{"id":5587,"date":"2021-09-22T02:26:53","date_gmt":"2021-09-21T20:56:53","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5587"},"modified":"2021-11-14T17:20:13","modified_gmt":"2021-11-14T11:50:13","slug":"lagranges-mean-value-theorem","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/lagranges-mean-value-theorem\/","title":{"rendered":"Lagrange’s Mean Value Theorem"},"content":{"rendered":"

Here you will learn lagrange’s mean value theorem statement, its geometrical and physical interpretation with examples.<\/p>\n

Let’s begin – <\/p>\n

Lagrange’s Mean Value Theorem (LMVT)<\/h2>\n
\n

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

Geometrical Interpretation :<\/strong><\/h4>\n
\n

Geometrically, the lagrange’s mean value theorem says that somewhere between A and B the curve has atleast on tangent parallel to chord AB.<\/p>\n<\/blockquote>\n

Physical Interpretation :<\/strong><\/h4>\n
\n

If we think  of the number (f(b) – f(a))\/(b – a) as the average change in f over [a, b] and f'(c) as an instantaneous change, then the mean value theorem says that at some interior point the instantaneous change must equal the average change over the entire interval.<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : find c of the Lagranges mean value theorem for the function f(x) = \\(3x^2\\) + 5x + 7 in the interval [1, 3].<\/p>\n

Solution<\/strong><\/span> : Given f(x) = \\(3x^2\\) + 5x + 7<\/p>\n

\\(\\implies\\) f(1) = 3 + 5 + 7 = 15 and f(3) = 27 + 15 + 7 = 49<\/p>\n

Now, Differentiating f(x) with respect to x,<\/p>\n

\\(\\implies\\) f'(x) = 6x + 5<\/p>\n

Here a = 1, b = 3<\/p>\n

Now from lagrange’s mean value theorem <\/p>\n

f'(c) = \\(f(b) – f(a)\\over b – a\\) \\(\\implies\\) 6c + 5 = \\(49 – 15\\over 2\\) = 17<\/p>\n

\\(\\implies\\) c = 2<\/p>\n

Example<\/strong><\/span> : If f(x) is continuous and differentiable over [-2, 5] and -4 < f'(x) < 3 for all x in (-2, 5), then the greatest possible value of f(5) – f(-2) is –<\/p>\n

Solution<\/strong><\/span> : Applying Lagranges mean value theorem (LMVT),<\/p>\n

f'(x) = \\(f(5) – f(-2)\\over 5 -(-2)\\) for some x in (-2, 5) <\/p>\n

Now, -4 \\(\\le\\) \\(f(5) – f(-2)\\over 7\\) \\(\\le\\) 3<\/p>\n

-28 \\(\\le\\) f(5) – f(-2) \\(\\le\\) 21<\/p>\n

\\(\\therefore\\)  Greatest possible value of f(5) – f(-2) is 21.<\/p>\n\n\n

\n
Next – Slopes of Tangent and Normal to the Curve<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Rolle\u2019s Theorem \u2013 Statement & Examples<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn lagrange’s mean value theorem statement, its geometrical and physical interpretation with examples. Let’s begin –  Lagrange’s Mean Value Theorem (LMVT) Statement : Let f be a function that satisfies the following conditions : (i) f is continuous in [a, b] (ii) f is differentiable in (a, b) Then there is a …<\/p>\n

Lagrange’s Mean Value Theorem<\/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":[76,104,105,107,106],"yoast_head":"\nLagrange's Mean Value Theorem - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn lagrange's mean value theorem statement, its geometrical and physical interpretation 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\/lagranges-mean-value-theorem\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Lagrange's Mean Value Theorem - 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