{"id":48,"date":"2023-09-17T10:55:22","date_gmt":"2023-09-17T10:55:22","guid":{"rendered":"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/"},"modified":"2023-09-17T10:55:22","modified_gmt":"2023-09-17T10:55:22","slug":"punti-di-flesso-di-una-funzione","status":"publish","type":"post","link":"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/","title":{"rendered":"Punti di flesso di una funzione"},"content":{"rendered":"<p>Qui spieghiamo cos&#8217;\u00e8 un punto di flesso di una funzione e come trovare tutti i punti di flesso di una funzione. Inoltre, troverai esercizi passo passo sulla curvatura e sui punti di flesso di una funzione. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-son-los-puntos-de-inflexion-de-una-funcion\"><\/span> Quali sono i punti di flesso di una funzione?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>I punti di flesso di una funzione sono i punti in cui il grafico della funzione cambia curvatura, cio\u00e8 in un punto di flesso una funzione cambia da concava a convessa o viceversa.<\/strong> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-saber-si-una-funcion-tiene-un-punto-de-inflexion\"><\/span> Come sapere se una funzione ha un punto di flesso<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Data la definizione di punto di flesso, vediamo come sapere se un certo punto \u00e8 un punto di flesso della funzione. <\/p>\n<div style=\"background-color:#FFF3E0; padding-top: 23px; padding-bottom: 0.5px; padding-right: 35px; padding-left: 30px; border-radius:30px;\">\n<p> Una funzione ha un <strong>punto di flesso<\/strong> nei punti che annullano la sua derivata seconda e la sua derivata terza \u00e8 diversa da zero.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0deb5fc13e20049e642bdc68a5c35a8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left.\\begin{array}{l}f''(a)=0\\\\[2ex]f'''(a)\\neq 0\\end{array}\\right\\} \\quad \\bm{\\longrightarrow} \\quad x=a \\text{ es un punto de inflexi\\'on}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"405\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<\/div>\n<p> Ad esempio, calcoleremo i punti di flesso della seguente funzione di terzo grado:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-37bed97da1ec1d3c4eefd018c3d75650_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^3-5x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"117\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Per prima cosa calcoliamo la derivata seconda e terza della funzione: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-379ab2238c65191bfc16a940f5a7375d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=3x^2-5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"119\" 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-86d5e98dcde33659284eaafb55050852_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)=6x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" 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-d65ebc63b5e7fa667d5e25e40448942c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'''(x)=6\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"79\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Ora impostiamo la derivata seconda uguale a 0 e risolviamo l&#8217;equazione risultante:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7878c5d5f08c25729d37b94ea643bdf7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6x=0\" 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-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>\n<p> Quindi, il punto x=0 sar\u00e0 un punto di flesso della funzione se la derivata terza \u00e8 diversa da zero in questo punto. Nel nostro caso la derivata terza \u00e8 sempre uguale a 6.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4e11c7a68a7805dd5d27f43c4d332d11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'''(0)=6\\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Pertanto, x=0 \u00e8 un punto di flesso della funzione. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-estudiar-la-curvatura-y-hallar-los-puntos-de-inflexion-de-una-funcion\"><\/span> Come studiare la curvatura e trovare i punti di flesso di una funzione<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Abbiamo appena visto un metodo per trovare punti di svolta. Tuttavia, normalmente tendiamo a studiare la curvatura di una funzione, cio\u00e8 a determinare la concavit\u00e0 e la convessit\u00e0 di una funzione, e da l\u00ec a calcolare i punti di flesso.<\/p>\n<p> Per trovare i punti di flesso di una funzione attraverso la sua curvatura, \u00e8 necessario eseguire i seguenti passaggi: <\/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:#101010;font-weight: normal;\">Trova i <strong>punti che non appartengono al dominio<\/strong> della funzione.<\/span><\/li>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Calcolare la derivata prima e <strong>la derivata seconda della funzione.<\/strong><\/span><\/li>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Trova le <strong>radici della derivata seconda<\/strong> , cio\u00e8 calcola i punti che annullano la derivata seconda risolvendo\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><\/span> .<\/li>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Crea <strong>intervalli<\/strong> con le radici della derivata e i punti che non appartengono al dominio della funzione.<\/span><\/li>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Calcola il valore della derivata seconda in un punto di ciascun intervallo.<\/span><\/li>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\"><strong>Il segno della derivata seconda<\/strong> determina la concavit\u00e0 o convessit\u00e0 della funzione in questo intervallo:<\/span>\n<ul style=\"color:#64B5F6; font-weight: bold; margin-top:8px; margin-left:8%\">\n<li style=\"margin-bottom:8px\"> <span style=\"color:#101010;font-weight: normal;\">Se la derivata seconda della funzione \u00e8 positiva, la funzione \u00e8 <strong>convessa<\/strong> su questo intervallo.<\/span><\/li>\n<li style=\"margin-bottom:8px\"> <span style=\"color:#101010;font-weight: normal;\">Se la derivata seconda della funzione \u00e8 negativa, la funzione \u00e8 <strong>concava<\/strong> su questo intervallo.<\/span><\/li>\n<\/ul>\n<\/li>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\"><strong>I punti di flesso<\/strong> sono i punti in cui la funzione cambia da convessa a concava o viceversa.<\/span><\/li>\n<\/ol>\n<\/div>\n<p> Affinch\u00e9 tu possa vedere come vengono calcolati i punti di flesso di una funzione utilizzando questa procedura, risolveremo un esempio passo dopo passo di seguito:<\/p>\n<ul>\n<li> Studia la curvatura e trova i punti di flesso della seguente funzione polinomiale:<\/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-1c8c498c40649602a0573f686b30f46d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^4-6x^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"124\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> La prima cosa da fare \u00e8 calcolare il dominio di definizione della funzione. \u00c8 una funzione polinomiale, quindi il dominio della funzione \u00e8 costituito da numeri reali, cio\u00e8 \u00e8 una funzione continua:<\/p>\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> Una volta calcolato il dominio della funzione, dobbiamo studiare in quali punti si realizza<\/p>\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> .<\/p>\n<p> Calcoliamo quindi prima la derivata prima della funzione:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7602daac91808a70b1b980676921f6d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^4-6x^2 \\ \\longrightarrow \\ f'(x)= 4x^3-12x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"314\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Successivamente calcoliamo la derivata seconda della funzione:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-53a801c5309b20a5ad6b5b5981fde19e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=4x^3-12x \\ \\longrightarrow \\ f''(x)= 12x^2-12\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"330\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> E ora impostiamo la derivata seconda uguale a 0 e risolviamo l&#8217;equazione: <\/p>\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-2ce944968a13592d994b5d76185e2ae9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"12x^2-12=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"107\" 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-e5db8208512c91e36a1ece605a967be0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"12x^2=12\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"75\" 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-22930ba9e08e00b2388ec1a767443147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=\\cfrac{12}{12}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"61\" 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-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-fa2841908f61047e2edf3a8b60ab5962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\pm1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Una volta calcolato il dominio della funzione 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-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> , rappresentiamo tutti i punti critici che si trovano sulla retta numerica: <\/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=\"312\" height=\"79\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> E ora valutiamo il segno della derivata seconda in ogni intervallo, per sapere se la funzione \u00e8 concava o convessa. Prendiamo quindi un punto in ogni intervallo (mai i punti critici) e guardiamo che segno ha la derivata seconda in questo punto: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3268d2c9fa6d3192f84d914bb9163680_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)=12x^2-12\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"141\" 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-290ddf1ed5524d38fa79c593052cca0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(-2) = 12\\cdot (-2)^2-12 =36 \\  \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"287\" 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-dc8cd9fb09f76eeb69fdeda81a49c674_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(0) = 12\\cdot 0^2-12 = -12 \\  \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"259\" 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-f35d5dc33f0ffbdf9b02b856ff70e1fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(2) = 12\\cdot 2^2-12=36 \\  \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"245\" 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-1-1-positif-negatif-positif.webp\" alt=\"\" class=\"wp-image-2597\" width=\"315\" height=\"140\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Se la derivata seconda \u00e8 positiva significa che la funzione \u00e8 convessa.<\/p>\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 la derivata seconda \u00e8 negativa significa che la funzione \u00e8 concava<\/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> . Pertanto, gli intervalli di concavit\u00e0 e convessit\u00e0 della funzione sono:<\/p>\n<p class=\"has-text-align-center\"> <strong>Convesso<\/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-9f11bddd110e2869bf60761b066d16c5_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>Concavo<\/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> Inoltre, in x=-1 la funzione passa da convessa a concava, quindi <strong>x=-1 \u00e8 un punto di flesso<\/strong> della funzione <strong>.<\/strong> E in x=1, la funzione passa da concava a convessa, quindi <strong>x=1 \u00e8 anche un punto di flesso<\/strong> della funzione.<\/p>\n<p> Infine, sostituiamo i punti trovati nella funzione originale per trovare la coordinata Y dei punti di flesso:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-da3f60ecfa8eb1666821a1a26a103825_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(-1)=(-1)^4-6(-1)^2 = 1-6 \\cdot 1 = -5 \\ \\longrightarrow \\ (-1,-5)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"437\" 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-d8544d4624c7030bd065734e22efd792_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1)=1^4-6\\cdot 1^2 = 1-6 \\cdot 1 = -5 \\ \\longrightarrow \\ (1,-5)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"367\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> I punti di svolta della funzione sono quindi:<\/p>\n<p class=\"has-text-align-center\"> <strong>Punti di svolta:<\/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-2576ee14a4d19421150e7b696f83c31f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-1,-5)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>E<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2220ee8c6ddd44fd21975caeace894d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(1,-5)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Di seguito potete vedere la rappresentazione grafica della funzione studiata: <\/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\/points-dinflexion-dune-fonction.webp\" alt=\"punti di flesso di una funzione\" class=\"wp-image-2598\" width=\"457\" height=\"493\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Come puoi vedere dal grafico, la funzione va da convessa<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d2bb599fff4b55075f6de7628a35f822_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\cup)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> essere concavo<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-69b1c740acf744d34bf12f12dedabf65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\cap)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> Di<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1bf0660982e8c9a229e9bd39f6a71341_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1,-5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -5px;\"><\/p>\n<p> poich\u00e9 la sua curvatura cambia. E d&#8217;altra parte la funzione va da concava<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-69b1c740acf744d34bf12f12dedabf65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\cap)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> essere convesso<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d2bb599fff4b55075f6de7628a35f822_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\cup)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> Di<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-97e5c7dfce41ba68b73f6246d1554039_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(1,-5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<p> . <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-puntos-de-inflexion\"><\/span> Esercizi di svolta risolti<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Esercizio 1<\/h3>\n<p> Calcola gli intervalli di concavit\u00e0 e convessit\u00e0 nonch\u00e9 i punti di flesso della seguente funzione esponenziale: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-175df6eaf87d8a34e6c7f52ffcf4dde7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x) = xe^x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"84\" 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>Vedi la soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> La prima cosa da fare \u00e8 calcolare il dominio di definizione della funzione. La funzione \u00e8 composta da una funzione polinomiale (x), il cui dominio \u00e8 costituito solo da numeri reali, e da una funzione esponenziale (e <sup>x<\/sup> ), il cui dominio \u00e8 costituito anch&#8217;esso da numeri reali. Pertanto, il dominio della funzione \u00e8 costituito da numeri reali:<\/p>\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\"> Ora calcoliamo la derivata della funzione. In questo caso la funzione \u00e8 composta dal prodotto di due funzioni, quindi per derivare la funzione dobbiamo applicare la formula per la derivata di un prodotto: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-996eb43fbe77816d37d7bfa7f35e1e63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=1 \\cdot e^x+ x \\cdot e^x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"162\" 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-a3d95f44577d8448674370dd53e02726_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=e^x +xe^x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"127\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Successivamente calcoliamo la derivata seconda della funzione: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-956f25a80e8ccc50a8888c8adc1e134e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= e^x + 1 \\cdot e^x+ x \\cdot e^x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"204\" 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-551f7a75a4b0338b53e7decaf413e976_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)=e^x +e^x + xe^x  = 2e^x +xe^x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"267\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Impostiamo la derivata seconda uguale a 0 e risolviamo l&#8217;equazione: <\/p>\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-dc9977afe23305adab3b6c332e4232bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2e^x+xe^x= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"107\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Estraiamo il fattore comune:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bf04b8cea9a81435b3106d94a8bba193_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"e^x(2+x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"104\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Perch\u00e9 la moltiplicazione sia uguale a 0, uno dei due elementi della moltiplicazione deve essere zero. Pertanto, impostiamo ciascun fattore uguale 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-e4b369d45e5559de1f7069b49db2d173_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle e^x\\cdot(2+x) =0 \\longrightarrow \\begin{cases} e^x=0 \\ \\color{red}\\bm{\\times}\\color{black}  \\\\[2ex] 2+x=0 \\ \\longrightarrow \\ x= - 2 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"350\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Un numero elevato a un altro non pu\u00f2 mai dare come risultato 0. Pertanto, l&#8217;equazione<\/p>\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> Non c&#8217;\u00e8 soluzione.<\/p>\n<p class=\"has-text-align-left\"> Rappresentiamo tutti i punti singolari ottenuti a destra: <\/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-2-1.webp\" alt=\"\" class=\"wp-image-2602\" width=\"199\" height=\"78\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> E ora valutiamo il segno della derivata seconda in ciascun intervallo per sapere se la funzione \u00e8 concava o convessa. Per fare ci\u00f2, prendiamo un punto in ciascun intervallo e guardiamo quale segno ha la derivata seconda in quel punto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e5deb7fd9574dd41f2571836c131654d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(-3)= 2e^{-3} +(-3)\\cdot e^{-3} = 0,1 - 0,15 = -0,05\\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"442\" 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-fc38d25b16bf677962f89d5562265437_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(0)= 2e^0 +0\\cdot e^0 = 2 \\cdot 1 + 0 \\cdot 1 =2+0= 2 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"404\" 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-2-courbure.webp\" alt=\"\" class=\"wp-image-2603\" width=\"201\" height=\"135\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Se la derivata seconda \u00e8 positiva significa che la funzione \u00e8 convessa.<\/p>\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 la derivata seconda \u00e8 negativa significa che la funzione \u00e8 concava<\/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> . Gli intervalli di concavit\u00e0 e convessit\u00e0 sono quindi:<\/p>\n<p class=\"has-text-align-center\"> <strong>Convesso<\/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-087de0300a010f86265ccd4f69d8570e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-2,+\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"74\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Concavo<\/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-266c9537df2a55b66ad1c2868728fddf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty,-2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"74\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Inoltre, la funzione cambia da concava a convessa in x=-2, quindi <strong>x=-2 \u00e8 un punto di flesso<\/strong> della funzione.<\/p>\n<p class=\"has-text-align-left\"> Infine, sostituiamo il punto di flesso trovato nella funzione originale per trovare la coordinata Y del punto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3aee30bfafb6ab70bdc2bc672a61a780_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(-2) = (-2)\\cdot e^{-2} =-2e^{-2} \\ \\longrightarrow \\ (-2,-2e^{-2})\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"362\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> In conclusione gli unici punti di svolta della funzione sono:<\/p>\n<p class=\"has-text-align-center\"> <strong>Punti di svolta:<\/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-676ef444bb4f5ed3116dc104f451295d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-2,-2e^{-2})}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"92\" 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\">Esercizio 2<\/h3>\n<p> Studia gli intervalli di concavit\u00e0 e convessit\u00e0 e trova i punti di flesso della seguente funzione razionale: <\/p>\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>Vedi la soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Per prima cosa dobbiamo calcolare il dominio della funzione. Trattandosi di una funzione razionale, poniamo il denominatore uguale a zero per vedere quali numeri non appartengono al dominio della funzione: <\/p>\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-left\"> Ci\u00f2 significa che quando x \u00e8 -2 o +2, il denominatore sar\u00e0 0. E quindi la funzione non esister\u00e0. Il dominio della funzione \u00e8 quindi composto da tutti i numeri tranne x=-2 e x=+2.<\/p>\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\"> In secondo luogo, calcoliamo la derivata prima della funzione: <\/p>\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\"> E poi risolviamo la derivata seconda: <\/p>\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-left\"> Tutti i termini vengono moltiplicati per<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1e0f0d9a63183e28c50a5cedcddeddd3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x^2-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"60\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Possiamo quindi semplificare la frazione: <\/p>\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\"> Ora calcoliamo le radici della derivata seconda della funzione: <\/p>\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-left\"> Il termine<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d8590a90fab5aef55d7b45cd89e01943_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(x^2-4\\right)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"70\" style=\"vertical-align: -7px;\"><\/p>\n<p> Ci\u00f2 comporta la divisione dell&#8217;intero lato sinistro, quindi possiamo moltiplicarlo per l&#8217;intero lato destro: <\/p>\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-left\"> Estraiamo il fattore comune:<\/p>\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-left\"> Perch\u00e9 la moltiplicazione sia uguale a 0, uno dei due elementi della moltiplicazione deve essere zero. Pertanto, impostiamo ciascun fattore uguale 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-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> Non esiste una soluzione poich\u00e9 non esiste una radice negativa di un numero reale.<\/p>\n<p class=\"has-text-align-left\"> Rappresentiamo ora sulla retta tutti i punti critici ottenuti, cio\u00e8 i punti che non appartengono al dominio (x=-2 e x=+2) e quelli che annullano la derivata seconda (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=\"385\" height=\"75\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> E valutiamo il segno della derivata seconda in ogni intervallo, per sapere se la funzione \u00e8 concava o convessa. Quindi prendiamo un punto in ogni intervallo e guardiamo quale segno ha la derivata seconda in quel punto: <\/p>\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-3814746f7f9e8aa3920e3f84cd0ff0eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(1)=\\cfrac{8\\cdot1^3+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=\"383\" height=\"129\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Se la derivata seconda \u00e8 positiva significa che la funzione \u00e8 convessa.<\/p>\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 la derivata seconda \u00e8 negativa significa che la funzione \u00e8 concava<\/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> . Gli intervalli di concavit\u00e0 e convessit\u00e0 sono quindi:<\/p>\n<p class=\"has-text-align-center\"> <strong>Convesso<\/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>Concavo<\/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\"> La funzione cambia curvatura in tre punti, quindi la funzione razionale avrebbe in linea di principio tre punti di flesso, che sono x=-2, x=0 e x=2. Tuttavia, sebbene vi sia un cambiamento nella curvatura in x=-2 e in x=+2, questi non sono punti di flesso perch\u00e9 non appartengono al dominio della funzione. D&#8217;altra parte, in x=0 c&#8217;\u00e8 un cambiamento nella curvatura e questo appartiene alla funzione, quindi <strong>x=0 \u00e8 l&#8217;unico punto di flesso della funzione.<\/strong><\/p>\n<p class=\"has-text-align-left\"> Non resta che calcolare la coordinata Y del punto di flesso:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ba8c4d5b9201e0971f4a036ea6b0f887_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(0)=\\frac{0^3}{0^2-4} =\\frac{0}{-4}=0\\ \\longrightarrow \\ (0,0)\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"279\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> In breve, l&#8217;unico punto di flesso della funzione razionale \u00e8 l&#8217;origine delle coordinate:<\/p>\n<p class=\"has-text-align-center\"> <strong>Punti di svolta:<\/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<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Esercizio 3<\/h3>\n<p> Sappiamo che la funzione<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a047814c11fd2f3da04a21a9d489da58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^3+ax^2+bx+c\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"194\" style=\"vertical-align: -5px;\"><\/p>\n<p> passare per il punto<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1542b88e9f0c10166380ce26011f4d14_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(3,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<p> , ha un estremo relativo in<\/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> e una svolta decisiva<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3976ff8b81cbf5060581dc2ccd19c5c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<p> . Da queste informazioni, calcolare i valori dei parametri<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0357ced152d91599aefcf60b48861b74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a, b\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"25\" 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-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"c\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> . <\/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>Vedi la soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Lascia che la funzione abbia un punto di flesso in<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2a9b760ebc3dca7954a7b6656a30c8cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<p> significa che<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b9c643154cd44808d027e645761f5921_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(2)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"74\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Pertanto, calcoliamo la derivata seconda della funzione in<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2a9b760ebc3dca7954a7b6656a30c8cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= 2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<p> e lo impostiamo uguale 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-a452ebc56e45c70455952e82f24bc213_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x) = x^3+ax^2+bx+c \\ \\longrightarrow \\  f'(x)=3x^2+2ax+b\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"413\" 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-88269f2eaaf1326ed88cec634a839505_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=3x^2+2ax+b \\ \\longrightarrow \\ f''(x)= 6x+2a\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"344\" 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-428d0d2aa58a4f0bee3155e72060aee4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{l} f''(2)=6\\cdot 2+2a\\\\[2ex] f''(2)=0\\end{array} \\right\\} \\longrightarrow 6\\cdot 2+2a=0\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"301\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E risolviamo l&#8217;equazione ottenuta per trovare il valore del parametro 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-08eaf911dd0de580da1db41c3a8a25d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6\\cdot 2+2a=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"103\" 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-4127c25ec4efd2c7b05070600fca7040_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"12+2a=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"89\" 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-21de66a92427317edcb4a2475fcbb3f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2a=-12\" 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-313aff06eefaaf9af361841519334293_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a=\\cfrac{-12}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"75\" 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-81f76a9a366a0e75d702129c2a0cf565_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{a=-6}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> La funzione sar\u00e0 quindi:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f82d6e87e465fabd9889eec2ed94b7ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^3+ax^2+bx+c \\ \\xrightarrow{a \\ = \\ -6}\\ f(x)=x^3-6x^2+bx+c\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"467\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Inoltre, la funzione ha un estremo in<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5a0a069e4bb53cf46ff9611c104b13f0_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> , Che significa che<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ba772a3114b7ee64171db5cc258b34b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"70\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Pertanto, calcoliamo la derivata prima della funzione in<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5a0a069e4bb53cf46ff9611c104b13f0_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> e lo impostiamo uguale 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-347dc6fd1ded234104fd25f2e787c4b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^3-6x^2+bx+c \\ \\longrightarrow \\ f'(x)=3x^2-12x+b\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"412\" 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-561b9c2aa6ba34d90df560c5a97e3a92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{l} f'(1)=3\\cdot 1^2-12\\cdot 1+b\\\\[2ex] f'(1)=0\\end{array} \\right\\} \\longrightarrow 3\\cdot 1^2-12\\cdot 1+b=0\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"413\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E risolviamo l&#8217;equazione ottenuta per trovare il valore dell&#8217;incognita 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-3362d7ea7f2e10163031aaa7ed824c52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3\\cdot 1^2-12\\cdot 1+b=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"161\" 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-2e6fe2c6a93ee7e3cda9732cced98c54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3 \\cdot 1 -12 + b = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"132\" 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-a73022570e08c83b3c148066c88595fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3 -12 + b = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"110\" 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-4cb3fc18020c592a99e0d3db8a57955f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b=+12-3\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"93\" 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-a73ce8b725442e911ac1a46aceb20457_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{b=9}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"40\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> La funzione sar\u00e0 quindi:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9a7549d54df3579050c71e4e06eeb053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^3-6x^2+bx+c \\ \\xrightarrow{b \\ = \\ 9} \\ f(x)=x^3-6x^2+9x+c\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"455\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ci dicono invece che la funzione passa per il punto (3,1). Questo \u00e8 da dire,<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-27c6a831fde145aea6f8d30359e84f03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(3)=1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Pertanto, possiamo applicare questa condizione per trovare il valore del parametro c:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b496beb319ccab8292181ec1387ba9f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{l} f(3)=3^3-6\\cdot 3^2+9\\cdot3+c \\\\[2ex] f(3)=1 \\end{array} \\right\\} \\longrightarrow 3^3-6\\cdot 3^2+9\\cdot 3+c = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"467\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E risolviamo l&#8217;equazione ottenuta per trovare il valore di <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4586bc1b16791cf732fc00ee37db4357_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" 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-410da829676625edf3c06a986ce96acb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3^3-6\\cdot 3^2+9\\cdot 3+c = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"190\" 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-f860f41d277c58f8714e45bda45c1a27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"27-6\\cdot 9+27+c = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"170\" 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-74e9b29b4c5de8c37cad4b293f196fca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"27-54+27+c = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"157\" 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-41daca67d3a063fdd50d16d2a3e43bb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"c=1-27+54-27\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"159\" 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-5825f757c3f04143a8c5598e0468cb33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{c=1}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"39\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> La funzione sar\u00e0 quindi:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dd400bd7ca4f75d8ae12b7aa9e9680ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^3-6x^2+9x+c \\ \\xrightarrow{c \\ = \\ 1} \\ f(x)=x^3-6x^2+9x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"457\" 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>Qui spieghiamo cos&#8217;\u00e8 un punto di flesso di una funzione e come trovare tutti i punti di flesso di una funzione. Inoltre, troverai esercizi passo passo sulla curvatura e sui punti di flesso di una funzione. Quali sono i punti di flesso di una funzione? I punti di flesso di una funzione sono i punti &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/\"> <span class=\"screen-reader-text\">Punti di flesso di una funzione<\/span> Leggi altro &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":[17],"tags":[],"class_list":["post-48","post","type-post","status-publish","format-standard","hentry","category-rappresentazione-delle-funzioni"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Punti di flesso di una funzione -<\/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\/it\/punti-di-flesso-di-una-funzione\/\" \/>\n<meta property=\"og:locale\" content=\"it_IT\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Punti di flesso di una funzione -\" \/>\n<meta property=\"og:description\" content=\"Qui spieghiamo cos&#8217;\u00e8 un punto di flesso di una funzione e come trovare tutti i punti di flesso di una funzione. Inoltre, troverai esercizi passo passo sulla curvatura e sui punti di flesso di una funzione. Quali sono i punti di flesso di una funzione? I punti di flesso di una funzione sono i punti &hellip; Punti di flesso di una funzione Leggi altro &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-09-17T10:55:22+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0deb5fc13e20049e642bdc68a5c35a8c_l3.png\" \/>\n<meta name=\"author\" content=\"Squadra di Mathority\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Scritto da\" \/>\n\t<meta name=\"twitter:data1\" content=\"Squadra di Mathority\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tempo di lettura stimato\" \/>\n\t<meta name=\"twitter:data2\" content=\"8 minuti\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/\"},\"author\":{\"name\":\"Squadra di Mathority\",\"@id\":\"https:\/\/mathority.org\/it\/#\/schema\/person\/8d6f69ffbe48aea8b43675a9a3ddb9c8\"},\"headline\":\"Punti di flesso di una funzione\",\"datePublished\":\"2023-09-17T10:55:22+00:00\",\"dateModified\":\"2023-09-17T10:55:22+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/\"},\"wordCount\":1536,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/it\/#organization\"},\"articleSection\":[\"Rappresentazione delle funzioni\"],\"inLanguage\":\"it-IT\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/\",\"url\":\"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/\",\"name\":\"Punti di flesso di una funzione -\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/it\/#website\"},\"datePublished\":\"2023-09-17T10:55:22+00:00\",\"dateModified\":\"2023-09-17T10:55:22+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/#breadcrumb\"},\"inLanguage\":\"it-IT\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathority.org\/it\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Punti di flesso di una funzione\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathority.org\/it\/#website\",\"url\":\"https:\/\/mathority.org\/it\/\",\"name\":\"Mathority\",\"description\":\"Dove la curiosit\u00e0 incontra il calcolo!\",\"publisher\":{\"@id\":\"https:\/\/mathority.org\/it\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathority.org\/it\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"it-IT\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathority.org\/it\/#organization\",\"name\":\"Mathority\",\"url\":\"https:\/\/mathority.org\/it\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"it-IT\",\"@id\":\"https:\/\/mathority.org\/it\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/mathority.org\/it\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"contentUrl\":\"https:\/\/mathority.org\/it\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"width\":703,\"height\":151,\"caption\":\"Mathority\"},\"image\":{\"@id\":\"https:\/\/mathority.org\/it\/#\/schema\/logo\/image\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathority.org\/it\/#\/schema\/person\/8d6f69ffbe48aea8b43675a9a3ddb9c8\",\"name\":\"Squadra di Mathority\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"it-IT\",\"@id\":\"https:\/\/mathority.org\/it\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"caption\":\"Squadra di Mathority\"},\"sameAs\":[\"http:\/\/mathority.org\/it\"]}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Punti di flesso di una funzione -","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/","og_locale":"it_IT","og_type":"article","og_title":"Punti di flesso di una funzione -","og_description":"Qui spieghiamo cos&#8217;\u00e8 un punto di flesso di una funzione e come trovare tutti i punti di flesso di una funzione. Inoltre, troverai esercizi passo passo sulla curvatura e sui punti di flesso di una funzione. Quali sono i punti di flesso di una funzione? I punti di flesso di una funzione sono i punti &hellip; Punti di flesso di una funzione Leggi altro &raquo;","og_url":"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/","article_published_time":"2023-09-17T10:55:22+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0deb5fc13e20049e642bdc68a5c35a8c_l3.png"}],"author":"Squadra di Mathority","twitter_card":"summary_large_image","twitter_misc":{"Scritto da":"Squadra di Mathority","Tempo di lettura stimato":"8 minuti"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/"},"author":{"name":"Squadra di Mathority","@id":"https:\/\/mathority.org\/it\/#\/schema\/person\/8d6f69ffbe48aea8b43675a9a3ddb9c8"},"headline":"Punti di flesso di una funzione","datePublished":"2023-09-17T10:55:22+00:00","dateModified":"2023-09-17T10:55:22+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/"},"wordCount":1536,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/it\/#organization"},"articleSection":["Rappresentazione delle funzioni"],"inLanguage":"it-IT","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/","url":"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/","name":"Punti di flesso di una funzione -","isPartOf":{"@id":"https:\/\/mathority.org\/it\/#website"},"datePublished":"2023-09-17T10:55:22+00:00","dateModified":"2023-09-17T10:55:22+00:00","breadcrumb":{"@id":"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/#breadcrumb"},"inLanguage":"it-IT","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathority.org\/it\/punti-di-flesso-di-una-funzione\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathority.org\/it\/"},{"@type":"ListItem","position":2,"name":"Punti di flesso di una funzione"}]},{"@type":"WebSite","@id":"https:\/\/mathority.org\/it\/#website","url":"https:\/\/mathority.org\/it\/","name":"Mathority","description":"Dove la curiosit\u00e0 incontra il calcolo!","publisher":{"@id":"https:\/\/mathority.org\/it\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathority.org\/it\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"it-IT"},{"@type":"Organization","@id":"https:\/\/mathority.org\/it\/#organization","name":"Mathority","url":"https:\/\/mathority.org\/it\/","logo":{"@type":"ImageObject","inLanguage":"it-IT","@id":"https:\/\/mathority.org\/it\/#\/schema\/logo\/image\/","url":"https:\/\/mathority.org\/it\/wp-content\/uploads\/2023\/10\/mathority-logo.png","contentUrl":"https:\/\/mathority.org\/it\/wp-content\/uploads\/2023\/10\/mathority-logo.png","width":703,"height":151,"caption":"Mathority"},"image":{"@id":"https:\/\/mathority.org\/it\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/mathority.org\/it\/#\/schema\/person\/8d6f69ffbe48aea8b43675a9a3ddb9c8","name":"Squadra di Mathority","image":{"@type":"ImageObject","inLanguage":"it-IT","@id":"https:\/\/mathority.org\/it\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","caption":"Squadra di Mathority"},"sameAs":["http:\/\/mathority.org\/it"]}]}},"yoast_meta":{"yoast_wpseo_title":"","yoast_wpseo_metadesc":"","yoast_wpseo_canonical":""},"_links":{"self":[{"href":"https:\/\/mathority.org\/it\/wp-json\/wp\/v2\/posts\/48","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathority.org\/it\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathority.org\/it\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathority.org\/it\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathority.org\/it\/wp-json\/wp\/v2\/comments?post=48"}],"version-history":[{"count":0,"href":"https:\/\/mathority.org\/it\/wp-json\/wp\/v2\/posts\/48\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathority.org\/it\/wp-json\/wp\/v2\/media?parent=48"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathority.org\/it\/wp-json\/wp\/v2\/categories?post=48"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathority.org\/it\/wp-json\/wp\/v2\/tags?post=48"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}