{"id":43,"date":"2023-09-17T10:56:20","date_gmt":"2023-09-17T10:56:20","guid":{"rendered":"https:\/\/mathority.org\/pt\/maximos-minimos-de-uma-funcao-extremos-relativos\/"},"modified":"2023-09-17T10:56:20","modified_gmt":"2023-09-17T10:56:20","slug":"maximos-minimos-de-uma-funcao-extremos-relativos","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/maximos-minimos-de-uma-funcao-extremos-relativos\/","title":{"rendered":"M\u00e1ximo e m\u00ednimo de uma fun\u00e7\u00e3o (extremos relativos)"},"content":{"rendered":"<p>Neste artigo voc\u00ea descobrir\u00e1 como calcular o m\u00e1ximo e o m\u00ednimo de uma fun\u00e7\u00e3o, explicamos resolvendo dois exemplos passo a passo. Al\u00e9m disso, voc\u00ea poder\u00e1 praticar exerc\u00edcios passo a passo sobre os m\u00e1ximos e m\u00ednimos de uma fun\u00e7\u00e3o. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-son-los-maximos-y-minimos-de-una-funcion\"><\/span> Quais s\u00e3o o m\u00e1ximo e o m\u00ednimo de uma fun\u00e7\u00e3o?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Os m\u00e1ximos de uma fun\u00e7\u00e3o s\u00e3o os maiores valores da fun\u00e7\u00e3o e os m\u00ednimos de uma fun\u00e7\u00e3o s\u00e3o os menores valores da fun\u00e7\u00e3o.<\/strong> Os m\u00e1ximos e m\u00ednimos de uma fun\u00e7\u00e3o s\u00e3o <strong>extremos relativos<\/strong> quando representam apenas os maiores ou menores valores em seu ambiente, mas s\u00e3o <strong>extremos absolutos<\/strong> quando representam os maiores ou menores valores de toda 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\/maximums-et-minimums-d-une-fonction.webp\" alt=\"m\u00e1ximos e m\u00ednimos de uma fun\u00e7\u00e3o\" class=\"wp-image-2437\" width=\"512\" height=\"420\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Voc\u00ea tamb\u00e9m pode identificar extremos relativos estudando o <strong>crescimento e a diminui\u00e7\u00e3o da fun\u00e7\u00e3o<\/strong> :<\/p>\n<ul>\n<li> Um ponto \u00e9 um <strong>m\u00e1ximo relativo<\/strong> quando a fun\u00e7\u00e3o vai de crescente para decrescente.<\/li>\n<li> Um ponto \u00e9 um <strong>m\u00ednimo relativo<\/strong> quando a fun\u00e7\u00e3o passa de decrescente para crescente. <\/li>\n<\/ul>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-hallar-los-maximos-y-minimos-de-una-funcion\"><\/span> Como encontrar o m\u00e1ximo e o m\u00ednimo de uma fun\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A partir da primeira e da segunda derivada de uma fun\u00e7\u00e3o, podemos saber se uma fun\u00e7\u00e3o tem um extremo relativo em um ponto e se esse ponto \u00e9 um m\u00e1ximo relativo ou um m\u00ednimo relativo: <\/p>\n<div style=\"background-color:#FFF3E0; padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 10px; border-radius:30px;\">\n<ul style=\"color:#64B5F6; font-weight: bold;\">\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Uma fun\u00e7\u00e3o tem um <strong>extremo em rela\u00e7\u00e3o<\/strong> aos pontos que cancelam sua primeira derivada.<\/span><\/li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d6dcc4b4bb7f26cf48a025c8e0dddf83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(a)=0 \\quad \\bm{\\longrightarrow} \\quad x=a \\text{ es un extremo relativo}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"357\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">E o sinal da segunda derivada da fun\u00e7\u00e3o determina se o ponto \u00e9 m\u00e1ximo ou m\u00ednimo:<\/span>\n<ul style=\"color:#64B5F6; font-weight: bold; margin-left:7%; list-style-type:circle\">\n<li style=\"margin-bottom:10px\"> <span style=\"color:#101010;font-weight: normal;\">Se a segunda derivada for negativa, a fun\u00e7\u00e3o ter\u00e1 um <strong>m\u00e1ximo relativo<\/strong> nesse ponto.<\/span><\/li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-211c91be3bfe6e5a91f048684198c70a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(a)<0 \\quad \\bm{\\longrightarrow} \\quad x=a \\text{ es un m\\'aximo relativo}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"360\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<li style=\"margin-bottom:10px\"> <span style=\"color:#101010;font-weight: normal;\">Se a segunda derivada for positiva, a fun\u00e7\u00e3o ter\u00e1 um <strong>m\u00ednimo relativo<\/strong> nesse ponto.<\/span> <\/li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-089b7ae49fe440e3b4db19e0b17d8815_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(a)>0 \\quad \\bm{\\longrightarrow} \\quad x=a \\text{ es un m\\&#8217;inimo relativo}&#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;19&#8243; width=&#8221;356&#8243; style=&#8221;vertical-align: -5px;&#8221;><\/p>\n<\/p>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-1-como-calcular-los-maximos-y-minimos-de-una-funcion\"><\/span> Exemplo 1: Como calcular o m\u00e1ximo e o m\u00ednimo de uma fun\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Depois de vermos as defini\u00e7\u00f5es de m\u00e1ximo e m\u00ednimo de uma fun\u00e7\u00e3o, resolveremos um exemplo passo a passo para que voc\u00ea possa ver como s\u00e3o calculados o m\u00e1ximo e o m\u00ednimo de uma fun\u00e7\u00e3o.<\/p>\n<ul>\n<li> Calcule os extremos relativos da seguinte fun\u00e7\u00e3o e determine se s\u00e3o m\u00e1ximos ou m\u00ednimos:<\/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-1d76dfe92202a4fa44057a7f4576c97a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^3-3x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"117\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Os extremos relativos da fun\u00e7\u00e3o ser\u00e3o os pontos que satisfazem<\/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> . Portanto, primeiro calculamos a 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-cad79f4ba702585c8bece2546419bd83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^3-3x \\ \\longrightarrow \\ f'(x)=3x^2-3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"287\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> E agora igualamos a derivada da fun\u00e7\u00e3o a zero e resolvemos 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-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-02ff903613a3329eb87c4943c4cb135b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x^2-3=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"90\" 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-af1930d8be7faaf020a4103a17e484b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x^2=3\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"59\" 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-e8367eba5249bf4e7b1e395d86bb91be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=\\cfrac{3}{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"52\" 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-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-f0428d6e7e3932e00d3e6a7ab1a779d2_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> Portanto, os extremos relativos da fun\u00e7\u00e3o s\u00e3o x=+1 e x=-1.<\/p>\n<p> Depois de conhecermos os extremos relativos da fun\u00e7\u00e3o, podemos saber se s\u00e3o m\u00e1ximo ou m\u00ednimo com o sinal da segunda derivada. Portanto, 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-a2a906b229d6f4d4d03f59828f327fdb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=3x^2-3 \\ \\longrightarrow \\ f''(x)=6x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"256\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> E agora avaliamos na segunda derivada os extremos relativos que encontramos antes, para saber se s\u00e3o m\u00e1ximo ou m\u00ednimo relativo:<\/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-eb0c79376c0c8816492173e5f109809f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(1)=6\\cdot 1 = 6 \\ \\longrightarrow\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"167\" style=\"vertical-align: -5px;\"><\/p>\n<p> M\u00ednimo relativo<\/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-8d51f54d437345293be122b03b5fff03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(-1)=6\\cdot (-1) = -6 \\ \\longrightarrow\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"222\" style=\"vertical-align: -5px;\"><\/p>\n<p> Relativo m\u00e1ximo<\/p>\n<p> A segunda derivada em x=1 \u00e9 positiva, ent\u00e3o <strong>x=1 \u00e9 um m\u00ednimo relativo<\/strong> . Por outro lado, a segunda derivada em x=-1 \u00e9 negativa, ent\u00e3o <strong>x=-1 \u00e9 um m\u00e1ximo relativo<\/strong> .<\/p>\n<p> Finalmente, substitu\u00edmos os pontos encontrados na fun\u00e7\u00e3o original para encontrar a coordenada Y dos extremos relativos:<\/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-65771f8b9ce10ad863604fe6e6dca867_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1)=1^3-3\\cdot 1=-2 \\ \\longrightarrow \\ (1,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"275\" 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-04df1e45775b1318795100d7211f3b32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(-1)=(-1)^3-3\\cdot(-1)= 2 \\ \\longrightarrow \\ (-1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"330\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Em conclus\u00e3o, os extremos relativos da fun\u00e7\u00e3o s\u00e3o:<\/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-a80e956c137aedb103a56acc0cf510e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(1,-2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<\/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-92defeed12f15814813d53b8a24be9ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-1,2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-2-estudiar-la-monotonia-y-los-maximos-y-minimos-de-una-funcion\"><\/span> Exemplo 2: Estudando a monotonicidade e os m\u00e1ximos e m\u00ednimos de uma fun\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Agora vamos ver como se resolve outro tipo de exerc\u00edcio. Neste caso explicaremos como encontrar o m\u00e1ximo e o m\u00ednimo da monotonicidade de uma fun\u00e7\u00e3o.<\/p>\n<ul>\n<li> Estude a monotonicidade e calcule os extremos relativos da seguinte fun\u00e7\u00e3o:<\/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 calcular o dom\u00ednio de defini\u00e7\u00e3o da fun\u00e7\u00e3o. Sendo uma fun\u00e7\u00e3o racional, 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-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-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> Uma vez calculado o dom\u00ednio de defini\u00e7\u00e3o da fun\u00e7\u00e3o, precisamos estudar quais pontos cancelam a primeira derivada. Portanto, derivamos a 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-19cfa0164864d95d9b2d51743bf7c0d1_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}{\\left(x-1\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"364\" 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-8a280e333342dbe57e5d18839a1c9c0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=\\cfrac{2x^2-2x - x^2}{\\left(x-1\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"172\" 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-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> 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-faf06fb85062e758f99800d1ffa0788b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^2-2x}{\\left(x-1\\right)^2}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"95\" style=\"vertical-align: -20px;\"><\/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 as duas 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> Depois de calcularmos o dom\u00ednio da fun\u00e7\u00e3o e<\/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> , representamos todos os pontos cr\u00edticos encontrados na linha: <\/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=\"399\" 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. Para fazer isso, pegamos um ponto em cada intervalo (nunca os pontos cr\u00edticos) e observamos qual sinal a derivada tem naquele 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-84e1013672adc10e9447af5478f592a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(0,5) = \\cfrac{0,5^2-2\\cdot0,5}{\\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-110911aebecc81132e3d726e00be1fcc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1,5) = \\cfrac{1,5^2-2\\cdot1,5}{\\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-b0a995a54cd7d661f6431cdc3d0d0eda_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(3) = \\cfrac{3^2-2\\cdot3}{\\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=\"400\" height=\"138\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Se a derivada for positiva, significa que a fun\u00e7\u00e3o est\u00e1 aumentando, mas 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 <strong>x=0 \u00e9 um m\u00e1ximo relativo<\/strong> da fun\u00e7\u00e3o <strong>.<\/strong> E em x=2, a fun\u00e7\u00e3o vai de decrescente para crescente, ent\u00e3o <strong>x=2 \u00e9 um m\u00ednimo relativo<\/strong> da fun\u00e7\u00e3o.<\/p>\n<p> E por fim, substitu\u00edmos os pontos encontrados na fun\u00e7\u00e3o original para encontrar a coordenada Y das extremidades:<\/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> Resumindo, os extremos relativos da fun\u00e7\u00e3o s\u00e3o:<\/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<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-maximos-y-minimos-de-una-funcion\"><\/span> Exerc\u00edcios resolvidos sobre m\u00e1ximos e m\u00ednimos de uma fun\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Calcule os extremos relativos da seguinte fun\u00e7\u00e3o polinomial e determine se eles s\u00e3o m\u00e1ximos ou m\u00ednimos: <\/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-930d724a5ca23ed9152211f24dc2340b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^3-3x^2-9x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"165\" 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\"> Os extremos relativos da fun\u00e7\u00e3o ser\u00e3o os pontos em que a primeira derivada da fun\u00e7\u00e3o \u00e9 igual a zero. Portanto, calculamos a 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-f353678aff2ff5f19c53042f35ef8a19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^3-3x^2-9x \\ \\longrightarrow \\  f'(x)=3x^2-6x-9\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"376\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E agora 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-58dcd049349f740f082d583dfd9e364c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=0:\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"80\" 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-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-77ac7ac1a36d5c8591235d8400eb68cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x^2-6x-9=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"131\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Temos uma equa\u00e7\u00e3o quadr\u00e1tica, ent\u00e3o aplicamos a f\u00f3rmula geral para resolv\u00ea-la:<\/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-8e4a1d5ede3779d54c8b9b66571a3394_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} x &amp;=\\cfrac{-b \\pm \\sqrt{b^2-4ac}}{2a} =\\cfrac{-(-6) \\pm \\sqrt{(-6)^2-4\\cdot 3 \\cdot (-9)}}{2\\cdot 3}=\\\\[1.5ex]&amp;=\\cfrac{6 \\pm \\sqrt{144}}{6}=\\cfrac{6 \\pm 12}{6} =\\begin{cases} \\cfrac{6 + 12}{6}=\\cfrac{18}{6}= 3 \\\\[4ex] \\cfrac{6 - 12}{6}=\\cfrac{-6}{6}=-1 \\end{cases} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"168\" width=\"451\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, os extremos relativos da fun\u00e7\u00e3o s\u00e3o os pontos x=3 e x=-1.<\/p>\n<p class=\"has-text-align-left\"> Depois de conhecermos os extremos relativos da fun\u00e7\u00e3o, podemos saber se s\u00e3o m\u00e1ximo ou m\u00ednimo com o sinal da segunda derivada. Portanto, diferenciamos a fun\u00e7\u00e3o novamente:<\/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-b2ddaaae5740b93b84eb1db4c4e12f69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=3x^2-6x-9 \\ \\longrightarrow \\  f''(x)=6x-6\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"327\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E agora avaliamos os pontos que calculamos antes na 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-b97f883eb74286ab41179d4353161816_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(3)=6(3)-6=18-6 = +12 \\ \\longrightarrow \\ \\text{M\\'inimo}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"365\" 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-b6efbeb4ad4b54c03aa440dcafb7dc4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(-1)=6(-1)-6=-6-6 = -12 \\ \\longrightarrow \\ \\text{M\\'aximo}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"400\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A segunda derivada em x=3 \u00e9 positiva, ent\u00e3o <strong>x=3 \u00e9 um m\u00ednimo<\/strong> . E a segunda derivada em x=-1 \u00e9 negativa, ent\u00e3o <strong>x=-1 \u00e9 m\u00e1ximo<\/strong> .<\/p>\n<p class=\"has-text-align-left\"> E por fim, substitu\u00edmos os pontos encontrados na fun\u00e7\u00e3o original para encontrar a coordenada Y das extremidades: <\/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-0b885b81db85c9d12caeed0e046f14ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(3)=3^3-3\\cdot 3^2-9\\cdot3=-27 \\ \\longrightarrow \\ (3,-27)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"353\" 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-668346639ebe571949cd8e8939c8a4a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(-1)=(-1)^3-3(-1)^2-9(-1)=5 \\ \\longrightarrow \\ (-1,5)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"392\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Resumindo, os extremos relativos da fun\u00e7\u00e3o s\u00e3o:<\/p>\n<p class=\"has-text-align-center\"> <strong>M\u00ednimo relativo ao 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-1b572e4ffbdfe59c16e4e1a30b9ac82a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(3,-27)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"61\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>M\u00e1ximo em rela\u00e7\u00e3o ao 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-c7aca1cad23e01f6998ce87ff4f73734_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-1,5)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\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> Calcule os extremos relativos da seguinte fun\u00e7\u00e3o exponencial e determine se eles s\u00e3o m\u00e1ximos ou m\u00ednimos: <\/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-6e82a7f154d9620b6fdcd2d134cbf20a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=e^x(x-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" 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\"> Primeiro, precisamos diferenciar a fun\u00e7\u00e3o. Para fazer isso, aplicamos a f\u00f3rmula da derivada de um produto: <\/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-5508d7a8f5ef73fd09e2c8a013513229_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=e^x\\cdot (x-1)+ e^x\\cdot 1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"206\" 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-6d1f83cd5953e56070c9f8dea5a03ea5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=xe^x -e^x +e^x = xe^x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"216\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E agora 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-58dcd049349f740f082d583dfd9e364c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=0:\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"80\" 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-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-9da9b4b19b1c7985bf785b693009de95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"xe^x=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"60\" 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-f5c0d99b3aa4115c0415e0e57f5df2a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x\\cdot e^x =0 \\longrightarrow \\begin{cases} \\bm{x=0} \\\\[2ex] e^x=0 \\ \\color{red}\\bm{\\times} \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"220\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Um n\u00famero elevado a outro nunca pode resultar em 0. 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-0108040ee23df4da2db681c9ffb2decc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"e^x=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"49\" style=\"vertical-align: 0px;\"><\/p>\n<p> n\u00e3o tem solu\u00e7\u00e3o e o \u00fanico extremo relativo \u00e9<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_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> .<\/p>\n<p class=\"has-text-align-left\"> Agora calculamos a segunda derivada da fun\u00e7\u00e3o para saber se o extremo relativo \u00e9 m\u00e1ximo ou m\u00ednimo:<\/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-2e3186824ec757b18335f7c6b93e6068_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= xe^x \\ \\longrightarrow \\ f''(x)= 1\\cdot e^x + x \\cdot e^x = e^x+xe^x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"394\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E agora avaliamos na segunda derivada o extremo que encontramos antes, para ver se \u00e9 m\u00e1ximo ou m\u00ednimo:<\/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-a333c6f36d372595070b5cf10ef06659_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(0)= e^{0}+0\\cdot e^{0} = 1+0\\cdot 1 = 1 \\ \\longrightarrow \\ \\text{M\\'inimo}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"368\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Como a segunda derivada em x=0 \u00e9 positiva, <strong>x=0 \u00e9 um m\u00ednimo relativo ou local<\/strong> .<\/p>\n<p class=\"has-text-align-left\"> Finalmente, substitu\u00edmos o ponto encontrado na fun\u00e7\u00e3o original para encontrar a outra coordenada final:<\/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-b262d8c03601983b5497fc165bab677a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=e^{0}(0-1) =1\\cdot (-1)=-1 \\ \\longrightarrow \\ (0,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"357\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O \u00fanico extremo relativo da fun\u00e7\u00e3o \u00e9, portanto:<\/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-737e35e6d1698a9e89986af90d34722e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,-1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\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> Estude a monotonicidade e encontre os extremos relativos 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-03ea07dcfe35eeade4235b3325681c2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=\\frac{x -1 }{x^2+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"109\" style=\"vertical-align: -14px;\"><\/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\"> Primeiro, determinamos o dom\u00ednio da fun\u00e7\u00e3o. Para fazer isso, igualamos o denominador da fra\u00e7\u00e3o a zero e resolvemos 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-13670326c7cf3ae27c79e8e2ea4f438b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+1 = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"81\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A express\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-b35b5141a240a76c5fc0e3c75ab5689d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+1\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"47\" style=\"vertical-align: -2px;\"><\/p>\n<p> Nunca ser\u00e1 0, pois o resultado de x <sup>2<\/sup> ser\u00e1 sempre um n\u00famero positivo ou 0. Portanto, somar 1 nunca dar\u00e1 0. O dom\u00ednio da fun\u00e7\u00e3o \u00e9, portanto, composto apenas por 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\"> A seguir, estudamos quais pontos se encontram<\/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-3d9516fab46f301bc09e336a12418ad4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=0.\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<p> Diferenciamos a fun\u00e7\u00e3o usando a regra do quociente: <\/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-2a21eceaf556455939314d569b69f365_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x -1 }{x^2+1} \\ \\longrightarrow \\ f'(x)= \\cfrac{1 \\cdot (x^2+1) - (x-1) \\cdot 2x }{\\left(x^2+1}\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"415\" 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-6104d51f83e54587e198db396734fec1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= \\cfrac{x^2+1-(2x^2-2x)}{\\left(x^2+1\\right)^2} = \\cfrac{x^2+1-2x^2+2x}{\\left(x^2+1\\right)^2}= \\cfrac{-x^2+2x+1}{\\left(x^2+1\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"515\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> 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-501383d3407e95ff1980351452e414f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{-x^2+2x+1}{\\left(x^2+1\\right)^2}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"143\" 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-348057b71ce15780c2f47bd8053e4cd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-x^2+2x+1=0\\cdot \\left(x^2+1\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"219\" 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-579207dec3599e2925ad24d2e951cb47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-x^2+2x+1=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"134\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Temos uma equa\u00e7\u00e3o quadr\u00e1tica, ent\u00e3o usamos a f\u00f3rmula geral para resolv\u00ea-la:<\/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-836d878f15098c1fe997fbb0392b8733_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}x &amp;=\\cfrac{-b \\pm \\sqrt{b^2-4ac}}{2a} =\\cfrac{-2 \\pm \\sqrt{2^2-4\\cdot (-1) \\cdot 1}}{2\\cdot (-1)} = \\\\[1.5ex]&amp;=\\cfrac{-2 \\pm \\sqrt{8}}{-2} =\\begin{cases} \\cfrac{-2 + \\sqrt{8}}{-2}= -0,41 \\\\[4ex] \\cfrac{-2 - \\sqrt{8}}{-2}= 2,41\\end{cases} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"174\" width=\"396\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Depois de calcularmos o dom\u00ednio da fun\u00e7\u00e3o e<\/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> , representamos todos os pontos singulares encontrados na reta num\u00e9rica: <\/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-041-241.webp\" alt=\"\" class=\"wp-image-2451\" width=\"319\" height=\"83\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> E agora avaliamos o sinal da derivada em cada intervalo, para descobrir se a fun\u00e7\u00e3o \u00e9 crescente ou decrescente. 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-6c9e9690a2834e9ba455ebe711bfba4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1)= \\cfrac{-(-1)^2+2(-1)+1}{\\left((-1)^2+1\\right)^2}}= \\cfrac{-2}{+4} =-0,5 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"412\" 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-6c280aa192cd61431df6a1ade0389ed2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(0)= \\cfrac{-0^2+2(0)+1}{\\left(0^2+1\\right)^2}}= \\cfrac{+1}{+1} =+1 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"340\" 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-b40b424c7a763aa9849f33d850a10a1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(3)= \\cfrac{-3^2+2\\cdot 3+1}{\\left(3^2+1\\right)^2}}= \\cfrac{-2}{+100} =-0,02 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"375\" 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-041-241-negatif-positif-negatif.webp\" alt=\"\" class=\"wp-image-2453\" width=\"319\" height=\"150\" 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 nesse intervalo, mas 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-262e8d5f95ee4afe2dacc0037d8f334c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-0,41 \\ , \\ 2,41)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" 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-33c3a6bfd3dbfbdd2eff5fc4b70aea5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty \\ , \\ -0,41)\\cup (2,41 \\ , \\ +\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"231\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A fun\u00e7\u00e3o muda de decrescente para crescente em x=-0,41, ent\u00e3o <strong>x=-0,41 \u00e9 um m\u00ednimo local<\/strong> da fun\u00e7\u00e3o. E a fun\u00e7\u00e3o vai de crescente para decrescente em x=2,41, ent\u00e3o <strong>x=2,41 \u00e9 um m\u00e1ximo local<\/strong> 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-605a64bba8103c7ee0015a92b60273b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(-0,41)=\\cfrac{-0,41 -1 }{(-0,41)^2+1} = \\cfrac{-1,41}{1,17}= -1,21 \\ \\longrightarrow \\ (-0,41 \\ , \\ -1,21)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"532\" 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-02ff38a48dc66cf3a658619cf41803c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2,41)=\\cfrac{2,41 -1 }{2,41^2+1} = \\cfrac{1,41}{6,81}= 0,21 \\ \\longrightarrow \\ (2,41 \\ , \\ 0,21)\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"427\" 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\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-906261d9a75f4bc2766c65fc0ac5a363_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-0,41 \\ , \\ -1,21)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"128\" style=\"vertical-align: -5px;\"><\/p>\n<\/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-bf7380f280bd665935068801c9c0d83d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{ (2,41 \\ , \\ 0,21)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"101\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 4<\/h3>\n<p> Sabemos que a 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-bc79cbc1c7886fe5d95d2db47d1635f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+ax+b\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"147\" style=\"vertical-align: -5px;\"><\/p>\n<p> passar pelo ponto<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8f97f2fdc4d62902377daa83ebbd005b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(1,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<p> e tem um extremo relativo em<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3d8c9d56ee018947d8f054cd237e8c06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -1 .\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"61\" style=\"vertical-align: 0px;\"><\/p>\n<p> Determine o valor das inc\u00f3gnitas<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> e o valor de <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-73a3ea89ad967f2efadeb096bd87bdb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b .\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\"><\/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\"> Deixe a fun\u00e7\u00e3o ter um extremo relativo em<\/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-ee848c4f59793dbd8bd705b4e2411c8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<p> isso significa que foi realizado<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6f57b701a1080acd4db5681249566b5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1)=0.\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"88\" style=\"vertical-align: -5px;\"><\/p>\n<p> Portanto, calculamos a derivada da fun\u00e7\u00e3o em<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ee848c4f59793dbd8bd705b4e2411c8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<p> e definimos como igual a 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-41316f1389c40d8634eb0ad596956ca2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x) = x^2+ax+b \\ \\longrightarrow \\ f'(x)=2x+a\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"309\" 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-1786074f9a3b69a0c2a13a0db7a67895_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{l} f'(-1)=2(-1)+a\\\\[2ex] f'(-1)=0\\end{array} \\right\\} \\longrightarrow 2(-1)+a=0\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E resolvemos a equa\u00e7\u00e3o obtida para encontrar o valor do par\u00e2metro a: <\/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-f5e35d6b05179bb3e4db43f738b6da29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2(-1)+a=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" 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-729fe80252784c84b2a49624e59b2ac7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-2+a=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"85\" 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-85d78ced37b5f76e83a3c9c24a8b3eca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{a=2}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A fun\u00e7\u00e3o ser\u00e1 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-bbea1a44d5027753ebc196d004e5671d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+ax+b \\ \\xrightarrow{a \\ = \\ 2} \\ f(x)=x^2+2x+b\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"362\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Por outro lado, dizem-nos que a fun\u00e7\u00e3o passa pelo 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-702446e4df1457eff7e83e00a8709824_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(1,-2) .\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"57\" style=\"vertical-align: -5px;\"><\/p>\n<p> Isso \u00e9 para dizer,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-304a45b518ffaec62b95f169ad647688_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1)=-2 .\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"84\" style=\"vertical-align: -5px;\"><\/p>\n<p> Portanto, podemos aplicar esta condi\u00e7\u00e3o para encontrar o valor da vari\u00e1vel b:<\/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-6c09bb57a4a4fd3eb5d72f5d35d3c539_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{l} f(1)=1^2+2\\cdot1+b \\\\[2ex] f(1)=-2 \\end{array} \\right\\} \\longrightarrow 1^2+2\\cdot 1+b = -2\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"361\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E resolvemos a equa\u00e7\u00e3o obtida para encontrar o valor do par\u00e2metro b: <\/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-4e05b252664d0ea2da72627e779469d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1^2+2\\cdot1+b=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"143\" 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-626f95581121d205b149c2323e711759_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1+2+b=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"113\" 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-625149278867f4929d813258055868e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b=-2-1-2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"114\" 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-2e925fb7c5eaa30caa970c92688ede93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{b=-5}\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"53\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A fun\u00e7\u00e3o \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-2becf12662d9dc5f68cb13dd248f3e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+2x+b \\ \\xrightarrow{b \\ = \\ -5} \\ f(x)=x^2+2x-5\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"371\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Neste artigo voc\u00ea descobrir\u00e1 como calcular o m\u00e1ximo e o m\u00ednimo de uma fun\u00e7\u00e3o, explicamos resolvendo dois exemplos passo a passo. Al\u00e9m disso, voc\u00ea poder\u00e1 praticar exerc\u00edcios passo a passo sobre os m\u00e1ximos e m\u00ednimos de uma fun\u00e7\u00e3o. Quais s\u00e3o o m\u00e1ximo e o m\u00ednimo de uma fun\u00e7\u00e3o? Os m\u00e1ximos de uma fun\u00e7\u00e3o s\u00e3o os &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/maximos-minimos-de-uma-funcao-extremos-relativos\/\"> <span class=\"screen-reader-text\">M\u00e1ximo e m\u00ednimo de uma fun\u00e7\u00e3o (extremos relativos)<\/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-43","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>M\u00e1ximo e m\u00ednimo de uma fun\u00e7\u00e3o (extremos relativos) -<\/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\/maximos-minimos-de-uma-funcao-extremos-relativos\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"M\u00e1ximo e m\u00ednimo de uma fun\u00e7\u00e3o (extremos relativos) -\" \/>\n<meta property=\"og:description\" content=\"Neste artigo voc\u00ea descobrir\u00e1 como calcular o m\u00e1ximo e o m\u00ednimo de uma fun\u00e7\u00e3o, explicamos resolvendo dois exemplos passo a passo. Al\u00e9m disso, voc\u00ea poder\u00e1 praticar exerc\u00edcios passo a passo sobre os m\u00e1ximos e m\u00ednimos de uma fun\u00e7\u00e3o. Quais s\u00e3o o m\u00e1ximo e o m\u00ednimo de uma fun\u00e7\u00e3o? Os m\u00e1ximos de uma fun\u00e7\u00e3o s\u00e3o os &hellip; M\u00e1ximo e m\u00ednimo de uma fun\u00e7\u00e3o (extremos relativos) Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/maximos-minimos-de-uma-funcao-extremos-relativos\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-09-17T10:56:20+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/maximums-et-minimums-d-une-fonction.webp\" \/>\n<meta name=\"author\" content=\"Equipe Mathoridade\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"Equipe Mathoridade\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. tempo de leitura\" \/>\n\t<meta name=\"twitter:data2\" content=\"9 minutos\" \/>\n<script type=\"application\/ld+json\" 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