{"id":322,"date":"2023-07-06T08:31:48","date_gmt":"2023-07-06T08:31:48","guid":{"rendered":"https:\/\/mathority.org\/pt\/matriz-hessiana-hessiana-2x2-3x3\/"},"modified":"2023-07-06T08:31:48","modified_gmt":"2023-07-06T08:31:48","slug":"matriz-hessiana-hessiana-2x2-3x3","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/matriz-hessiana-hessiana-2x2-3x3\/","title":{"rendered":"Matriz hessiana (ou hesse)"},"content":{"rendered":"<p>Esta p\u00e1gina \u00e9 certamente a explica\u00e7\u00e3o mais completa da matriz Hessiana que existe. Aqui \u00e9 explicado o conceito de matriz Hessiana, como calcul\u00e1-la com exemplos e h\u00e1 ainda v\u00e1rios exerc\u00edcios resolvidos para praticar. Al\u00e9m disso, voc\u00ea poder\u00e1 aprender como s\u00e3o calculados os valores m\u00e1ximo e m\u00ednimo de uma fun\u00e7\u00e3o multivari\u00e1vel, bem como se \u00e9 uma fun\u00e7\u00e3o c\u00f4ncava ou convexa. E, finalmente, voc\u00ea tamb\u00e9m encontrar\u00e1 os utilit\u00e1rios e aplicativos da matriz Hessiana.<\/p>\n<h2 class=\"wp-block-heading\"> Qual \u00e9 a matriz Hessiana?<\/h2>\n<p> A defini\u00e7\u00e3o da matriz Hessiana (ou Hessiana) \u00e9 a seguinte:<\/p>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> A <strong>matriz Hessiana<\/strong> \u00e9 uma matriz quadrada de dimens\u00e3o n \u00d7 n composta pelas segundas derivadas parciais de uma fun\u00e7\u00e3o de n vari\u00e1veis.<\/p>\n<p> Essa matriz tamb\u00e9m \u00e9 conhecida como Hessiana, ou mesmo em alguns livros de matem\u00e1tica \u00e9 chamada de Discriminante. Mas a forma mais comum de cham\u00e1-la \u00e9 a matriz Hessiana.<\/p>\n<p> A f\u00f3rmula para a matriz Hessiana \u00e9, portanto, a seguinte: <\/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\/formule-matricielle-hessienne.webp\" alt=\"F\u00f3rmula de matriz hessiana ou hessiana\" class=\"wp-image-2445\" width=\"541\" height=\"345\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Portanto, a matriz Hessiana ser\u00e1 sempre uma matriz quadrada cuja dimens\u00e3o ser\u00e1 igual ao n\u00famero de vari\u00e1veis da fun\u00e7\u00e3o. Por exemplo, se a fun\u00e7\u00e3o tiver 3 vari\u00e1veis, a matriz Hessiana ter\u00e1 dimens\u00e3o 3\u00d73.<\/p>\n<p> Al\u00e9m disso, <strong>o teorema de Schwarz<\/strong> (ou teorema de Clairaut) diz que a ordem de diferencia\u00e7\u00e3o n\u00e3o importa, ou seja, deriva parcialmente primeiro em rela\u00e7\u00e3o \u00e0 vari\u00e1vel<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-01a7b7b5dca66cb33a1207e1f39c1140_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_1\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\"><\/p>\n<p> ent\u00e3o em rela\u00e7\u00e3o \u00e0 vari\u00e1vel<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f1cd6be340b4fce14489cf5b565a169e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_2\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<p> equivale a diferenciar parcialmente em rela\u00e7\u00e3o a<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f1cd6be340b4fce14489cf5b565a169e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_2\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<p> ent\u00e3o respeite<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-01a7b7b5dca66cb33a1207e1f39c1140_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x_1\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\"><\/p>\n<p> .<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8e5cc28564c40c6588680df48d8255ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x_i\\partial x_j} = \\cfrac{\\partial^2 f}{\\partial x_j\\partial x_i}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"133\" style=\"vertical-align: -18px;\"><\/p>\n<\/p>\n<p> Portanto, a matriz Hessiana \u00e9 uma matriz <strong>sim\u00e9trica<\/strong> , ou seja, possui uma simetria cujo eixo \u00e9 a sua diagonal principal.<\/p>\n<p> A t\u00edtulo de curiosidade, a matriz Hessiana leva o nome de Ludwig Otto Hesse, um matem\u00e1tico alem\u00e3o do s\u00e9culo XIX que fez contribui\u00e7\u00f5es muito importantes ao campo da \u00e1lgebra linear.<\/p>\n<h2 class=\"wp-block-heading\"> Exemplo de c\u00e1lculo da matriz Hessiana<\/h2>\n<p> Vejamos um exemplo de como encontrar uma matriz Hessiana de dimens\u00e3o 2 \u00d7 2:<\/p>\n<ul>\n<li> Calcule a matriz Hessiana no ponto (1,0) 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-03bcc33e393ddc95defdc3cc04da35c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  f(x,y)=y^4+x^3+3x^2+ 4y^2 -4xy -5y +8\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"348\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Primeiro, precisamos calcular as derivadas parciais de primeira ordem:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fd57722dfa592f129cac21ba8f183e05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial x} = 3x^2 +6x -4y\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"152\" 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-1847d0b9ecd032b396d64ac27e5e3eeb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial y} = 4y^3+8y -4x -5\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"181\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Uma vez que j\u00e1 conhecemos as primeiras derivadas, calculamos todas as derivadas parciais de segunda ordem:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-294ce075a54a240e012d30770fa10e3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x^2} = 6x +6\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"102\" 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-895c605e3c401f96a0a9b07ea105522c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial y^2} =12y^2 +8\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"118\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2ea4b2979928198e9fb91592d9e38874_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x \\partial y} = \\cfrac{\\partial^2 f}{\\partial y \\partial x}= -4\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"153\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p>Portanto, podemos agora encontrar a matriz Hessiana a partir da f\u00f3rmula para matrizes 2 \u00d7 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-926f350fe0ac3184ec0b563b57fd6041_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f (x,y)=\\begin{pmatrix}\\cfrac{\\partial^2 f}{\\partial x^2} &amp; \\cfrac{\\partial^2 f}{\\partial x \\partial y} \\\\[4ex] \\cfrac{\\partial^2 f}{\\partial y \\partial x} &amp; \\cfrac{\\partial^2 f}{\\partial y^2} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"111\" width=\"215\" 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-b7f3d45918645a5b6019896ed45eda75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f (x,y)=\\begin{pmatrix}6x +6 &amp;-4 \\\\[2ex] -4 &amp; 12y^2+8 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"246\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Assim a matriz Hessiana avaliada no ponto (1,0) ser\u00e1: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bdccfc61f7befe6c75f66c8a4658f3e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f (1,0)=\\begin{pmatrix}6(1) +6 &amp;-4 \\\\[2ex] -4 &amp; 12(0)^2+8 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"270\" style=\"vertical-align: 0px;\"><\/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\/exemple-matrice-hessienne-de-dimension-22152.webp\" alt=\"exemplos de matrizes de serapilheira ou juta\" class=\"wp-image-2487\" width=\"230\" height=\"80\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"> Problemas resolvidos de matrizes Hessianas<\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Calcule a matriz Hessiana da seguinte fun\u00e7\u00e3o com 2 vari\u00e1veis no ponto (1,1): <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d1e63eb1506d471fb8fdbc0e6db8de0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  f(x,y)=x^2y+y^2x\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"151\" 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__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro, precisamos encontrar as derivadas parciais de primeira ordem 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-9786ccaea4ad9305eb8b50d98f7f2626_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial x} = 2xy+y^2\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"111\" 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-7b68a8a666732d228ad06ebd8bade063_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial y} = x^2+2yx\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"112\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Depois de j\u00e1 termos calculado as primeiras derivadas, procedemos \u00e0 resolu\u00e7\u00e3o de todas as derivadas parciais de segunda ordem: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef3e5008f32d204cae79f8faceafb6ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x^2} = 2y\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"70\" 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-0e3f3f68ddab03ad22776b84f441cb7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial y^2} =2x\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"71\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e6383bbe85e851dd1e573981b52a0158_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x \\partial y} = \\cfrac{\\partial^2 f}{\\partial y \\partial x}=2x+2y\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"188\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Assim a matriz Hessiana \u00e9 definida da seguinte forma: <\/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\/exercice-resolu-de-matrice-hessian-22152-1.webp\" alt=\"Exerc\u00edcio resolvido da matriz Hessiana ou Hessiana de dimens\u00e3o 2x2\" class=\"wp-image-2492\" width=\"290\" height=\"81\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Finalmente, resta avaliar a matriz Hessiana no ponto (1,1): <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e5353c0229942269e07455047284f92b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f (1,1)=\\begin{pmatrix}2\\cdot 1 &amp;2 \\cdot 1+2\\cdot 1 \\\\[1.5ex] 2\\cdot 1+2\\cdot 1 &amp; 2\\cdot 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"292\" 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-cf00fccdb37a19388e76b5a84a408d02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{H_f (1,1)}=\\begin{pmatrix}\\bm{2} &amp; \\bm{4} \\\\[1.1ex] \\bm{4} &amp; \\bm{2} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"145\" style=\"vertical-align: 0px;\"><\/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 o Hessian no ponto (1,1) da seguinte fun\u00e7\u00e3o em duas vari\u00e1veis: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d577030e612ef7b6ab7b59aea4469539_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  f(x,y)= e^{y\\ln x}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"115\" 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__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro, precisamos calcular as derivadas parciais de primeira ordem 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-1025a749344fdf59c8c016387b7e2c37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial x} = e^{y\\ln x} \\cdot \\cfrac{y}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"110\" 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-2ddfad6313a350d43404b19a2278d771_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial y} = e^{y\\ln x} \\cdot \\ln x\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"125\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Assim que tivermos as primeiras derivadas, calculamos as derivadas parciais de segunda ordem 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-edcdf882fac053dc938c4f0d10060d30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x^2} = e^{y\\ln x} \\cdot \\cfrac{y^2}{x^2} - e^{y\\ln x} \\cdot \\cfrac{y}{x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"219\" 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-1f59a0f7a852311293edd2231235ae7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial y^2} =e^{y\\ln x} \\cdot \\ln ^2 x\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"141\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b760e29d744c2f294cf9cec33d61d1e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x \\partial y}=\\cfrac{\\partial^2 f}{\\partial y \\partial x} =e^{y\\ln x} \\cdot \\cfrac{y}{x}\\cdot \\ln x + e^{y\\ln x}\\cdot \\cfrac{1}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"323\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Assim, a matriz Hessiana da fun\u00e7\u00e3o \u00e9 uma matriz quadrada de dimens\u00e3o 2\u00d72: <\/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\/exercice-resolu-de-hesse-de-dimension-22152-1.webp\" alt=\"Exerc\u00edcio Hessiano resolvido ou matriz Hessiana de dimens\u00e3o 2x2\" class=\"wp-image-2516\" width=\"620\" height=\"135\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Finalmente, resta avaliar a matriz Hessiana no ponto (1,1): <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c316cc61e6d007e5d034274e0f494520_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f (1,1)=\\begin{pmatrix} e^{1\\ln (1)} \\displaystyle \\cdot \\cfrac{1^2}{1^2} - e^{1\\ln (1)} \\cdot \\cfrac{1}{1^2}&amp; e^{1\\ln (1)} \\cdot \\cfrac{1}{1}\\cdot \\ln (1) + e^{1\\ln (1)}\\cdot \\cfrac{1}{1} \\\\[3ex] e^{1\\ln (1)} \\cdot \\cfrac{1}{1}\\cdot \\ln (1) + e^{1\\ln (1)}\\cdot \\cfrac{1}{1} &amp; e^{1\\ln (1)} \\cdot \\ln ^2 (1) \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"99\" width=\"557\" 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-6e456b856c722a140d73ade63f13ec9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f (1,1)=\\begin{pmatrix}e^{0} \\cdot 1 - e^{0} \\cdot 1&amp; e^{0} \\cdot 1\\cdot 0 + e^{0}\\cdot 1 \\\\[2ex] e^{0} \\cdot 1\\cdot 0 + e^{0}\\cdot 1 &amp; e^{0} \\cdot 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"366\" 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-929fbf6e7f0f90110d11d4ccd51fd51a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f (1,1)=\\begin{pmatrix}1 - 1&amp; 0+ 1 \\\\[1.5ex] 0 +1 &amp; 1 \\cdot 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"206\" 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-ce780ddb8c09515afccfb2da2d842584_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{H_f (1,1)}=\\begin{pmatrix}\\bm{0} &amp; \\bm{1} \\\\[1.1ex] \\bm{1} &amp; \\bm{0} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"145\" style=\"vertical-align: 0px;\"><\/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> Encontre a matriz Hessiana no ponto<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b324eec3855aa704cfe7cef3a72713f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,1,\\pi)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"><\/p>\n<p> da seguinte fun\u00e7\u00e3o com 3 vari\u00e1veis: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-07614221c6aa91fe6591ecfdd7ee064b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  f(x,y,z)= e^{-x}\\cdot \\text{sen}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"189\" 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__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro, calculamos as derivadas parciais de primeira ordem 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-f3403bbb8643fb0c3f01dc93031683d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial x} = -e^{-x}\\cdot \\text{sen}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"155\" 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-6e77e08ad03b7a454f9a9f5c320d4902_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial y} = ze^{-x}\\cdot \\text{cos}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"149\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-912b948c3339efba02beb10da8853e89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial z} = ye^{-x}\\cdot \\text{cos}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"149\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Assim que tivermos as primeiras derivadas, calculamos as derivadas parciais de segunda ordem 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-831ffaaf0208ac80132c58e97c10696d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x^2} =e^{-x}\\cdot \\text{sen}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"149\" 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-f8c77e5f03861145b5370a208642c199_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x \\partial y}=\\cfrac{\\partial^2 f}{\\partial y \\partial x} =-ze^{-x}\\cdot \\text{cos}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"248\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f3a5180d638ffbedb5028da56c5c2b9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x \\partial z}=\\cfrac{\\partial^2 f}{\\partial z \\partial x} =-ye^{-x}\\cdot \\text{cos}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"247\" 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-001a3978b472e9085c4658d19e976079_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial y^2} =-z^2e^{-x}\\cdot \\text{sen}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"179\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d8c2c222f469c688e41ed773c1721834_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial y \\partial z}=\\cfrac{\\partial^2 f}{\\partial z \\partial y} =e^{-x}\\cdot \\text{cos}(yz)-yze^{-x}\\cdot \\text{sen}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"360\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-98dcbe5198ef8336355c89a80bb981ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial^2 z} = -y^2e^{-x}\\cdot \\text{sen}(yz)\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"179\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Para que a matriz Hessiana da fun\u00e7\u00e3o seja uma matriz quadrada de dimens\u00e3o 3\u00d73: <\/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\/exemple-de-matrice-de-hesse-en-3-dimensions-32153-1.webp\" alt=\"exemplo de serapilheira ou matriz de serapilheira de dimens\u00e3o 3x3\" class=\"wp-image-2537\" width=\"857\" height=\"109\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Por fim, substitu\u00edmos as vari\u00e1veis pelos seus respectivos valores no ponto <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cf0e780dfde4cefb749a03fe14266290_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,1,\\pi):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" 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-4e198192f67babd81228caa53b66e8a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f(0,1,\\pi)=\\begin{pmatrix}e^{-0}\\cdot \\text{sen}(1\\pi) &amp; -\\pi e^{-0}\\cdot \\text{cos}(1\\pi) &amp;-1e^{-0}\\cdot \\text{cos}(1\\pi) \\\\[1.5ex] -\\pi e^{-0}\\cdot \\text{cos}(1 \\pi)&amp;-\\pi^2e^{-0}\\cdot \\text{sen}(1 \\pi) &amp;e^{-0}\\cdot \\text{cos}(1 \\pi)-1 \\pi e^{-0}\\cdot \\text{sen}(1 \\pi) \\\\[1.5ex] -1e^{-0}\\cdot \\text{cos}(1 \\pi)&amp; e^{-0}\\cdot \\text{cos}(1 \\pi)-1 \\pi e^{-0}\\cdot \\text{sen}(1 \\pi)&amp; -1^2e^{-0}\\cdot \\text{sen}(1 \\pi) \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"87\" width=\"756\" 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-4ce9c6b4cfcddfb0c2eb51db1189c653_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f(0,1,\\pi)=\\begin{pmatrix}1\\cdot 0 &amp; -\\pi \\cdot 1 \\cdot (-1)&amp;-1\\cdot 1 \\cdot (-1) \\\\[1.5ex] -\\pi \\cdot 1 \\cdot (-1) &amp;-\\pi^2\\cdot 1\\cdot 0 &amp;1 \\cdot (-1)-\\pi \\cdot 1\\cdot 0 \\\\[1.5ex] -1\\cdot 1 \\cdot (-1) &amp; 1\\cdot (-1) - \\pi \\cdot 1\\cdot 0 &amp; -1\\cdot 1 \\cdot 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"527\" style=\"vertical-align: 0px;\"><\/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\/exercice-resolu-de-matrice-hessiana-32153-1.webp\" alt=\"exerc\u00edcio resolvido passo a passo da matriz Hessiana ou Hessiana de dimens\u00e3o 3x3\" class=\"wp-image-2536\" width=\"291\" height=\"109\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\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> Determine a matriz Hessiana no ponto (2,-1,1,-1) da seguinte fun\u00e7\u00e3o com 4 vari\u00e1veis: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4cf4dde4cb282d4f494142475a514b6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  f(x,y,z,w)= 2x^3y^4zw^2 - 2y^3w^4+ 3x^2z^2\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"320\" 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__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> O primeiro passo \u00e9 encontrar as derivadas parciais de primeira ordem 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-d559c24695adc4e5bf25a07162b0b82c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial x} =6x^2y^4zw^2 + 6xz^2\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"175\" 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-7e4b630227d3c64561c2a87f729612b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial y} =8x^3y^3zw^2 - 6y^2w^4\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"186\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c473febcf510cf11a4b30fad46fb3d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial z} = 2x^3y^4w^2 + 6x^2z\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"166\" 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-afc9ac6f800c06919a5bee6d6ea20a38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial f}{\\partial w} =4x^3y^4zw - 8y^3w^3\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"181\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora resolvemos as derivadas parciais de segunda ordem 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-55333bf53a5934f849b6eea4d5a4f64c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x^2} =12xy^4zw^2 + 6z^2\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"174\" 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-ecd6fb72d35cb8c8b09e14dd2337bdfd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x \\partial y}=\\cfrac{\\partial^2 f}{\\partial y \\partial x}=24x^2y^3zw^2\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"211\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-02885e698f3ae93bbf9515970de030ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x \\partial z}=\\cfrac{\\partial^2 f}{\\partial z \\partial x}=6x^2y^4w^2 + 12xz\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"252\" 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-0a43e7868272e5f9981fd6961c4d2078_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial x \\partial w} = \\cfrac{\\partial^2 f}{\\partial w \\partial x}=12x^2y^4zw\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"212\" 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-fd1617aa3759b2cab9621b821d06f42d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial y^2} =24x^3y^2zw^2 - 12yw^4\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"204\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c019a5c8dd396e6d0ee491d1e96b0f42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial y \\partial z}=\\cfrac{\\partial^2 f}{\\partial y \\partial z}=8x^3y^3w^2\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"191\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-83dc31061c77785623e54e55f39c7bea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial y \\partial w} = \\cfrac{\\partial^2 f}{\\partial w \\partial y}=16x^3y^3zw - 24y^2w^3\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"287\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-57b2a8d35948c0ac623709a00922f2a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial^2 z} =6x^2\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"78\" 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-260eacc428eb6db6a9a5ad0b7c7e9649_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial z \\partial w} = \\cfrac{\\partial^2 f}{\\partial w \\partial z}=4x^3y^4w\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"192\" 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-7d4cb2a8938dfd2d17cc68cd9260eae5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial^2 f}{\\partial^2 w} =4x^3y^4z - 24y^3w^2\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"184\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Assim, a express\u00e3o da matriz Hessiana 4\u00d74 obtida resolvendo todas as derivadas parciais \u00e9 a seguinte: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exemple-de-toile-de-jute-ou-matrice-de-toile-de-jute-de-dimension-44.webp\" alt=\"exemplo resolvido passo a passo de serapilheira ou matriz de serapilheira de dimens\u00e3o 4x4\" width=\"854\" height=\"180\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Por fim, substitu\u00edmos as inc\u00f3gnitas pelos seus respectivos valores de pontos (2,-1,1,-1) e fazemos os c\u00e1lculos: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-de-hesse-ou-matrice-hessienne-de-dimension-44.webp\" alt=\"Exerc\u00edcio resolvido passo a passo da matriz Hessiana ou Hessiana de dimens\u00e3o 4x4\" width=\"457\" height=\"182\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<h2 class=\"wp-block-heading\"> Como saber quando a matriz Hessiana \u00e9 positiva, negativa ou indefinida?<\/h2>\n<p> Como veremos mais tarde, saber se a matriz Hessiana \u00e9 uma matriz positiva semidefinida, positiva definida, negativa semidefinida, negativa definida ou indefinida \u00e9 muito \u00fatil. Ent\u00e3o, vamos ver como podemos descobrir:<\/p>\n<h3 class=\"wp-block-heading\"> Crit\u00e9rio de autovalores (ou autovalores)<\/h3>\n<p> Uma forma de saber que tipo de matriz \u00e9 \u00e9 percorrer os autovalores (ou autovalores) da matriz Hessiana:<\/p>\n<ul>\n<li> A matriz Hessiana \u00e9 <span style=\"color:#1976d2;\"><strong>positiva semidefinida<\/strong><\/span> se tiver autovalores (ou autovalores) iguais e maiores que zero. Ou seja, possui autovalores positivos e pelo menos um igual a 0:<\/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-21d0afe89f5d1545ccda3d2bd0d8660a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda \\geq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"43\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<ul>\n<li> A matriz Hessiana \u00e9 <span style=\"color:#1976d2;\"><strong>positiva definida<\/strong><\/span> se todos os seus autovalores (ou autovalores) forem exclusivamente maiores que 0 (positivo):<\/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-f203da6c7fff0b1487c2084e3d90966b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda > 0&#8243; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;14&#8243; width=&#8221;43&#8243; style=&#8221;vertical-align: -2px;&#8221;><\/p>\n<\/p>\n<ul>\n<li> A matriz Hessiana \u00e9 <span style=\"color:#1976d2;\"><strong>negativa semidefinida<\/strong><\/span> se tiver autovalores (ou autovalores) iguais e menores que zero. Ou seja, possui autovalores negativos e pelo menos um igual a 0:<\/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-dfdf139bb553d680b38761b565dd3db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda \\leq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"43\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<ul>\n<li> A matriz Hessiana \u00e9 <span style=\"color:#1976d2;\"><strong>definida negativa<\/strong><\/span> se todos os seus autovalores (ou autovalores) forem menores que 0 (negativos):<\/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-1fdc8fd3de7c5d7643ca8e4dbfe3704d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda < 0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"43\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<ul>\n<li> A matriz Hessiana \u00e9 <span style=\"color:#1976d2;\"><strong>indefinida<\/strong><\/span> quando possui autovalores positivos e negativos (ou autovalores):<\/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-fa9eea2abe5f790e65f0c5afab8c3adb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda > 0 \\qquad \\lambda <0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"121\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"> Crit\u00e9rio de Sylvester<\/h3>\n<p> Outra forma de saber de que tipo \u00e9 a matriz Hessiana \u00e9 utilizar o crit\u00e9rio de Sylvester, embora este teorema apenas nos permita saber se \u00e9 definida positiva, definida negativa ou indefinida. Mas \u00e0s vezes pode ser muito mais r\u00e1pido de usar porque os c\u00e1lculos geralmente s\u00e3o mais f\u00e1ceis. <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-118\"><\/div>\n<\/div>\n<p> Assim, o <strong>crit\u00e9rio de Sylvester<\/strong> \u00e9 o seguinte:<\/p>\n<ul>\n<li> Se todos os menores principais da matriz Hessiana forem maiores que 0, \u00e9 uma matriz <span style=\"color:#1976d2;\"><strong>definida positiva<\/strong><\/span> .<\/li>\n<\/ul>\n<ul>\n<li> Se os principais menores da matriz Hessiana com \u00edndice par forem maiores que 0 e aqueles com \u00edndice \u00edmpar forem menores que 0, \u00e9 uma matriz <span style=\"color:#1976d2;\"><strong>definida negativa<\/strong><\/span> .<\/li>\n<\/ul>\n<ul>\n<li> Se todos os menores principais da matriz Hessiana forem diferentes de 0 e nenhuma das duas condi\u00e7\u00f5es anteriores for atendida, \u00e9 uma matriz <span style=\"color:#1976d2;\"><strong>indefinida<\/strong><\/span> . <\/li>\n<\/ul>\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\/majeurs-mineurs-d-une-matrice.webp\" alt=\"principais menores de uma matriz hessiana\" class=\"wp-image-2583\" width=\"417\" height=\"361\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Obviamente, o m\u00e1ximo menor principal da matriz Hessiana coincidir\u00e1 sempre com o seu determinante. Apenas para fins informativos, o determinante da matriz Hessiana tamb\u00e9m \u00e9 chamado de \u201co Hessiano\u201d, embora n\u00e3o o fa\u00e7amos aqui para evitar confus\u00e3o.<\/p>\n<h2 class=\"wp-block-heading\"> Como calcular o m\u00e1ximo ou m\u00ednimo de uma fun\u00e7\u00e3o com a matriz Hessiana<\/h2>\n<p> Depois de saber como calcular a matriz Hessiana, voc\u00ea provavelmente est\u00e1 se perguntando: e para que serve essa matriz?<\/p>\n<p> Pois bem, uma das aplica\u00e7\u00f5es da matriz Hessiana \u00e9 encontrar o m\u00e1ximo ou m\u00ednimo de uma fun\u00e7\u00e3o com mais de uma vari\u00e1vel. Aqui est\u00e1 uma explica\u00e7\u00e3o passo a passo de como calcular m\u00e1ximos e m\u00ednimos:<\/p>\n<ol style=\"color:#1976d2; font-weight: bold;>\n<li><span style=\" color:#262626;font-weight:=\"\" normal;\"=\"\">\n<li style=\"margin-bottom:18px\"><span style=\"color:#262626;font-weight: normal;\">Primeiro, s\u00e3o calculados os <strong>pontos cr\u00edticos<\/strong> da fun\u00e7\u00e3o multivari\u00e1vel. Para isso, calculamos o gradiente ou a <a href=\"https:\/\/mathority.org\/pt\/matriz-jacobiana-jacobiana\/\">matriz Jacobiana<\/a> da fun\u00e7\u00e3o, igualamos-o a 0 e resolvemos as equa\u00e7\u00f5es.<\/span><\/li>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#262626;font-weight: normal;\">A matriz Hessiana \u00e9 calculada.<\/span><\/li>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#262626;font-weight: normal;\">Os pontos cr\u00edticos encontrados na etapa 1 s\u00e3o substitu\u00eddos na matriz Hessiana. Obteremos assim tantas matrizes Hessianas quantos pontos cr\u00edticos a fun\u00e7\u00e3o tiver.<\/span><\/li>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#262626;font-weight: normal;\">Vemos que tipo de matriz \u00e9 cada matriz Hessiana. Ou seja, procuramos ver se \u00e9 definido positivo, definido negativo, indefinido, etc.<\/span>\n<ul>\n<li style=\"margin-left:30px; margin-bottom:10px; margin-top:10px\"> <span style=\"color:#262626;font-weight: normal;\">Se a matriz Hessiana for positiva definida, o ponto cr\u00edtico \u00e9 um <span style=\"color:#1976d2;\"><strong>m\u00ednimo relativo<\/strong><\/span> da fun\u00e7\u00e3o.<\/span><\/li>\n<li style=\"margin-left:30px; margin-bottom:10px\"> <span style=\"color:#262626;font-weight: normal;\">Se a matriz Hessiana for definida negativa, o ponto cr\u00edtico \u00e9 um <span style=\"color:#1976d2;\"><strong>m\u00e1ximo relativo<\/strong><\/span> da fun\u00e7\u00e3o.<\/span><\/li>\n<li style=\"margin-left:30px;\"> <span style=\"color:#262626;font-weight: normal;\">Se a matriz Hessiana for indefinida, o ponto cr\u00edtico \u00e9 um <span style=\"color:#1976d2;\"><strong>ponto de sela<\/strong><\/span> .<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<h3 class=\"wp-block-heading\"> Exemplo de c\u00e1lculo de m\u00e1ximos e m\u00ednimos de uma fun\u00e7\u00e3o multivari\u00e1vel<\/h3>\n<p> Para ver como isso \u00e9 feito, aqui est\u00e1 um exemplo de c\u00e1lculo e classifica\u00e7\u00e3o dos extremos relativos de uma fun\u00e7\u00e3o usando a matriz Hessiana:<\/p>\n<ul>\n<li> Encontre todos os extremos relativos da seguinte fun\u00e7\u00e3o multivari\u00e1vel:<\/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-f422cf8ac57ec69b69af091986d534da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  f(x,y)=x^2-y^2+2xy+ 4x-4y\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"262\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A primeira coisa que precisamos fazer \u00e9 calcular a matriz Jacobiana da fun\u00e7\u00e3o, que neste caso coincidir\u00e1 com o gradiente por ser uma fun\u00e7\u00e3o escalar:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b583925d8f0f0a93afa22c409ae00aa8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\nabla f (x,y)=(2x+2y+4 \\ , \\ -2y+2x-4 )\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"315\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Devemos agora determinar os pontos cr\u00edticos, para isso igualamos as equa\u00e7\u00f5es obtidas a 0 e resolvemos o sistema de equa\u00e7\u00f5es:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0b72f879adb6df18c19610c21eba3887_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\nabla f (x,y)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" 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-d851eb626a9bd385aec8f68c9df71a39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left. \\begin{array}{l} 2x+2y+4 =0 \\\\[2ex] -2y+2x-4=0 \\end{array}\\right\\} \\longrightarrow \\left. \\begin{array}{c} x = 0 \\\\[1.1ex] y = -2 \\end{array}\\right\\} \\longrightarrow \\ (0,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"383\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Portanto, o ponto cr\u00edtico que encontramos \u00e9 (0,-2).<\/p>\n<p> Uma vez encontrado o ponto cr\u00edtico da fun\u00e7\u00e3o, devemos calcular a matriz Hessiana:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-702fa5f5c3e3d872e1ec0dad0e3216c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f (x,y)=\\begin{pmatrix}2 &amp; 2 \\\\[1.1ex] 2 &amp; -2  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"160\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> E, obviamente, a matriz Hessiana avaliada no ponto cr\u00edtico \u00e9 a mesma:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7b56ffff28d1a9b98c9848891ae924eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f (0,-2)=\\begin{pmatrix}2 &amp; 2 \\\\[1.1ex] 2 &amp; -2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"172\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Para ver que tipo de matriz \u00e9, usaremos o crit\u00e9rio de Sylvester. Portanto, resolvemos os principais menores da matriz:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-226cf6a18d27a9da300823c13158d56a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix}2 \\end{vmatrix} = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"50\" 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-d66efe9fca481475009bb1703939e4f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix}2 &amp; 2 \\\\[1.1ex] 2 &amp; -2 \\end{vmatrix} = -8\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"104\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> O menor principal 1 (\u00edmpar) \u00e9 positivo e o menor principal 2 (par) \u00e9 negativo, portanto, pelo crit\u00e9rio de Sylvester, \u00e9 uma <strong>matriz indefinida.<\/strong> E, portanto, o ponto cr\u00edtico (0,-2) \u00e9 um <strong>ponto de sela.<\/strong> <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-119\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"> Determinando a concavidade ou convexidade de uma fun\u00e7\u00e3o com a matriz Hessiana<\/h2>\n<p> Outro uso da matriz Hessiana \u00e9 saber se uma fun\u00e7\u00e3o \u00e9 c\u00f4ncava ou convexa. E isso pode ser determinado pelo seguinte teorema:<\/p>\n<p> Ser<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e87f01194fba5ba72beb64431139ece0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A \\subseteq \\mathbb{R}^n\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"58\" style=\"vertical-align: -3px;\"><\/p>\n<p> um conjunto aberto e<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4fa98607bbf0ec2af91778a78a134c97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f \\colon A \\to \\mathbb{R}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\"><\/p>\n<p> uma fun\u00e7\u00e3o cujas segundas derivadas s\u00e3o cont\u00ednuas, sua concavidade e convexidade s\u00e3o definidas pela matriz Hessiana:<\/p>\n<ul style=\"color:#1976d2; font-weight: bold;\">\n<li style=\"margin-bottom:18px\"> <span style=\"color:#262626;font-weight: normal;\">Fun\u00e7\u00e3o\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p><\/span> \u00e9 <span style=\"color:#1976d2;\"><strong>convexo<\/strong><\/span> em toda parte<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> se e somente se sua matriz Hessiana for positiva semidefinida em todos os pontos do conjunto.<\/li>\n<li> <span style=\"color:#262626;font-weight: normal;\">Fun\u00e7\u00e3o\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p><\/span> \u00e9 <span style=\"color:#1976d2;\"><strong>estritamente convexo<\/strong><\/span> em todo<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> se e somente se sua matriz Hessiana for positiva definida em todos os pontos do conjunto.<\/li>\n<\/ul>\n<p> <strong><span style=\"color:#1976d2;\">\u2713<\/span><\/strong> Portanto, se<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p> \u00e9 uma fun\u00e7\u00e3o convexa em um ponto onde a matriz Jacobiana tamb\u00e9m desaparece, este ponto \u00e9 um <strong>m\u00ednimo local<\/strong> .<\/p>\n<ul style=\"color:#1976d2; font-weight: bold;\">\n<li style=\"margin-bottom:18px\"> <span style=\"color:#262626;font-weight: normal;\">Fun\u00e7\u00e3o\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p><\/span> \u00e9 <span style=\"color:#1976d2;\"><strong>c\u00f4ncavo<\/strong><\/span> em geral<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> se e somente se sua matriz Hessiana for negativa semidefinida em todos os pontos do conjunto.<\/li>\n<li> <span style=\"color:#262626;font-weight: normal;\">Fun\u00e7\u00e3o\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p><\/span> \u00e9 <span style=\"color:#1976d2;\"><strong>estritamente c\u00f4ncavo<\/strong><\/span> em geral<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> se e somente se sua matriz Hessiana for negativa definida em todos os pontos do conjunto.<\/li>\n<\/ul>\n<p> <strong><span style=\"color:#1976d2;\">\u2713<\/span><\/strong> Portanto, se<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p> \u00e9 uma fun\u00e7\u00e3o c\u00f4ncava em um ponto onde a matriz Jacobiana tamb\u00e9m desaparece, este ponto \u00e9 um <strong>m\u00e1ximo local<\/strong> .<\/p>\n<p> Abaixo voc\u00ea tem um exemplo de fun\u00e7\u00e3o convexa e outro de fun\u00e7\u00e3o c\u00f4ncava representada em um espa\u00e7o tridimensional: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-7\">\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center has-text-color has-medium-font-size\" style=\"color:#1976d2\"> fun\u00e7\u00e3o convexa <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/convexite-et-concavite-d-une-fonction-avec-la-matrice-hessienne.webp\" alt=\"determinar a fun\u00e7\u00e3o convexa ou c\u00f4ncava com a matriz Hessiana\" width=\"411\" height=\"308\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center has-text-color has-medium-font-size\" style=\"color:#1976d2\"> <strong>fun\u00e7\u00e3o c\u00f4ncava<\/strong> <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/concavite-et-convexite-d-une-fonction-avec-la-matrice-hessienne.webp\" alt=\"A imagem possui um atributo ALT vazio; seu nome de arquivo \u00e9 concavidade-e-convexidade-de-uma-fun\u00e7\u00e3o-com-a-matriz-hessiana-1024x768.jpg\" width=\"411\" height=\"307\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"> Mais aplica\u00e7\u00f5es da matriz Hessiana<\/h2>\n<p> Os principais usos da matriz Hessiana s\u00e3o os que j\u00e1 vimos, por\u00e9m, ela tamb\u00e9m possui outras aplica\u00e7\u00f5es. Explicamos-los abaixo para os mais curiosos.<\/p>\n<h3 class=\"wp-block-heading\"> Polin\u00f4mio de Taylor<\/h3>\n<p> A expans\u00e3o do <strong>polin\u00f4mio de Taylor<\/strong> para fun\u00e7\u00f5es de 2 ou mais vari\u00e1veis no ponto<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-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> come\u00e7a assim:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-813a551888f2fe0d61201e11c9cf83da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  T(x) = f(a) + (x-a)^T \\nabla f(a) + \\frac{1}{2}(x-a)^T \\operatorname{H}_f(a)(x-a) + \\ldots\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"475\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Como voc\u00ea pode ver, os termos de segunda ordem da expans\u00e3o de Taylor s\u00e3o dados pela matriz Hessiana avaliada no ponto de expans\u00e3o do polin\u00f4mio.<\/p>\n<h3 class=\"wp-block-heading\"> Matriz de serapilheira com bordas<\/h3>\n<p> Outro uso da matriz Hessiana \u00e9 calcular os m\u00ednimos e m\u00e1ximos de uma fun\u00e7\u00e3o multivariada<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d9f627258270ef54673906bdea5bc47c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x,y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\"><\/p>\n<p> restrito a outra fun\u00e7\u00e3o<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-629bf228b3d2b0003d598c1591ec6000_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(x,y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"49\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Para resolver este problema, utiliza-se a <strong>matriz Hessiana limitada<\/strong> e segue-se o seguinte procedimento:<\/p>\n<p> <strong>Passo 1:<\/strong> Calcula-se a fun\u00e7\u00e3o de Lagrange, que \u00e9 definida pela seguinte 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-7253e3e437e00468c7dc9b5e4546991a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle L(x,y,\\lambda) = f(x,y)+ \\lambda \\cdot g(x,y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"241\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> <strong>Passo 2:<\/strong> Os pontos cr\u00edticos da fun\u00e7\u00e3o de Lagrange s\u00e3o encontrados. Para fazer isso, calculamos o gradiente da fun\u00e7\u00e3o de Lagrange, igualamos as equa\u00e7\u00f5es a 0 e resolvemos as equa\u00e7\u00f5es.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-372152b499747854f2da5a2c8c211ce4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\nabla L = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"59\" style=\"vertical-align: -1px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6d76d99126f67279a1302a805e2b12e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cfrac{\\partial L}{\\partial x} = 0 \\qquad \\cfrac{\\partial L}{\\partial y}=0 \\qquad \\cfrac{\\partial L}{\\partial \\lambda}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"240\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> <strong>Etapa 3:<\/strong> Para cada ponto encontrado, calculamos o Hessiano limitado, que \u00e9 definido pela seguinte matriz:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3d1b2b04de9559a521e6704151c27bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H(f,g) = \\begin{pmatrix}0 &amp; \\cfrac{\\partial g}{\\partial x_1} &amp; \\cfrac{\\partial g}{\\partial x_2} &amp; \\cdots &amp; \\cfrac{\\partial g}{\\partial x_n} \\\\[4ex] \\cfrac{\\partial g}{\\partial x_1} &amp; \\cfrac{\\partial^2 f}{\\partial x_1^2} &amp; \\cfrac{\\partial^2 f}{\\partial x_1\\,\\partial x_2} &amp; \\cdots &amp; \\cfrac{\\partial^2 f}{\\partial x_1\\,\\partial x_n} \\\\[4ex] \\cfrac{\\partial g}{\\partial x_2} &amp; \\cfrac{\\partial^2 f}{\\partial x_2\\,\\partial x_1} &amp; \\cfrac{\\partial^2 f}{\\partial x_2^2} &amp; \\cdots &amp; \\cfrac{\\partial^2 f}{\\partial x_2\\,\\partial x_n} \\\\[3ex] \\vdots &amp; \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\[3ex] \\cfrac{\\partial g}{\\partial x_n} &amp; \\cfrac{\\partial^2 f}{\\partial x_n\\,\\partial x_1} &amp; \\cfrac{\\partial^2 f}{\\partial x_n\\,\\partial x_2} &amp; \\cdots &amp; \\cfrac{\\partial^2 f}{\\partial x_n^2}\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"289\" width=\"415\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> <strong>Etapa 4:<\/strong> Determinamos para cada ponto cr\u00edtico se \u00e9 m\u00e1ximo ou m\u00ednimo:<\/p>\n<ul>\n<li> Este ser\u00e1 um m\u00e1ximo local da fun\u00e7\u00e3o\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p> sob restri\u00e7\u00f5es de fun\u00e7\u00e3o<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> se o \u00faltimo <em>nm<\/em> (onde <em>n<\/em> \u00e9 o n\u00famero de vari\u00e1veis <em>em<\/em> o n\u00famero de restri\u00e7\u00f5es) os principais menores da matriz Hessiana com borda avaliada no ponto cr\u00edtico t\u00eam sinais alternados come\u00e7ando com o sinal negativo.<\/li>\n<\/ul>\n<ul>\n<li> Este ser\u00e1 um m\u00ednimo local da fun\u00e7\u00e3o\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p> sob restri\u00e7\u00f5es de fun\u00e7\u00e3o<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> se o \u00faltimo <em>nm<\/em> (onde <em>n<\/em> \u00e9 o n\u00famero de vari\u00e1veis <em>em<\/em> o n\u00famero de restri\u00e7\u00f5es), os principais menores da matriz Hessiana acentuada avaliada no ponto cr\u00edtico t\u00eam todos sinais negativos.<\/li>\n<\/ul>\n<p> Deve-se ter em mente que os m\u00ednimos ou m\u00e1ximos relativos de uma fun\u00e7\u00e3o restrita para outra n\u00e3o precisam necessariamente ser assim para a fun\u00e7\u00e3o irrestrita. A matriz Hessiana com borda \u00e9, portanto, \u00fatil apenas para este tipo de problema.<\/p>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<h3 class=\"wp-block-heading\"> Relacionamento com outras opera\u00e7\u00f5es<\/h3>\n<p> Por fim, a matriz Hessiana tamb\u00e9m est\u00e1 ligada a outras opera\u00e7\u00f5es ou matrizes importantes, principalmente com a matriz Jacobiana e com o operador de Laplace.<\/p>\n<h4 class=\"wp-block-heading\"> Rela\u00e7\u00e3o com a matriz Jacobiana<\/h4>\n<p> A matriz Hessiana de uma 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-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p> \u00e9 a <strong>matriz Jacobiana<\/strong> do gradiente da mesma 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-96ab1054f3c447eedac17f9ce04b4606_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle H_f = J(\\nabla f)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"97\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<h4 class=\"wp-block-heading\"> Operador Laplace<\/h4>\n<p> O tra\u00e7o da matriz Hessiana \u00e9 equivalente ao <strong>operador Laplace<\/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-77f16c5285ca6a9ac5899d6e832e9a40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle tr( H_f) = \\Delta f\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"101\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p> Esta igualdade pode ser facilmente comprovada, pois a defini\u00e7\u00e3o do operador de Laplace \u00e9 a diverg\u00eancia do gradiente de uma 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-5a3e06f41c5bd51c969601d46507a9c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\Delta f =\\nabla \\cdot (\\nabla f) = (\\nabla \\cdot \\nabla )f = \\nabla^2 f\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"262\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Sua express\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-c7e1ec07ef9178ea4dd56d9cae72d275_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\Delta f = \\cfrac{\\partial ^2 f}{\\partial^2 x_1} +\\cfrac{\\partial ^2 f}{\\partial^2 x_2} + \\cfrac{\\partial ^2 f}{\\partial^2 x_3}+ \\ldots +\\cfrac{\\partial ^2 f}{\\partial^2 x_n}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"310\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<p> E esta soma \u00e9 apenas o tra\u00e7o da matriz Hessiana, ent\u00e3o a equival\u00eancia est\u00e1 provada.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Esta p\u00e1gina \u00e9 certamente a explica\u00e7\u00e3o mais completa da matriz Hessiana que existe. Aqui \u00e9 explicado o conceito de matriz Hessiana, como calcul\u00e1-la com exemplos e h\u00e1 ainda v\u00e1rios exerc\u00edcios resolvidos para praticar. Al\u00e9m disso, voc\u00ea poder\u00e1 aprender como s\u00e3o calculados os valores m\u00e1ximo e m\u00ednimo de uma fun\u00e7\u00e3o multivari\u00e1vel, bem como se \u00e9 uma &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/matriz-hessiana-hessiana-2x2-3x3\/\"> <span class=\"screen-reader-text\">Matriz hessiana (ou hesse)<\/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":[12],"tags":[],"class_list":["post-322","post","type-post","status-publish","format-standard","hentry","category-determinante-de-uma-matriz"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Matriz Hessiana (ou Hesse) -<\/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\/matriz-hessiana-hessiana-2x2-3x3\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Matriz Hessiana (ou Hesse) -\" \/>\n<meta property=\"og:description\" content=\"Esta p\u00e1gina \u00e9 certamente a explica\u00e7\u00e3o mais completa da matriz Hessiana que existe. 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Al\u00e9m disso, voc\u00ea poder\u00e1 aprender como s\u00e3o calculados os valores m\u00e1ximo e m\u00ednimo de uma fun\u00e7\u00e3o multivari\u00e1vel, bem como se \u00e9 uma &hellip; Matriz hessiana (ou hesse) Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/matriz-hessiana-hessiana-2x2-3x3\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T08:31:48+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/formule-matricielle-hessienne.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=\"5 minutos\" \/>\n<script type=\"application\/ld+json\" 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Aqui \u00e9 explicado o conceito de matriz Hessiana, como calcul\u00e1-la com exemplos e h\u00e1 ainda v\u00e1rios exerc\u00edcios resolvidos para praticar. 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