{"id":35391,"date":"2025-01-01T13:00:22","date_gmt":"2025-01-01T13:00:22","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35391"},"modified":"2025-12-14T23:46:16","modified_gmt":"2025-12-14T23:46:16","slug":"maxima-et-minima-functionis","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/maxima-et-minima-functionis\/","title":{"rendered":"Maxima et Minima Functionis"},"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 Functionis<\/h1>\n<p style=\"text-align:center\"><em>Ubi est punctum \u201coptimum\u201d functionis: maximum quod assequi vis an minimum quod vitare debes? Haec quaestio, quae in optimizatione, physica, oeconomia et ingeniaria occurrit, una ex praecipuis applicationibus calculi differentialis est. Hic autem momentum apparet: <a href=\"https:\/\/toposuranos.com\/material\/la\/theorema-weierstrassii-de-valoribus-extremis\/\" rel=\"noopener\" target=\"_blank\">Theorema Weierstrassii<\/a> tibi confirmat quod, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> continua est et in intervallo clauso et finito operaris, tunc <strong>extrema absoluta exsistunt<\/strong>. Inde ludus fit practicus: discere <strong>extrema localia<\/strong> deprehendere per <strong>puncta critica<\/strong> (<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f&#039;(x)=0<\/span><\/span> vel non exsistit) atque uti instrumentis ut Rolle et Valor Medius, ut inquisitio \u201ccaeca\u201d in methodum claram, verificabilem et efficientem convertatur.<\/em><\/p>\n<p style=\"text-align:center\">\n<strong>Proposita Discendi:<\/strong>\n<\/p>\n<ol>\n<li><b>Exsequi<\/b> processum completum ad extrema absoluta in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span> inveniendum: aestimare <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> in punctis criticis interioribus et in extremis intervalli, atque valores comparare ad maximum et minimum absolutum statuendum.\n  <\/li>\n<li><strong>Comparare<\/strong> valorem condicionis necessariae contra sufficientem: agnoscere quod \u201c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f&#039;(x_0)=0<\/span><\/span>\u201d extremum locale non praestat, et statuere quae argumenta addita (comparatio valorum, analysis signorum, mores locales) in singulis casibus sint opportuna.\n  <\/li>\n<li><strong>Determinare<\/strong> rationem efficacissimam secundum genus problematis: extrema absoluta in intervallis compactis (Weierstrass + aestimatio finita) contra extrema localia in punctis interioribus (puncta critica + analysis localis), electione rationabiliter iustificata.\n  <\/li>\n<\/ol>\n<p style=\"text-align:center;\">\n<strong><u>INDEX CONTINENTIUM<\/u>:<\/strong><br \/>\n<a href=\"#1\"><strong>Maxima et minima, extrema absoluta et localia<\/strong><\/a><br \/>\n<a href=\"#2\">Criterium Primae Derivatae<\/a><br \/>\n<a href=\"#3\"><strong>Theorema Rollii<\/strong><\/a><br \/>\n<a href=\"#4\"><strong>Theorema Valorii Medii Differentialis<\/strong><\/a><br \/>\n<a href=\"#5\">Intervalla incrementi et decrementi<\/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  <a href=\"https:\/\/toposuranos.com\/material\/la\/theorema-weierstrassii-de-valoribus-extremis\/\" rel=\"noopener\" target=\"_blank\">Theorema Weierstrassii<\/a> nobis confirmat quod, si functio realis definita est et continua in subcollectione clausa et finita <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}<\/span>, tum necessario valores maximum et minimum attingit (extrema absoluta). Investigatio maximorum et minimorum functionis est id quod <strong>problema optimizationis<\/strong> appellatur, atque theorema Weierstrassii nobis existentiam solutionum in sensu extremorum absolutorum praestat, dummodo functio continua sit et dominium compactum. Existentia igitur iam confirmata, nunc solum restat rationes evolvere quae has solutiones reperire permittant.\n<\/p>\n<p><a name=\"1\"><\/a><\/br><\/p>\n<h2>Maxima et minima, extrema absoluta et localia<\/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;\">Antequam incipiamus recensere<\/span><\/a> rationes ad maxima et minima quaerenda, clare definiamus quid sit id quod quaerere volumus.<\/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>DEFINITIO:<\/strong><\/span><br \/>\n          Sit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> functio cum dominio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">D<\/span><\/span>. Dicemus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> attingere <strong>maximum absolutum<\/strong> in puncto <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          atque attinget <strong>minimum absolutum<\/strong> in <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  Similiter definiuntur extrema localia (relativa ad dominium).\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>DEFINITIO:<\/strong><\/span><br \/>\n          Sit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> functio cum dominio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">D<\/span><\/span> et sit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in D<\/span><\/span>. Dicemus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> attingere <strong>maximum locale<\/strong> in <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          atque attinget <strong>minimum locale<\/strong> in <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  Ex his possumus enuntiare sequens eventum:\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>THEOREMA:<\/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;\">Sit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> punctum<\/span><\/strong><\/a> interius intervalli compacti <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> attingit maximum vel minimum <strong>locale<\/strong> in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x_0)<\/span><\/span> exsistit, tunc <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>DEMONSTRATIO:<\/strong><\/span><br \/>\n          Supponamus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> attingere maximum locale in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>. Tunc exsistit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h_0 \\gt 0<\/span><\/span> tale ut, pro omni <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h<\/span><\/span> cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|h|\\lt h_0<\/span><\/span> et cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0+h\\in I<\/span><\/span>, valeat:\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          quod aequivalet:\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          Nunc duo casus consideremus:\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>, tunc:\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>, tunc:\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> exsistit, tunc limes quotientis incrementalis cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h\\to 0<\/span><\/span> exsistit et utrique inaequalitati congruere debet, quod cogit ut:\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          Quod erat demonstrandum.\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\n  Animadvertendum est hanc demonstrationem etiam pro minimis localibus valere. Hoc in casu initium sumitur a: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x_0+h)\\ge f(x_0)<\/span><\/span> pro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|h|<\/span><\/span> satis parvo.\n<\/p>\n<p><a name=\"2\"><\/a><\/br><\/p>\n<h3>Criterium Derivatae Primae<\/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;\">Eventus quem modo recensuimus<\/span><\/strong><\/a> in sequenti implicatione summari potest:<\/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{ attingit}\\\\ \\text{extremum locale in }x_0 \\end{matrix}\\right\\}\n\n  \\Longrightarrow\n\n  \\left\\{\\begin{matrix} \\displaystyle f^\\prime(x_0) = 0 \\\\ \\\\ \\vee \\\\ \\\\ \\text{Derivata non exsistit in }x_0 \\end{matrix}\\right\\}\n\n  <\/span><\/span>\n<\/p>\n<p style=\"text-align: justify;\">\n  Quamquam reciprocum huius implicationis generaliter non valet, tamen admodum utile est ad inquisitionem extremorum localium circumscribendam. Ex hoc definiuntur puncta critica derivatae primae.\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>DEFINITIO:<\/strong><\/span><br \/>\n          Dicitur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> esse <strong>punctum criticum derivatae primae<\/strong> si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x_0)=0<\/span><\/span> aut si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x_0)<\/span><\/span> non exsistit.\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\n  Puncta critica derivatae primae sunt momenti, quia omne punctum in quo functio extremizat (localiter vel absolute) ad collectionem punctorum criticorum pertinere debet:\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{puncta quae}\\\\ \\text{absolute extremizant}\\end{matrix}\\right\\}\n\n  \\subseteq\n\n  \\left\\{\\begin{matrix}\\text{puncta quae}\\\\ \\text{localiter extremizant}\\end{matrix}\\right\\}\n\n  \\subseteq\n\n  \\left\\{\\begin{matrix}\\text{puncta critica}\\\\ \\text{derivatae primae}\\end{matrix}\\right\\}\n\n  <\/span><\/span>\n<\/p>\n<p style=\"text-align: justify;\">\n  Hoc est quod <strong>criterium derivatae primae<\/strong> appellamus, intellectum ut condicionem necessariam ad existentiam extremorum localium in punctis interioribus.\n<\/p>\n<p><a name=\"3\"><\/a><\/br><\/p>\n<h2>Theorema Rollii<\/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;\">Iam vidimus determinationem<\/span><\/strong><\/a> punctorum criticorum derivatae primae esse praecipuam in investigatione extremorum localium. Quapropter naturale est inquirere sub quibus condicionibus existentia talium punctorum criticorum certari possit. Progressus in hoc sensu per theorema Rollii obtinetur.<\/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>THEOREMA:<\/strong><\/span><br \/>\n          Sit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> functio definita et continua in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span>, atque derivabilis in <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>, tunc exsistit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span> tale ut <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>DEMONSTRATIO:<\/strong><\/span><br \/>\n          Duas possibilitates considerabimus:\n        <\/p>\n<ol style=\"text-align: justify;\">\n<li>\n            Si pro omni <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]a,b[<\/span><\/span> valet <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=f(a)=f(b)<\/span><\/span>, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> est constans et, consequenter, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(x)=0<\/span><\/span> pro omni <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]a,b[<\/span><\/span>. In particulari, exsistit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span> cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c)=0<\/span><\/span>.\n          <\/li>\n<li>\n            Si exsistit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]a,b[<\/span><\/span> tale ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)\\neq f(a)=f(b)<\/span><\/span>, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> non est constans. Cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> sit continua in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span>, per theorema Weierstrassii maximum absolutum et minimum absolutum in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span> attingit.<\/p>\n<p>Praeterea, cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(a)=f(b)<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> non sit constans, saltem unum ex his extremis in interiori <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span> contingere debet.<\/p>\n<p>Itaque, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span> est punctum interius ubi <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> extremum locale attingit, cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> derivabilis sit in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span>, in particulari <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c)<\/span><\/span> exsistit, et per theorema superius concluditur <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>Theorema Valorii Medii Differentialis<\/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;\">Aliud eventum quod est directa consequentia<\/span><\/strong><\/a> eorum quae modo recensuimus, atque quod informationem utilem ad studium functionum affert, est theorema valoris medii ad calculum differentialem.\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>THEOREMA:<\/strong><\/span><br \/>\n          Sit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> functio definita et continua in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span>, atque derivabilis in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span>. Tunc exsistit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span> tale ut:\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>DEMONSTRATIO:<\/strong><\/span><br \/>\n          Sit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> functio definita per:\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          Haec functio est continua in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span> et derivabilis in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span> quia <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> quoque ita est. Praeterea, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(a)=F(b)<\/span><\/span>, ita ut theorema Rollii adhibere possimus ad concludendum exsistere punctum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]a,b[<\/span><\/span> tale ut <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          Nunc, derivando <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> obtinetur:\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          Aestimando in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> et utens <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          Inde:\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          Quod erat demonstrandum.\n        <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"5\"><\/a><\/br><\/p>\n<h3>Intervalla incrementi et decrementi<\/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>THEOREMA:<\/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 functio<\/span><\/strong><\/a> talis ut <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>, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> est stricte crescens in <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 functio talis ut <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>, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> est stricte decrescens in <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>DEMONSTRATIO:<\/strong><\/span><br \/>\n          Sint <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_1,x_2\\in ]a,b[<\/span><\/span> tales ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_1 \\lt x_2<\/span><\/span>. Cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> derivabilis sit in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]a,b[<\/span><\/span>, theorema valoris medii ad <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> super intervallum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[x_1,x_2]\\subset ]a,b[<\/span><\/span> applicare possumus. Quapropter exsistit punctum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in]x_1,x_2[<\/span><\/span> tale ut:\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          Ex hoc sequitur:\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>, tunc <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 \/>\nQuapropter <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> est crescens.\n          <\/li>\n<li>\n            Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f^\\prime(c) \\lt 0<\/span><\/span>, tunc <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 \/>\nQuapropter <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> est decrescens.\n          <\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\nStudium maximorum et minimorum non solum est \u201cderivatas computare\u201d, sed discere quomodo inquisitio diffusa in proceduram cum cautionibus et criteriis claris convertatur. Weierstrass indicat quando fidere possis optimum in intervallo compacto exsistere, dum criterium derivatae primae, theorema Rollii et Theorema Valorii Medii tibi tradunt quasi mappam ad candidatos inveniendos et conclusiones iustificandas: ubi functio extremizare possit, quando illa condicio tantum necessaria sit, et quomodo signum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f&#039;<\/span><\/span> incrementum et decrementum detegat. Si hanc idearum seriem dominas, a contemplatione graphorum sola intuitione ad solutionem problematum optimizationis cum argumentis verificabilibus transis, quod est prorsus discrimen inter \u201cputo hic esse punctum optimum\u201d et \u201cscio cur hic esse debeat\u201d.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Maxima et Minima Functionis Ubi est punctum \u201coptimum\u201d functionis: maximum quod assequi vis an minimum quod vitare debes? Haec quaestio, quae in optimizatione, physica, oeconomia et ingeniaria occurrit, una ex praecipuis applicationibus calculi differentialis est. Hic autem momentum apparet: Theorema Weierstrassii tibi confirmat quod, si continua est et in intervallo clauso et finito operaris, tunc [&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":[1328,1298],"tags":[],"class_list":["post-35391","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-calculus-differentialis","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Maxima et Minima Functionis - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Maxima et minima domina per prospectum modernum calculi differentialis: Weierstrass, puncta critica, theorema Rollii et valor medius ad optimizationem methodo, non sola intuitione.\" \/>\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\/la\/maxima-et-minima-functionis\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Maxima et Minima Functionis\" \/>\n<meta property=\"og:description\" content=\"Maxima et minima domina per prospectum modernum calculi differentialis: Weierstrass, puncta critica, theorema Rollii et valor medius ad optimizationem methodo, non sola intuitione.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/la\/maxima-et-minima-functionis\/\" \/>\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=\"2025-01-01T13:00:22+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-12-14T23:46:16+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=\"Maxima et minima domina per prospectum modernum calculi differentialis: Weierstrass, puncta critica, theorema Rollii et valor medius ad optimizationem methodo, non sola intuitione.\" \/>\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=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\/\/toposuranos.com\/material\/la\/maxima-et-minima-functionis\/#article\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/maxima-et-minima-functionis\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Maxima et Minima Functionis\",\"datePublished\":\"2025-01-01T13:00:22+00:00\",\"dateModified\":\"2025-12-14T23:46:16+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/maxima-et-minima-functionis\/\"},\"wordCount\":1648,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/maxima-et-minima-functionis\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/01\/maximum.jpg\",\"articleSection\":[\"Calculus Differentialis\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\/\/toposuranos.com\/material\/la\/maxima-et-minima-functionis\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\/\/toposuranos.com\/material\/la\/maxima-et-minima-functionis\/\",\"url\":\"http:\/\/toposuranos.com\/material\/la\/maxima-et-minima-functionis\/\",\"name\":\"Maxima et Minima Functionis - 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