1fico. Como representar uma fun\u00e7\u00e3o em um gr\u00e1fico Para representar uma fun\u00e7\u00e3o em um gr\u00e1fico, as seguintes etapas devem ser executadas: Encontre o dom\u00ednio da fun\u00e7\u00e3o. Calcule &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/representacao-de-funcoes\/\"> <span class=\"screen-reader-text\">Representa\u00e7\u00e3o de fun\u00e7\u00e3o<\/span> Leia mais &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[22],"tags":[],"class_list":["post-46","post","type-post","status-publish","format-standard","hentry","category-representacao-de-funcao"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Representa\u00e7\u00e3o de fun\u00e7\u00f5es -<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/pt\/representacao-de-funcoes\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Representa\u00e7\u00e3o de fun\u00e7\u00f5es -\" \/>\n<meta property=\"og:description\" content=\"Neste artigo veremos como representar qualquer tipo de fun\u00e7\u00e3o em um gr\u00e1fico. Al\u00e9m disso, voc\u00ea encontrar\u00e1 exerc\u00edcios passo a passo resolvidos sobre a representa\u00e7\u00e3o de fun\u00e7\u00f5es em um gr\u00e1fico. Como representar uma fun\u00e7\u00e3o em um gr\u00e1fico Para representar uma fun\u00e7\u00e3o em um gr\u00e1fico, as seguintes etapas devem ser executadas: Encontre o dom\u00ednio da fun\u00e7\u00e3o. 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