{"id":5583,"date":"2021-09-22T00:24:36","date_gmt":"2021-09-21T18:54:36","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5583"},"modified":"2021-11-14T17:58:52","modified_gmt":"2021-11-14T12:28:52","slug":"mean-value-theorems-class-12","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/mean-value-theorems-class-12\/","title":{"rendered":"Mean Value Theorems Class 12"},"content":{"rendered":"

Here you will learn mean value theorems i.e rolle’s theorem, lagrange’s theorem and extreme value theorem.<\/p>\n

Let’s begin –<\/p>\n

Mean Value Theorems<\/h2>\n

(a) Rolle’s Theorem<\/a><\/strong><\/h4>\n

Let f be a real valued function defined on the closed interval [a, b] such that<\/p>\n

\n

(i) it is continuous on the closed interval [a, b],<\/p>\n

(ii) it is differentiable on the open interval (a, b)<\/p>\n

(iii) f(a) = f(b)<\/p>\n

Then, there exist a real number c \\(\\in\\) (a, b) such that f'(c) = 0<\/p>\n<\/blockquote>\n

Note<\/strong> : If f is differentiable function then between any two consecutive roots of f(x) = 0, there is atleast one root of the equation f'(x) = 0<\/p>\n

(b) Lagrange’s Mean Value Theorem (LMVT)<\/a><\/strong><\/h4>\n

Let f be a function that satisfies the following conditions :<\/p>\n

\n

(i) f is continuous in [a, b]<\/p>\n

(ii) f is differentiable in (a, b).<\/p>\n

Then there is a number c in (a, b) such that f'(c) = \\(f(b) – f(a)\\over b – a\\)<\/p>\n<\/blockquote>\n

Extreme Value Theorem\u00a0<\/h3>\n

If f is continuous on [a, b] then f takes on,\u00a0 a least value m and a greatest value M on this interval.<\/p>\n

Note<\/strong> : Continuity throught the interval [a, b] is essential for the validity of this theorem.<\/p>\n

(a) If a continuous function y = f(x) is increasing in the closed interval [a, b] , then f(a) is the least value and f(b) is the greatest value of f(x) in [a, b]<\/p>\n

(b) If a continuous function y = f(x) is decreasing in the closed interval [a, b] , then f(b) is the least value and f(a) is the greatest value of f(x) in [a, b]<\/p>\n

(c) If a continuous function y = f(x) is increasing\/decreasing in the (a, b) , then no greatest and least value exist.<\/p>\n\n\n

\n
Next – Rolle\u2019s Theorem \u2013 Statement & Examples<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Differentials Errors and Approximations<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn mean value theorems i.e rolle’s theorem, lagrange’s theorem and extreme value theorem. Let’s begin – Mean Value Theorems (a) Rolle’s Theorem Let f be a real valued function defined on the closed interval [a, b] such that (i) it is continuous on the closed interval [a, b], (ii) it is differentiable …<\/p>\n

Mean Value Theorems Class 12<\/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":[76,106,113,112],"yoast_head":"\nMean Value Theorems Class 12 - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn mean value theorems i.e rolle's theorem and lagrange's theorem of class 12.\" \/>\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\/mean-value-theorems-class-12\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Mean Value Theorems Class 12 - 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