{"id":35386,"date":"2025-01-01T13:00:12","date_gmt":"2025-01-01T13:00:12","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35386"},"modified":"2025-12-14T23:41:55","modified_gmt":"2025-12-14T23:41:55","slug":"maxima-et-minima-dune-fonction","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/maxima-et-minima-dune-fonction\/","title":{"rendered":"Maxima et Minima d\u2019une Fonction"},"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>Maxima et Minima d\u2019une Fonction<\/h1>\n<p style=\"text-align:center\"><em>O\u00f9 se situe le \u00ab meilleur \u00bb point d\u2019une fonction : le maximum que l\u2019on souhaite atteindre ou le minimum que l\u2019on doit \u00e9viter ? Cette question, qui appara\u00eet en optimisation, en physique, en \u00e9conomie et en ing\u00e9nierie, constitue l\u2019une des principales applications du calcul diff\u00e9rentiel. Et voici l\u2019essentiel : le <a href=\"https:\/\/toposuranos.com\/material\/fr\/theoreme-de-weierstrass-des-valeurs-extremes\/\" rel=\"noopener\" target=\"_blank\">th\u00e9or\u00e8me de Weierstrass<\/a> garantit que, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> est continue et que l\u2019on travaille sur un intervalle ferm\u00e9 et born\u00e9, alors <strong>les extr\u00eames absolus existent<\/strong>. \u00c0 partir de l\u00e0, la d\u00e9marche devient pratique : apprendre \u00e0 d\u00e9tecter des <strong>extr\u00eames locaux<\/strong> \u00e0 l\u2019aide de <strong>points critiques<\/strong> (<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f&#039;(x)=0<\/span><\/span> ou n\u2019existe pas) et utiliser des outils tels que le th\u00e9or\u00e8me de Rolle et le th\u00e9or\u00e8me de la valeur moyenne pour transformer une recherche \u00ab \u00e0 l\u2019aveugle \u00bb en une m\u00e9thode claire, v\u00e9rifiable et efficace.<\/em><\/p>\n<p style=\"text-align:center\">\n<strong>Objectifs d\u2019apprentissage :<\/strong>\n<\/p>\n<ol>\n<li><b>Ex\u00e9cuter<\/b> une proc\u00e9dure compl\u00e8te pour d\u00e9terminer les extr\u00eames absolus sur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span> : \u00e9valuer <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> aux points critiques int\u00e9rieurs et aux extr\u00e9mit\u00e9s de l\u2019intervalle, puis comparer les valeurs afin de d\u00e9cider du maximum et du minimum absolus.\n  <\/li>\n<li><strong>Comparer<\/strong> la valeur d\u2019une condition n\u00e9cessaire par rapport \u00e0 une condition suffisante : reconna\u00eetre que \u00ab <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f&#039;(x_0)=0<\/span><\/span> \u00bb ne garantit pas un extr\u00eame local, et d\u00e9terminer quelles preuves suppl\u00e9mentaires (comparaison des valeurs, analyse des signes, comportement local) sont pertinentes dans chaque cas.\n  <\/li>\n<li><strong>D\u00e9terminer<\/strong> la strat\u00e9gie la plus efficace selon le type de probl\u00e8me : extr\u00eames absolus sur des intervalles compacts (Weierstrass + \u00e9valuation finie) versus extr\u00eames locaux en des points int\u00e9rieurs (points critiques + analyse locale), en justifiant le choix.\n  <\/li>\n<\/ol>\n<p style=\"text-align:center;\">\n<strong><u>INDEX DES CONTENUS<\/u> :<\/strong><br \/>\n<a href=\"#1\"><strong>Maxima et minima, extr\u00eames absolus et locaux<\/strong><\/a><br \/>\n<a href=\"#2\">Crit\u00e8re de la 1re d\u00e9riv\u00e9e<\/a><br \/>\n<a href=\"#3\"><strong>Le th\u00e9or\u00e8me de Rolle<\/strong><\/a><br \/>\n<a href=\"#4\"><strong>Le th\u00e9or\u00e8me de la valeur moyenne diff\u00e9rentielle<\/strong><\/a><br \/>\n<a href=\"#5\">Intervalles de croissance et de d\u00e9croissance<\/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  Le <a href=\"https:\/\/toposuranos.com\/material\/fr\/theoreme-de-weierstrass-des-valeurs-extremes\/\" rel=\"noopener\" target=\"_blank\">th\u00e9or\u00e8me de Weierstrass<\/a> nous assure que, si une fonction r\u00e9elle est d\u00e9finie et continue sur un sous-ensemble ferm\u00e9 et born\u00e9 de <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}<\/span>, alors elle atteint n\u00e9cessairement des valeurs maximale et minimale (extr\u00eames absolus). La recherche des maxima et des minima d\u2019une fonction correspond \u00e0 ce que l\u2019on appelle un <strong>probl\u00e8me d\u2019optimisation<\/strong>, et le th\u00e9or\u00e8me de Weierstrass garantit l\u2019existence de solutions au sens des extr\u00eames absolus, pourvu que la fonction soit continue et que le domaine soit compact. Une fois l\u2019existence assur\u00e9e, il ne reste plus qu\u2019\u00e0 d\u00e9velopper des strat\u00e9gies permettant de trouver effectivement ces solutions.\n<\/p>\n<p><a name=\"1\"><\/a><\/br><\/p>\n<h2>Maxima et minima, extr\u00eames absolus et locaux<\/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;\">Avant de commencer \u00e0 examiner<\/span><\/a> des strat\u00e9gies pour la recherche des maxima et des minima, d\u00e9finissons clairement ce que nous cherchons \u00e0 identifier.<\/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>D\u00c9FINITION :<\/strong><\/span><br \/>\nSoit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> une fonction de domaine <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">D<\/span><\/span>. Nous dirons que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> atteint un <strong>maximum absolu<\/strong> en un point <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in D<\/span><\/span> si :\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          et qu\u2019elle atteint un <strong>minimum absolu<\/strong> en <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> si :\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 mani\u00e8re analogue, on d\u00e9finit les extr\u00eames locaux (relatifs au domaine).\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>D\u00c9FINITION :<\/strong><\/span><br \/>\n          Soit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> une fonction de domaine <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">D<\/span><\/span> et soit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in D<\/span><\/span>. Nous dirons que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> atteint un <strong>maximum local<\/strong> en <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> si :\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          et qu\u2019elle atteint un <strong>minimum local<\/strong> en <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> si :\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  \u00c0 partir de cela, nous pouvons \u00e9noncer le r\u00e9sultat suivant :\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>TH\u00c9OR\u00c8ME :<\/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;\">Soit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> un point<\/span><\/strong><\/a> int\u00e9rieur d\u2019un intervalle compact <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I<\/span><\/span>. Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> atteint un maximum ou un minimum <strong>local<\/strong> en <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> et si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x_0)<\/span><\/span> existe, alors <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>D\u00c9MONSTRATION :<\/strong><\/span><br \/>\n          Supposons que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> atteigne un maximum local en <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>. Il existe alors <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h_0 \\gt 0<\/span><\/span> tel que, pour tout <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h<\/span><\/span> v\u00e9rifiant <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|h|\\lt h_0<\/span><\/span> et avec <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0+h\\in I<\/span><\/span>, on ait :\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          ce qui est \u00e9quivalent \u00e0 :\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          Consid\u00e9rons maintenant deux cas :\n        <\/p>\n<ul>\n<li>\n<p style=\"text-align: justify;\">\n              Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h&gt;0<\/span><\/span>, alors :\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              Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h\\lt 0<\/span><\/span>, alors :\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          Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x_0)<\/span><\/span> existe, alors la limite du quotient incr\u00e9mental lorsque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h\\to 0<\/span><\/span> existe et doit \u00eatre compatible avec les deux in\u00e9galit\u00e9s, ce qui impose 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          Ce qui \u00e9tait \u00e0 d\u00e9montrer.\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\n  Il convient de noter que cette d\u00e9monstration est \u00e9galement valable pour les minima locaux. Dans ce cas, on commence par : <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x_0+h)\\ge f(x_0)<\/span><\/span> pour <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|h|<\/span><\/span> suffisamment petit.\n<\/p>\n<p><a name=\"2\"><\/a><\/br><\/p>\n<h3>Crit\u00e8re de la 1re d\u00e9riv\u00e9e<\/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;\">Le r\u00e9sultat que nous venons d\u2019examiner<\/span><\/strong><\/a> peut \u00eatre r\u00e9sum\u00e9 par l\u2019implication suivante :<\/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{ atteint un}\\\\ \\text{extr\u00eame local en }x_0 \\end{matrix}\\right\\}\n\n  \\Longrightarrow\n\n  \\left\\{\\begin{matrix} \\displaystyle f^\\prime(x_0) = 0 \\\\ \\\\ \\vee \\\\ \\\\ \\text{La d\u00e9riv\u00e9e n\u2019existe pas en }x_0 \\end{matrix}\\right\\}\n\n  <\/span><\/span>\n<\/p>\n<p style=\"text-align: justify;\">\n  Bien que la r\u00e9ciproque de cette implication ne soit pas vraie en g\u00e9n\u00e9ral, elle est n\u00e9anmoins tr\u00e8s utile pour restreindre la recherche des extr\u00eames locaux. Sur cette base, on d\u00e9finit les points critiques de la premi\u00e8re d\u00e9riv\u00e9e.\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>D\u00c9FINITION :<\/strong><\/span><br \/>\n          On dit que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> est un <strong>point critique de la premi\u00e8re d\u00e9riv\u00e9e<\/strong> si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x_0)=0<\/span><\/span> ou si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x_0)<\/span><\/span> n\u2019existe pas.\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\n  Les points critiques de la premi\u00e8re d\u00e9riv\u00e9e sont pertinents, car tout point o\u00f9 la fonction atteint un extr\u00eame (local ou absolu) doit appartenir \u00e0 l\u2019ensemble des points critiques :\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{points qui}\\\\ \\text{extr\u00e9misent absolument}\\end{matrix}\\right\\}\n\n  \\subseteq\n\n  \\left\\{\\begin{matrix}\\text{points qui}\\\\ \\text{extr\u00e9misent localement}\\end{matrix}\\right\\}\n\n  \\subseteq\n\n  \\left\\{\\begin{matrix}\\text{points critiques de la}\\\\ \\text{premi\u00e8re d\u00e9riv\u00e9e}\\end{matrix}\\right\\}\n\n  <\/span><\/span>\n<\/p>\n<p style=\"text-align: justify;\">\n  C\u2019est ce que l\u2019on appelle le <strong>crit\u00e8re de la premi\u00e8re d\u00e9riv\u00e9e<\/strong>, entendu comme une condition n\u00e9cessaire \u00e0 l\u2019existence d\u2019extr\u00eames locaux en des points int\u00e9rieurs.\n<\/p>\n<p><a name=\"3\"><\/a><\/br><\/p>\n<h2>Le th\u00e9or\u00e8me 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;\">Nous avons d\u00e9j\u00e0 vu que la d\u00e9termination<\/span><\/strong><\/a> des points critiques de la premi\u00e8re d\u00e9riv\u00e9e est essentielle dans la recherche des extr\u00eames locaux. Pour cette raison, il est naturel d\u2019examiner sous quelles conditions on peut garantir l\u2019existence de tels points critiques. Une avanc\u00e9e en ce sens est apport\u00e9e par le th\u00e9or\u00e8me de Rolle.\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>TH\u00c9OR\u00c8ME :<\/strong><\/span><br \/>\n          Soit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> une fonction d\u00e9finie et continue sur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span>, et d\u00e9rivable sur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span>. Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(a)=f(b)<\/span><\/span>, alors il existe un <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span> tel 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>D\u00c9MONSTRATION :<\/strong><\/span><br \/>\n          Nous analyserons deux possibilit\u00e9s :\n        <\/p>\n<ol style=\"text-align: justify;\">\n<li>\n            Si, pour tout <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]a,b[<\/span><\/span>, on a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=f(a)=f(b)<\/span><\/span>, alors <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> est constante et, par cons\u00e9quent, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x)=0<\/span><\/span> pour tout <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]a,b[<\/span><\/span>. En particulier, il existe un <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span> tel que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c)=0<\/span><\/span>.\n          <\/li>\n<li>\n            S\u2019il existe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]a,b[<\/span><\/span> tel que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)\\neq f(a)=f(b)<\/span><\/span>, alors <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> n\u2019est pas constante. Comme <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> est continue sur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span>, elle atteint, d\u2019apr\u00e8s le th\u00e9or\u00e8me de Weierstrass, un maximum absolu et un minimum absolu sur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span>.<\/p>\n<p>En outre, comme <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(a)=f(b)<\/span><\/span> et que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> n\u2019est pas constante, au moins l\u2019un de ces extr\u00eames doit se produire \u00e0 l\u2019int\u00e9rieur de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span>.<\/p>\n<p>Ainsi, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span> est un point int\u00e9rieur o\u00f9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> atteint un extr\u00eame local, comme <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> est d\u00e9rivable sur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span>, en particulier <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c)<\/span><\/span> existe, et, d\u2019apr\u00e8s le th\u00e9or\u00e8me pr\u00e9c\u00e9dent, on conclut 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>Le th\u00e9or\u00e8me de la valeur moyenne diff\u00e9rentielle<\/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;\">Un autre r\u00e9sultat qui est une cons\u00e9quence directe<\/span><\/strong><\/a> de ceux que nous venons d\u2019examiner, et qui fournit des informations utiles pour l\u2019\u00e9tude des fonctions, est le th\u00e9or\u00e8me de la valeur moyenne en calcul diff\u00e9rentiel.\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>TH\u00c9OR\u00c8ME :<\/strong><\/span><br \/>\n          Soit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> une fonction d\u00e9finie et continue sur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span>, et d\u00e9rivable sur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span>. Il existe alors un <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span> tel 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>D\u00c9MONSTRATION :<\/strong><\/span><br \/>\n          Soit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> la fonction d\u00e9finie par :\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          Cette fonction est continue sur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span> et d\u00e9rivable sur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span> parce que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> l\u2019est \u00e9galement. De plus, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(a)=F(b)<\/span><\/span>, ce qui permet d\u2019utiliser le th\u00e9or\u00e8me de Rolle pour conclure qu\u2019il existe un point <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span> tel 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          En d\u00e9rivant maintenant <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span>, on obtient :\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          En \u00e9valuant en <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> et en utilisant <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          On obtient alors :\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          ce qui \u00e9tait \u00e0 d\u00e9montrer.\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"5\"><\/a><\/br><\/p>\n<h3>Intervalles de croissance et de d\u00e9croissance<\/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>TH\u00c9OR\u00c8ME :<\/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;\">Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> est une fonction<\/span><\/strong><\/a> telle 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>, alors <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> est strictement croissante sur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span>.\n          <\/li>\n<li>\n            Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> est une fonction telle 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>, alors <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> est strictement d\u00e9croissante sur <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>D\u00c9MONSTRATION :<\/strong><\/span><br \/>\n          Soient <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_1,x_2\\in ]a,b[<\/span><\/span> tels que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_1 \\lt x_2<\/span><\/span>. Comme <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> est d\u00e9rivable sur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span>, nous pouvons appliquer le th\u00e9or\u00e8me de la valeur moyenne \u00e0 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> sur l\u2019intervalle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[x_1,x_2]\\subset ]a,b[<\/span><\/span>. Par cons\u00e9quent, il existe un point <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]x_1,x_2[<\/span><\/span> tel 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          \u00c0 partir de cela :\n        <\/p>\n<ul style=\"text-align: justify;\">\n<li>\n            Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c) \\gt 0<\/span><\/span>, alors <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 \/>\nPar cons\u00e9quent, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> est croissante.\n          <\/li>\n<li>\n            Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c) \\lt 0<\/span><\/span>, alors <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 \/>\nPar cons\u00e9quent, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> est d\u00e9croissante.\n          <\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\n\u00c9tudier les maxima et les minima ne consiste pas seulement \u00e0 \u00ab faire des d\u00e9riv\u00e9es \u00bb, mais \u00e0 apprendre \u00e0 transformer une recherche diffuse en une proc\u00e9dure fond\u00e9e sur des garanties et des crit\u00e8res clairs. Weierstrass indique quand on peut avoir confiance dans l\u2019existence d\u2019un optimum sur un intervalle compact, tandis que le crit\u00e8re de la premi\u00e8re d\u00e9riv\u00e9e, le th\u00e9or\u00e8me de Rolle et le th\u00e9or\u00e8me de la valeur moyenne fournissent la carte permettant d\u2019identifier des candidats et de justifier les conclusions : o\u00f9 une fonction peut pr\u00e9senter des extr\u00eames, quand cette condition est seulement n\u00e9cessaire, et comment le signe de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f&#039;<\/span><\/span> r\u00e9v\u00e8le la croissance et la d\u00e9croissance. En ma\u00eetrisant cette cha\u00eene d\u2019id\u00e9es, on passe de l\u2019observation intuitive de graphiques \u00e0 la r\u00e9solution de probl\u00e8mes d\u2019optimisation \u00e0 l\u2019aide d\u2019arguments v\u00e9rifiables, ce qui constitue pr\u00e9cis\u00e9ment la diff\u00e9rence entre \u00ab je pense que le meilleur point est ici \u00bb et \u00ab je sais pourquoi il doit se trouver ici \u00bb.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Maxima et Minima d\u2019une Fonction O\u00f9 se situe le \u00ab meilleur \u00bb point d\u2019une fonction : le maximum que l\u2019on souhaite atteindre ou le minimum que l\u2019on doit \u00e9viter ? Cette question, qui appara\u00eet en optimisation, en physique, en \u00e9conomie et en ing\u00e9nierie, constitue l\u2019une des principales applications du calcul diff\u00e9rentiel. Et voici l\u2019essentiel : [&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":4,"footnotes":""},"categories":[866,569],"tags":[],"class_list":["post-35386","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-calcul-differentiel","category-mathematiques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Maxima et Minima d\u2019une Fonction - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Ma\u00eetrisez les maxima et les minima gr\u00e2ce \u00e0 une approche moderne du calcul diff\u00e9rentiel : Weierstrass, points critiques, Rolle et valeur moyenne pour optimiser avec m\u00e9thode, et non avec intuition.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/fr\/maxima-et-minima-dune-fonction\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Maxima et Minima d\u2019une Fonction\" \/>\n<meta property=\"og:description\" content=\"Ma\u00eetrisez les maxima et les minima gr\u00e2ce \u00e0 une approche moderne du calcul diff\u00e9rentiel : Weierstrass, points critiques, Rolle et valeur moyenne pour optimiser avec m\u00e9thode, et non avec intuition.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/fr\/maxima-et-minima-dune-fonction\/\" \/>\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=\"2025-01-01T13:00:12+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-12-14T23:41:55+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=\"Maxima et Minima d\u2019une Fonction\" \/>\n<meta name=\"twitter:description\" content=\"Ma\u00eetrisez les maxima et les minima gr\u00e2ce \u00e0 une approche moderne du calcul diff\u00e9rentiel : Weierstrass, points critiques, Rolle et valeur moyenne pour optimiser avec m\u00e9thode, et non avec intuition.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/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=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/maxima-et-minima-dune-fonction\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/maxima-et-minima-dune-fonction\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Maxima et Minima d\u2019une Fonction\",\"datePublished\":\"2025-01-01T13:00:12+00:00\",\"dateModified\":\"2025-12-14T23:41:55+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/maxima-et-minima-dune-fonction\\\/\"},\"wordCount\":2210,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/maxima-et-minima-dune-fonction\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2025\\\/01\\\/maximum.jpg\",\"articleSection\":[\"Calcul Diff\u00e9rentiel\",\"Math\u00e9matiques\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/maxima-et-minima-dune-fonction\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/maxima-et-minima-dune-fonction\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/maxima-et-minima-dune-fonction\\\/\",\"name\":\"Maxima et Minima d\u2019une Fonction - toposuranos.com\\\/material\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/maxima-et-minima-dune-fonction\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/maxima-et-minima-dune-fonction\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2025\\\/01\\\/maximum.jpg\",\"datePublished\":\"2025-01-01T13:00:12+00:00\",\"dateModified\":\"2025-12-14T23:41:55+00:00\",\"description\":\"Ma\u00eetrisez les maxima et les minima gr\u00e2ce \u00e0 une approche moderne du calcul diff\u00e9rentiel : Weierstrass, points critiques, Rolle et valeur moyenne pour optimiser avec m\u00e9thode, et non avec intuition.\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/maxima-et-minima-dune-fonction\\\/#breadcrumb\"},\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/maxima-et-minima-dune-fonction\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/maxima-et-minima-dune-fonction\\\/#primaryimage\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2025\\\/01\\\/maximum.jpg\",\"contentUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2025\\\/01\\\/maximum.jpg\",\"width\":1536,\"height\":1024},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/maxima-et-minima-dune-fonction\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Portada\",\"item\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/cursos-de-matematica-y-fisica\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Maxima et Minima d\u2019une Fonction\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#website\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/\",\"name\":\"toposuranos.com\\\/material\",\"description\":\"\",\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"es\"},{\"@type\":\"Organization\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\",\"name\":\"toposuranos.com\\\/material\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/logo\\\/image\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/logo.png\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/logo.png\",\"width\":2400,\"height\":2059,\"caption\":\"toposuranos.com\\\/material\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/logo\\\/image\\\/\"},\"sameAs\":[\"https:\\\/\\\/www.facebook.com\\\/groups\\\/toposuranos\",\"https:\\\/\\\/x.com\\\/topuranos\",\"https:\\\/\\\/www.youtube.com\\\/channel\\\/UC16yDm12cPcrwsE0fAM7X1g\",\"https:\\\/\\\/www.linkedin.com\\\/company\\\/69429190\"]},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\",\"name\":\"giorgio.reveco\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"caption\":\"giorgio.reveco\"},\"description\":\"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.\",\"sameAs\":[\"http:\\\/\\\/toposuranos.com\\\/material\"],\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/author\\\/giorgio-reveco\\\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Maxima et Minima d\u2019une Fonction - toposuranos.com\/material","description":"Ma\u00eetrisez les maxima et les minima gr\u00e2ce \u00e0 une approche moderne du calcul diff\u00e9rentiel : Weierstrass, points critiques, Rolle et valeur moyenne pour optimiser avec m\u00e9thode, et non avec intuition.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/toposuranos.com\/material\/fr\/maxima-et-minima-dune-fonction\/","og_locale":"es_ES","og_type":"article","og_title":"Maxima et Minima d\u2019une Fonction","og_description":"Ma\u00eetrisez les maxima et les minima gr\u00e2ce \u00e0 une approche moderne du calcul diff\u00e9rentiel : Weierstrass, points critiques, Rolle et valeur moyenne pour optimiser avec m\u00e9thode, et non avec intuition.","og_url":"https:\/\/toposuranos.com\/material\/fr\/maxima-et-minima-dune-fonction\/","og_site_name":"toposuranos.com\/material","article_publisher":"https:\/\/www.facebook.com\/groups\/toposuranos","article_published_time":"2025-01-01T13:00:12+00:00","article_modified_time":"2025-12-14T23:41:55+00:00","og_image":[{"url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/01\/maximum-1024x683.jpg","type":"","width":"","height":""}],"author":"giorgio.reveco","twitter_card":"summary_large_image","twitter_title":"Maxima et Minima d\u2019une Fonction","twitter_description":"Ma\u00eetrisez les maxima et les minima gr\u00e2ce \u00e0 une approche moderne du calcul diff\u00e9rentiel : Weierstrass, points critiques, Rolle et valeur moyenne pour optimiser avec m\u00e9thode, et non avec intuition.","twitter_image":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/01\/maximum.jpg","twitter_creator":"@topuranos","twitter_site":"@topuranos","twitter_misc":{"Escrito por":"giorgio.reveco","Tiempo de lectura":"1 minuto"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/toposuranos.com\/material\/fr\/maxima-et-minima-dune-fonction\/#article","isPartOf":{"@id":"https:\/\/toposuranos.com\/material\/fr\/maxima-et-minima-dune-fonction\/"},"author":{"name":"giorgio.reveco","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1"},"headline":"Maxima et Minima d\u2019une Fonction","datePublished":"2025-01-01T13:00:12+00:00","dateModified":"2025-12-14T23:41:55+00:00","mainEntityOfPage":{"@id":"https:\/\/toposuranos.com\/material\/fr\/maxima-et-minima-dune-fonction\/"},"wordCount":2210,"commentCount":0,"publisher":{"@id":"https:\/\/toposuranos.com\/material\/#organization"},"image":{"@id":"https:\/\/toposuranos.com\/material\/fr\/maxima-et-minima-dune-fonction\/#primaryimage"},"thumbnailUrl":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/01\/maximum.jpg","articleSection":["Calcul Diff\u00e9rentiel","Math\u00e9matiques"],"inLanguage":"es","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/toposuranos.com\/material\/fr\/maxima-et-minima-dune-fonction\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/toposuranos.com\/material\/fr\/maxima-et-minima-dune-fonction\/","url":"https:\/\/toposuranos.com\/material\/fr\/maxima-et-minima-dune-fonction\/","name":"Maxima et Minima d\u2019une Fonction - toposuranos.com\/material","isPartOf":{"@id":"https:\/\/toposuranos.com\/material\/#website"},"primaryImageOfPage":{"@id":"https:\/\/toposuranos.com\/material\/fr\/maxima-et-minima-dune-fonction\/#primaryimage"},"image":{"@id":"https:\/\/toposuranos.com\/material\/fr\/maxima-et-minima-dune-fonction\/#primaryimage"},"thumbnailUrl":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/01\/maximum.jpg","datePublished":"2025-01-01T13:00:12+00:00","dateModified":"2025-12-14T23:41:55+00:00","description":"Ma\u00eetrisez les maxima et les minima gr\u00e2ce \u00e0 une approche moderne du calcul diff\u00e9rentiel : Weierstrass, points critiques, Rolle et valeur moyenne pour optimiser avec m\u00e9thode, et non avec intuition.","breadcrumb":{"@id":"https:\/\/toposuranos.com\/material\/fr\/maxima-et-minima-dune-fonction\/#breadcrumb"},"inLanguage":"es","potentialAction":[{"@type":"ReadAction","target":["https:\/\/toposuranos.com\/material\/fr\/maxima-et-minima-dune-fonction\/"]}]},{"@type":"ImageObject","inLanguage":"es","@id":"https:\/\/toposuranos.com\/material\/fr\/maxima-et-minima-dune-fonction\/#primaryimage","url":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/01\/maximum.jpg","contentUrl":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/01\/maximum.jpg","width":1536,"height":1024},{"@type":"BreadcrumbList","@id":"https:\/\/toposuranos.com\/material\/fr\/maxima-et-minima-dune-fonction\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Portada","item":"http:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/"},{"@type":"ListItem","position":2,"name":"Maxima et Minima d\u2019une Fonction"}]},{"@type":"WebSite","@id":"https:\/\/toposuranos.com\/material\/#website","url":"https:\/\/toposuranos.com\/material\/","name":"toposuranos.com\/material","description":"","publisher":{"@id":"https:\/\/toposuranos.com\/material\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/toposuranos.com\/material\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"es"},{"@type":"Organization","@id":"https:\/\/toposuranos.com\/material\/#organization","name":"toposuranos.com\/material","url":"https:\/\/toposuranos.com\/material\/","logo":{"@type":"ImageObject","inLanguage":"es","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","width":2400,"height":2059,"caption":"toposuranos.com\/material"},"image":{"@id":"https:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/groups\/toposuranos","https:\/\/x.com\/topuranos","https:\/\/www.youtube.com\/channel\/UC16yDm12cPcrwsE0fAM7X1g","https:\/\/www.linkedin.com\/company\/69429190"]},{"@type":"Person","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1","name":"giorgio.reveco","image":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","caption":"giorgio.reveco"},"description":"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.","sameAs":["http:\/\/toposuranos.com\/material"],"url":"https:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/"}]}},"_links":{"self":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/35386","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/comments?post=35386"}],"version-history":[{"count":0,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/35386\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media\/35349"}],"wp:attachment":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media?parent=35386"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/categories?post=35386"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/tags?post=35386"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}