{"id":35361,"date":"2025-01-01T13:00:38","date_gmt":"2025-01-01T13:00:38","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35361"},"modified":"2025-12-14T22:39:38","modified_gmt":"2025-12-14T22:39:38","slug":"maximos-e-minimos-de-uma-funcao","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/pt\/maximos-e-minimos-de-uma-funcao\/","title":{"rendered":"M\u00e1ximos e M\u00ednimos de uma Fun\u00e7\u00e3o"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>M\u00e1ximos e M\u00ednimos de uma Fun\u00e7\u00e3o<\/h1>\n<p style=\"text-align:center\"><em>Onde est\u00e1 o \u201cmelhor\u201d ponto de uma fun\u00e7\u00e3o: o m\u00e1ximo que voc\u00ea deseja alcan\u00e7ar ou o m\u00ednimo que precisa evitar? Essa pergunta, que aparece em otimiza\u00e7\u00e3o, f\u00edsica, economia e engenharia, \u00e9 uma das principais aplica\u00e7\u00f5es do c\u00e1lculo diferencial. E aqui vem o ponto central: o <a href=\"https:\/\/toposuranos.com\/material\/pt\/teorema-de-weierstrass-dos-valores-extremos\/\" rel=\"noopener\" target=\"_blank\">Teorema de Weierstrass<\/a> assegura que, se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u00e9 cont\u00ednua e voc\u00ea trabalha em um intervalo fechado e limitado, ent\u00e3o <strong>os extremos absolutos existem<\/strong>. A partir da\u00ed, o processo torna-se pr\u00e1tico: aprender a identificar <strong>extremos locais<\/strong> por meio de <strong>pontos cr\u00edticos<\/strong> (<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f&#039;(x)=0<\/span><\/span> ou n\u00e3o existe) e utilizar ferramentas como Rolle e o Teorema do Valor M\u00e9dio para transformar uma busca \u201c\u00e0s cegas\u201d em um m\u00e9todo claro, verific\u00e1vel e eficiente.<\/em><\/p>\n<p style=\"text-align:center\">\n<strong>Objetivos de Aprendizagem:<\/strong>\n<\/p>\n<ol>\n<li><b>Executar<\/b> um procedimento completo para encontrar extremos absolutos em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span>: avaliar <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> em pontos cr\u00edticos interiores e nos extremos do intervalo, e comparar valores para decidir m\u00e1ximo e m\u00ednimo absolutos.\n  <\/li>\n<li><strong>Contrastar<\/strong> o valor de uma condi\u00e7\u00e3o necess\u00e1ria versus suficiente: reconhecer que \u201c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f&#039;(x_0)=0<\/span><\/span>\u201d n\u00e3o garante extremo local, e decidir quais evid\u00eancias adicionais (compara\u00e7\u00e3o de valores, an\u00e1lise de sinais, comportamento local) s\u00e3o pertinentes em cada caso.\n  <\/li>\n<li><strong>Determinar<\/strong> a estrat\u00e9gia mais eficiente de acordo com o tipo de problema: extremos absolutos em intervalos compactos (Weierstrass + avalia\u00e7\u00e3o finita) versus extremos locais em pontos interiores (pontos cr\u00edticos + an\u00e1lise local), justificando a escolha.\n  <\/li>\n<\/ol>\n<p style=\"text-align:center;\">\n<strong><u>\u00cdNDICE DE CONTE\u00daDOS<\/u>:<\/strong><br \/>\n<a href=\"#1\"><strong>M\u00e1ximos e m\u00ednimos, extremos absolutos e locais<\/strong><\/a><br \/>\n<a href=\"#2\">Crit\u00e9rio da 1\u00aa Derivada<\/a><br \/>\n<a href=\"#3\"><strong>O Teorema de Rolle<\/strong><\/a><br \/>\n<a href=\"#4\"><strong>O Teorema do Valor M\u00e9dio Diferencial<\/strong><\/a><br \/>\n<a href=\"#5\">Intervalos de crescimento e decrescimento<\/a>\n<\/p>\n<div style=\"text-align: center;\">\n  <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/OscTlX3raaE\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe>\n<\/div>\n<p style=\"text-align: justify;\">\n  O <a href=\"https:\/\/toposuranos.com\/material\/pt\/teorema-de-weierstrass-dos-valores-extremos\/\" rel=\"noopener\" target=\"_blank\">Teorema de Weierstrass<\/a> assegura que, se uma fun\u00e7\u00e3o real est\u00e1 definida e \u00e9 cont\u00ednua em um subconjunto fechado e limitado de <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}<\/span>, ent\u00e3o necessariamente atinge valores m\u00e1ximo e m\u00ednimo (extremos absolutos). A busca por m\u00e1ximos e m\u00ednimos de uma fun\u00e7\u00e3o \u00e9 o que se conhece como um <strong>problema de otimiza\u00e7\u00e3o<\/strong>, e o teorema de Weierstrass garante a exist\u00eancia de solu\u00e7\u00f5es no sentido de extremos absolutos, desde que a fun\u00e7\u00e3o seja cont\u00ednua e o dom\u00ednio seja compacto. Tendo a exist\u00eancia assegurada, resta agora desenvolver estrat\u00e9gias que permitam encontrar essas solu\u00e7\u00f5es.\n<\/p>\n<p><a name=\"1\"><\/a><\/br><\/p>\n<h2>M\u00e1ximos e m\u00ednimos, extremos absolutos e locais<\/h2>\n<p style=\"text-align: justify;\">\n  <a href=\"https:\/\/www.youtube.com\/watch?v=OscTlX3raaE&amp;t=156s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">Antes de come\u00e7ar a revisar<\/span><\/a> estrat\u00e9gias para a busca de m\u00e1ximos e m\u00ednimos, definamos com clareza o que queremos procurar.<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0ffe0;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #000080;\"><strong>DEFINI\u00c7\u00c3O:<\/strong><\/span><br \/>\nSeja <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> uma fun\u00e7\u00e3o com dom\u00ednio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">D<\/span><\/span>. Diremos que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> atinge um <strong>m\u00e1ximo absoluto<\/strong> em um ponto <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in D<\/span><\/span> se:\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left( \\forall x \\in D \\right)\\bigl(f(x) \\leq f(x_0)\\bigr)<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          e atingir\u00e1 um <strong>m\u00ednimo absoluto<\/strong> em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> se:\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left( \\forall x \\in D \\right)\\bigl( f(x_0) \\leq f(x)\\bigr)<\/span><\/span>\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\n  De forma an\u00e1loga definem-se os extremos locais (relativos ao dom\u00ednio).\n<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0ffe0;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #000080;\"><strong>DEFINI\u00c7\u00c3O:<\/strong><\/span><br \/>\n          Seja <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> uma fun\u00e7\u00e3o com dom\u00ednio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">D<\/span><\/span> e seja <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in D<\/span><\/span>. Diremos que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> atinge um <strong>m\u00e1ximo local<\/strong> em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> se:\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\exists h&gt;0)\\left( \\forall x\\in [x_0-h, x_0+h] \\cap D \\right)\\bigl(f(x) \\leq f(x_0)\\bigr)<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          e atingir\u00e1 um <strong>m\u00ednimo local<\/strong> em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> se:\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\exists h&gt;0)\\left( \\forall x\\in [x_0-h, x_0+h] \\cap D \\right)\\bigl( f(x_0) \\leq f(x)\\bigr)<\/span><\/span>\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\n  A partir disso, podemos enunciar o seguinte resultado:\n<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #ffe0e0;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #800000;\"><strong>TEOREMA:<\/strong><\/span><br \/>\n          <a href=\"https:\/\/www.youtube.com\/watch?v=OscTlX3raaE&amp;t=833s\" target=\"_blank\" rel=\"noopener\"><br \/>\n            <strong><span style=\"color: #ff0000;\">Seja <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> um ponto<\/span><\/strong><\/a> interior de um intervalo compacto <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I<\/span><\/span>. Se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> atinge um m\u00e1ximo ou m\u00ednimo <strong>local<\/strong> em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x_0)<\/span><\/span> existe, ent\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x_0)=0<\/span><\/span>.\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0e0ff;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #000080;\"><strong>DEMONSTRA\u00c7\u00c3O:<\/strong><\/span><br \/>\n          Suponhamos que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> atinge um m\u00e1ximo local em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>. Ent\u00e3o existe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h_0 \\gt 0<\/span><\/span> tal que, para todo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h<\/span><\/span> com <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|h|\\lt h_0<\/span><\/span> e com <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0+h\\in I<\/span><\/span>, vale:\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x_0 + h)\\leq f(x_0)<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          o que \u00e9 equivalente a:\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x_0 + h) - f(x_0)\\leq 0<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          Consideremos agora dois casos:\n        <\/p>\n<ul>\n<li>\n<p style=\"text-align: justify;\">\n              Se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h&gt;0<\/span><\/span>, ent\u00e3o:\n            <\/p>\n<p style=\"text-align: center;\">\n              <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{f(x_0 + h) - f(x_0)}{h}\\leq 0<\/span><\/span>\n            <\/p>\n<\/li>\n<li>\n<p style=\"text-align: justify;\">\n              Se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h\\lt 0<\/span><\/span>, ent\u00e3o:\n            <\/p>\n<p style=\"text-align: center;\">\n              <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{f(x_0 + h) - f(x_0)}{h}\\geq 0<\/span><\/span>\n            <\/p>\n<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">\n          Se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x_0)<\/span><\/span> existe, ent\u00e3o o limite do quociente incremental quando <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h\\to 0<\/span><\/span> existe e deve ser compat\u00edvel com ambas as desigualdades, o que obriga que:\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f^\\prime(x_0)=\\lim_{h\\to 0}\\frac{f(x_0 + h) - f(x_0)}{h}= 0<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          Que \u00e9 o que se queria demonstrar.\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\n  Deve-se notar que esta demonstra\u00e7\u00e3o tamb\u00e9m \u00e9 v\u00e1lida para m\u00ednimos locais. Nesse caso, inicia-se com: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x_0+h)\\ge f(x_0)<\/span><\/span> para <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|h|<\/span><\/span> suficientemente pequeno.\n<\/p>\n<p><a name=\"2\"><\/a><\/br><\/p>\n<h3>Crit\u00e9rio da 1\u00aa Derivada<\/h3>\n<p style=\"text-align: justify;\">\n  <a href=\"https:\/\/www.youtube.com\/watch?v=OscTlX3raaE&amp;t=1257s\" target=\"_blank\" rel=\"noopener\"><br \/>\n    <strong><span style=\"color: #ff0000;\">O resultado que acabamos de revisar<\/span><\/strong><\/a> pode ser resumido na seguinte implica\u00e7\u00e3o:<\/p>\n<p style=\"text-align: center;\">\n  <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n  \\left\\{\\begin{matrix}f \\text{ atinge um}\\\\ \\text{extremo local em }x_0 \\end{matrix}\\right\\}\n\n  \\Longrightarrow\n\n  \\left\\{\\begin{matrix} \\displaystyle f^\\prime(x_0) = 0 \\\\ \\\\ \\vee \\\\ \\\\ \\text{A derivada n\u00e3o existe em }x_0 \\end{matrix}\\right\\}\n\n  <\/span><\/span>\n<\/p>\n<p style=\"text-align: justify;\">\n  Embora o rec\u00edproco dessa implica\u00e7\u00e3o n\u00e3o seja verdadeiro em geral, ela \u00e9 muito \u00fatil na delimita\u00e7\u00e3o da busca por extremos locais. Em fun\u00e7\u00e3o disso, definem-se os pontos cr\u00edticos da primeira derivada.\n<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0ffe0;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #000080;\"><strong>DEFINI\u00c7\u00c3O:<\/strong><\/span><br \/>\n          Diz-se que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u00e9 um <strong>ponto cr\u00edtico da primeira derivada<\/strong> se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x_0)=0<\/span><\/span> ou se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x_0)<\/span><\/span> n\u00e3o existe.\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\n  Os pontos cr\u00edticos da primeira derivada s\u00e3o relevantes porque todo ponto em que a fun\u00e7\u00e3o extremiza (local ou absolutamente) deve pertencer ao conjunto de pontos cr\u00edticos:\n<\/p>\n<p style=\"text-align: center;\">\n  <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\left\\{\\begin{matrix}\\text{pontos que}\\\\ \\text{extremizam absolutamente}\\end{matrix}\\right\\}\n\n  \\subseteq\n\n  \\left\\{\\begin{matrix}\\text{pontos que}\\\\ \\text{extremizam localmente}\\end{matrix}\\right\\}\n\n  \\subseteq\n\n  \\left\\{\\begin{matrix}\\text{pontos cr\u00edticos da}\\\\ \\text{primeira derivada}\\end{matrix}\\right\\}\n\n  <\/span><\/span>\n<\/p>\n<p style=\"text-align: justify;\">\n  Isso \u00e9 o que chamamos de <strong>crit\u00e9rio da primeira derivada<\/strong>, entendido como uma condi\u00e7\u00e3o necess\u00e1ria para a exist\u00eancia de extremos locais em pontos interiores.\n<\/p>\n<p><a name=\"3\"><\/a><\/br><\/p>\n<h2>O Teorema de Rolle<\/h2>\n<p style=\"text-align: justify;\">\n  <a href=\"https:\/\/www.youtube.com\/watch?v=OscTlX3raaE&amp;t=1454s\" target=\"_blank\" rel=\"noopener\"><br \/>\n    <strong><span style=\"color: #ff0000;\">J\u00e1 vimos que a determina\u00e7\u00e3o<\/span><\/strong><\/a> de pontos cr\u00edticos da primeira derivada \u00e9 fundamental na busca por extremos locais. Por esse motivo, \u00e9 natural investigar sob quais condi\u00e7\u00f5es se pode assegurar a exist\u00eancia de tais pontos cr\u00edticos. Um avan\u00e7o nesse sentido vem com o teorema de Rolle.<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #ffe0e0;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #800000;\"><strong>TEOREMA:<\/strong><\/span><br \/>\n          Seja <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> uma fun\u00e7\u00e3o definida e cont\u00ednua em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span>, e deriv\u00e1vel em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span>. Se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(a)=f(b)<\/span><\/span>, ent\u00e3o existe um <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span> tal que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c)=0<\/span><\/span>.\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0e0ff;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #000080;\"><strong>DEMONSTRA\u00c7\u00c3O:<\/strong><\/span><br \/>\n          Analisaremos duas possibilidades:\n        <\/p>\n<ol style=\"text-align: justify;\">\n<li>\n            Se, para todo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]a,b[<\/span><\/span>, vale <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=f(a)=f(b)<\/span><\/span>, ent\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u00e9 constante e, consequentemente, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x)=0<\/span><\/span> para todo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]a,b[<\/span><\/span>. Em particular, existe um <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span> com <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c)=0<\/span><\/span>.\n          <\/li>\n<li>\n            Se existe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]a,b[<\/span><\/span> tal que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)\\neq f(a)=f(b)<\/span><\/span>, ent\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> n\u00e3o \u00e9 constante. Como <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u00e9 cont\u00ednua em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span>, pelo teorema de Weierstrass atinge um m\u00e1ximo absoluto e um m\u00ednimo absoluto em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span>.<\/p>\n<p>Al\u00e9m disso, como <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(a)=f(b)<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> n\u00e3o \u00e9 constante, ao menos um desses extremos deve ocorrer no interior <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span>.<\/p>\n<p>Assim, se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span> \u00e9 um ponto interior onde <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> atinge um extremo local. Como <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u00e9 deriv\u00e1vel em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span>, em particular existe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c)<\/span><\/span>, e pelo teorema anterior conclui-se que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c)=0<\/span><\/span>.\n          <\/li>\n<\/ol>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"4\"><\/a><\/br><\/p>\n<h2>O Teorema do Valor M\u00e9dio Diferencial<\/h2>\n<p style=\"text-align: justify;\">\n  <a href=\"https:\/\/www.youtube.com\/watch?v=OscTlX3raaE&amp;t=1878s\" target=\"_blank\" rel=\"noopener\"><br \/>\n    <strong><span style=\"color: #ff0000;\">Outro resultado que \u00e9 consequ\u00eancia direta<\/span><\/strong><\/a> daqueles que acabamos de revisar, e que fornece informa\u00e7\u00e3o \u00fatil para o estudo das fun\u00e7\u00f5es, \u00e9 o teorema do valor m\u00e9dio para o c\u00e1lculo diferencial.\n<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #ffe0e0;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #800000;\"><strong>TEOREMA:<\/strong><\/span><br \/>\n          Seja <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> uma fun\u00e7\u00e3o definida e cont\u00ednua em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span>, e deriv\u00e1vel em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span>. Ent\u00e3o existe um <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span> tal que:\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c) =\\displaystyle \\frac{f(b) - f(a)}{b-a}<\/span><\/span>\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0e0ff;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #000080;\"><strong>DEMONSTRA\u00c7\u00c3O:<\/strong><\/span><br \/>\n          Seja <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> a fun\u00e7\u00e3o definida por:\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(x) = f(x) - \\displaystyle \\frac{f(b) - f(a)}{b-a}(x-a)<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          Essa fun\u00e7\u00e3o \u00e9 cont\u00ednua em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span> e deriv\u00e1vel em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span> porque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> tamb\u00e9m o \u00e9. Al\u00e9m disso, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(a)=F(b)<\/span><\/span>, de modo que podemos utilizar o teorema de Rolle para concluir que existe um ponto <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span> tal que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F^\\prime(c)=0<\/span><\/span>.\n        <\/p>\n<p style=\"text-align: justify;\">\n          Agora, derivando <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> obt\u00e9m-se:\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F^\\prime(x) = f^\\prime(x) - \\displaystyle\\frac{f(b) - f(a)}{b-a}<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          Avaliando em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> e utilizando <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F^\\prime(c)=0<\/span><\/span>:\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=F^\\prime(c) = f^\\prime(c) - \\displaystyle\\frac{f(b) - f(a)}{b-a}<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          Em seguida:\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c) = \\displaystyle\\frac{f(b) - f(a)}{b-a}<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          que \u00e9 o que se queria demonstrar.\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"5\"><\/a><\/br><\/p>\n<h3>Intervalos de crescimento e decrescimento<\/h3>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #ffe0e0;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #800000;\"><strong>TEOREMA:<\/strong><\/span>\n        <\/p>\n<ul style=\"text-align: justify;\">\n<li>\n            <a href=\"https:\/\/www.youtube.com\/watch?v=OscTlX3raaE&amp;t=2402s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u00e9 uma fun\u00e7\u00e3o<\/span><\/strong><\/a> tal que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x\\in ]a,b[)\\left(0\\lt f^\\prime(x)\\right)<\/span><\/span>, ent\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u00e9 estritamente crescente em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span>.\n          <\/li>\n<li>\n            Se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u00e9 uma fun\u00e7\u00e3o tal que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x\\in ]a,b[)\\left(f^\\prime(x)\\lt 0\\right)<\/span><\/span>, ent\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u00e9 estritamente decrescente em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span>.\n          <\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0e0ff;\">\n<p style=\"text-align: justify;\">\n          <span style=\"color: #000080;\"><strong>DEMONSTRA\u00c7\u00c3O:<\/strong><\/span><br \/>\n          Sejam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_1,x_2\\in ]a,b[<\/span><\/span> tais que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_1 \\lt x_2<\/span><\/span>. Como <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u00e9 deriv\u00e1vel em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span>, podemos aplicar o teorema do valor m\u00e9dio a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> sobre o intervalo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[x_1,x_2]\\subset ]a,b[<\/span><\/span>. Em consequ\u00eancia, existe um ponto <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]x_1,x_2[<\/span><\/span> tal que:\n        <\/p>\n<p style=\"text-align: center;\">\n          <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c) = \\displaystyle\\frac{f(x_2) - f(x_1)}{x_2 - x_1}<\/span><\/span>\n        <\/p>\n<p style=\"text-align: justify;\">\n          A partir disso:\n        <\/p>\n<ul style=\"text-align: justify;\">\n<li>\n            Se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c) \\gt 0<\/span><\/span>, ent\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x_2) - f(x_1) = f^\\prime(c)(x_2 - x_1) \\gt 0<\/span><\/span>.<br \/>\nPortanto, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u00e9 crescente.\n          <\/li>\n<li>\n            Se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c) \\lt 0<\/span><\/span>, ent\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x_2) - f(x_1) = f^\\prime(c)(x_2 - x_1) \\lt 0<\/span><\/span>.<br \/>\nPortanto, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u00e9 decrescente.\n          <\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\nEstudar m\u00e1ximos e m\u00ednimos n\u00e3o \u00e9 apenas \u201cfazer derivadas\u201d, mas aprender a transformar uma busca difusa em um procedimento com garantias e crit\u00e9rios claros. Weierstrass indica quando se pode confiar que o \u00f3timo existe em um intervalo compacto, enquanto o crit\u00e9rio da primeira derivada, Rolle e o Teorema do Valor M\u00e9dio fornecem o mapa para encontrar candidatos e justificar conclus\u00f5es: onde uma fun\u00e7\u00e3o pode extremizar, quando essa condi\u00e7\u00e3o \u00e9 apenas necess\u00e1ria e como o sinal de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f&#039;<\/span><\/span> revela crescimento e decrescimento. Ao dominar essa cadeia de ideias, passa-se de observar gr\u00e1ficos com intui\u00e7\u00e3o para resolver problemas de otimiza\u00e7\u00e3o com argumentos verific\u00e1veis, o que \u00e9 exatamente a diferen\u00e7a entre \u201cacho que aqui est\u00e1 o melhor ponto\u201d e \u201csei por que ele deve estar aqui\u201d.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>M\u00e1ximos e M\u00ednimos de uma Fun\u00e7\u00e3o Onde est\u00e1 o \u201cmelhor\u201d ponto de uma fun\u00e7\u00e3o: o m\u00e1ximo que voc\u00ea deseja alcan\u00e7ar ou o m\u00ednimo que precisa evitar? Essa pergunta, que aparece em otimiza\u00e7\u00e3o, f\u00edsica, economia e engenharia, \u00e9 uma das principais aplica\u00e7\u00f5es do c\u00e1lculo diferencial. E aqui vem o ponto central: o Teorema de Weierstrass assegura [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":35349,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":0,"footnotes":""},"categories":[856,571],"tags":[],"class_list":["post-35361","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-calculo-diferencial-pt","category-matematica-pt"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>M\u00e1ximos e M\u00ednimos de uma Fun\u00e7\u00e3o - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Domine m\u00e1ximos e m\u00ednimos com uma abordagem moderna do c\u00e1lculo diferencial: Weierstrass, pontos cr\u00edticos, Rolle e Valor M\u00e9dio para otimizar com m\u00e9todo, n\u00e3o com intui\u00e7\u00e3o.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/pt\/maximos-e-minimos-de-uma-funcao\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"M\u00e1ximos e M\u00ednimos de uma Fun\u00e7\u00e3o\" \/>\n<meta property=\"og:description\" content=\"Domine m\u00e1ximos e m\u00ednimos com uma abordagem moderna do c\u00e1lculo diferencial: Weierstrass, pontos cr\u00edticos, Rolle e Valor M\u00e9dio para otimizar com m\u00e9todo, n\u00e3o com intui\u00e7\u00e3o.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/pt\/maximos-e-minimos-de-uma-funcao\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2025-01-01T13:00:38+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-12-14T22:39:38+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/01\/maximum-1024x683.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"M\u00e1ximos e M\u00ednimos de uma Fun\u00e7\u00e3o\" \/>\n<meta name=\"twitter:description\" content=\"Domine m\u00e1ximos e m\u00ednimos com uma abordagem moderna do c\u00e1lculo diferencial: Weierstrass, pontos cr\u00edticos, Rolle e Valor M\u00e9dio para otimizar com m\u00e9todo, n\u00e3o com intui\u00e7\u00e3o.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/01\/maximum.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"7 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\/\/toposuranos.com\/material\/pt\/maximos-e-minimos-de-uma-funcao\/#article\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/pt\/maximos-e-minimos-de-uma-funcao\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"M\u00e1ximos e M\u00ednimos de uma Fun\u00e7\u00e3o\",\"datePublished\":\"2025-01-01T13:00:38+00:00\",\"dateModified\":\"2025-12-14T22:39:38+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/pt\/maximos-e-minimos-de-uma-funcao\/\"},\"wordCount\":1994,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/pt\/maximos-e-minimos-de-uma-funcao\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/01\/maximum.jpg\",\"articleSection\":[\"C\u00e1lculo Diferencial\",\"Matem\u00e1tica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\/\/toposuranos.com\/material\/pt\/maximos-e-minimos-de-uma-funcao\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\/\/toposuranos.com\/material\/pt\/maximos-e-minimos-de-uma-funcao\/\",\"url\":\"http:\/\/toposuranos.com\/material\/pt\/maximos-e-minimos-de-uma-funcao\/\",\"name\":\"M\u00e1ximos e M\u00ednimos de uma Fun\u00e7\u00e3o - 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