{"id":5670,"date":"2021-09-27T19:45:17","date_gmt":"2021-09-27T14:15:17","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5670"},"modified":"2021-11-13T17:27:37","modified_gmt":"2021-11-13T11:57:37","slug":"first-derivative-test-for-maxima-and-minima","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/","title":{"rendered":"First Derivative Test for Maxima and Minima"},"content":{"rendered":"

Here you will learn first derivative test for maxima and minima with examples.<\/p>\n

Let’s begin –<\/p>\n

First Derivative Test for Maxima and Minima<\/h2>\n

If f'(x) = 0 at a point (say x = a) and <\/p>\n

\n

(i) If f'(x) changes sign from positive to negative in the neighbourhood of x = a then x = a is said to be a point local maxima.<\/strong><\/p>\n

(ii) If f'(x) changes sign from negative to positive in the neighbourhood of x = a then x = a is said to be a point local minima.<\/strong><\/p>\n<\/blockquote>\n

Note<\/strong> – If f'(x)  does not change sign i.e. has the same sign in a certain complete neighbourhood of a, then f(x) is either increasing or decreasing throughout this neighbourhood implying that x = a is not a point of extremum of f.<\/p>\n

Example<\/strong><\/span> : Let f(x) = x + \\(1\\over x\\) ; x \\(\\ne\\) 0. Discuss the local maximum and local minimum values of f(x)<\/p>\n

Solution<\/span><\/strong> : Here f'(x) = 1 – \\(1\\over x^2\\) = \\(x^2 – 1\\over x^2\\)<\/p>\n

= \\((x – 1)(x + 1)\\over x^2\\)<\/p>\n

Using local number line rule, \"line\"<\/p>\n

f(x)  will have local maximum at x = – 1 and local minimum at x =1 <\/p>\n

\\(\\therefore\\) local maximum value of f(x) = -2 at x = -1<\/p>\n

and local minimum value of f(x) = 2 at x = 1<\/p>\n

Example<\/strong><\/span> : Let f(x) = \\(x^3 – 3x\\). Discuss the local maximum and local minimum values of f(x)<\/p>\n

Solution<\/span><\/strong> : Here f'(x) = \\(3x^2 -3\\) = 3(x + 1)(x – 1)<\/p>\n

Using local number line rule, \"line\"<\/p>\n

f(x)  will have local maximum at x = – 1 and local minimum at x =1 <\/p>\n

\\(\\therefore\\) local maximum value of f(x) = 2 at x = -1<\/p>\n

and local minimum value of f(x) = -2 at x = 1<\/p>\n\n\n

\n
Next – Second Derivative Test for Maxima and Minima<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Maxima and Minima Class 12<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn first derivative test for maxima and minima with examples. Let’s begin – First Derivative Test for Maxima and Minima If f'(x) = 0 at a point (say x = a) and  (i) If f'(x) changes sign from positive to negative in the neighbourhood of x = a then x = a …<\/p>\n

First Derivative Test for Maxima and Minima<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[37],"tags":[76,77,80,81,79],"yoast_head":"\nFirst Derivative Test for Maxima and Minima - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn first derivative test for maxima and minima with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"First Derivative Test for Maxima and Minima - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn first derivative test for maxima and minima with examples.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-09-27T14:15:17+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-11-13T11:57:37+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathemerize.com\/wp-content\/uploads\/2021\/09\/line-e1632580446507.png\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"First Derivative Test for Maxima and Minima\",\"datePublished\":\"2021-09-27T14:15:17+00:00\",\"dateModified\":\"2021-11-13T11:57:37+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/\"},\"wordCount\":301,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"keywords\":[\"calculus\",\"derivative\",\"first derivative test\",\"first derivative test for maxima and minima\",\"local maxima and minima class 12\"],\"articleSection\":[\"Application of Derivatives\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/\",\"url\":\"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/\",\"name\":\"First Derivative Test for Maxima and Minima - Mathemerize\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-09-27T14:15:17+00:00\",\"dateModified\":\"2021-11-13T11:57:37+00:00\",\"description\":\"In this post you will learn first derivative test for maxima and minima with examples.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"First Derivative Test for Maxima and Minima\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - Study Math Online\",\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathemerize.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathemerize.com\/#organization\",\"name\":\"Mathemerize\",\"url\":\"https:\/\/mathemerize.com\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"contentUrl\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"width\":140,\"height\":96,\"caption\":\"Mathemerize\"},\"image\":{\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.instagram.com\/mathemerize\/\"]},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\",\"name\":\"mathemerize\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"caption\":\"mathemerize\"},\"sameAs\":[\"https:\/\/mathemerize.com\"],\"url\":\"https:\/\/mathemerize.com\/author\/mathemerize\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"First Derivative Test for Maxima and Minima - Mathemerize","description":"In this post you will learn first derivative test for maxima and minima with examples.","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:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/","og_locale":"en_US","og_type":"article","og_title":"First Derivative Test for Maxima and Minima - Mathemerize","og_description":"In this post you will learn first derivative test for maxima and minima with examples.","og_url":"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/","og_site_name":"Mathemerize","article_published_time":"2021-09-27T14:15:17+00:00","article_modified_time":"2021-11-13T11:57:37+00:00","og_image":[{"url":"https:\/\/mathemerize.com\/wp-content\/uploads\/2021\/09\/line-e1632580446507.png"}],"author":"mathemerize","twitter_card":"summary_large_image","twitter_misc":{"Written by":"mathemerize","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"First Derivative Test for Maxima and Minima","datePublished":"2021-09-27T14:15:17+00:00","dateModified":"2021-11-13T11:57:37+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/"},"wordCount":301,"commentCount":0,"publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"keywords":["calculus","derivative","first derivative test","first derivative test for maxima and minima","local maxima and minima class 12"],"articleSection":["Application of Derivatives"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/","url":"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/","name":"First Derivative Test for Maxima and Minima - Mathemerize","isPartOf":{"@id":"https:\/\/mathemerize.com\/#website"},"datePublished":"2021-09-27T14:15:17+00:00","dateModified":"2021-11-13T11:57:37+00:00","description":"In this post you will learn first derivative test for maxima and minima with examples.","breadcrumb":{"@id":"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathemerize.com\/first-derivative-test-for-maxima-and-minima\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathemerize.com\/"},{"@type":"ListItem","position":2,"name":"First Derivative Test for Maxima and Minima"}]},{"@type":"WebSite","@id":"https:\/\/mathemerize.com\/#website","url":"https:\/\/mathemerize.com\/","name":"Mathemerize","description":"Maths Tutorials - Study Math Online","publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathemerize.com\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/mathemerize.com\/#organization","name":"Mathemerize","url":"https:\/\/mathemerize.com\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/","url":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","contentUrl":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","width":140,"height":96,"caption":"Mathemerize"},"image":{"@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.instagram.com\/mathemerize\/"]},{"@type":"Person","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df","name":"mathemerize","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","caption":"mathemerize"},"sameAs":["https:\/\/mathemerize.com"],"url":"https:\/\/mathemerize.com\/author\/mathemerize\/"}]}},"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/5670"}],"collection":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/comments?post=5670"}],"version-history":[{"count":7,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/5670\/revisions"}],"predecessor-version":[{"id":5717,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/5670\/revisions\/5717"}],"wp:attachment":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/media?parent=5670"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/categories?post=5670"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/tags?post=5670"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}