{"id":5672,"date":"2021-09-27T19:46:23","date_gmt":"2021-09-27T14:16:23","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5672"},"modified":"2021-11-13T17:12:10","modified_gmt":"2021-11-13T11:42:10","slug":"second-derivative-test-for-maxima-and-minima","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/second-derivative-test-for-maxima-and-minima\/","title":{"rendered":"Second Derivative Test for Maxima and Minima"},"content":{"rendered":"

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

Let’s begin –<\/p>\n

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

If f(x) is continuous and differentiable at x = a where f'(a) = 0 and f”(a) also exists then for ascertaining maxima\/minima at x = a, 2nd dervative can be used<\/p>\n

\n

(i) f”(a) > 0 \\(\\implies\\) x = a is a point of local minima<\/p>\n

(ii) f”(a) < 0 \\(\\implies\\) x = a is a point of local maxima<\/p>\n

(iii) f”(a) = 0 \\(\\implies\\) second derivative test fails.\u00a0 To identify maxima\/minima at this point either first derivative test or higher derivative test can be used.<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : find all the points of local maxima and minima and the corresponding maximum and minimum values of the function f(x) = \\(2x^3 – 21x^2 + 36x – 20\\).<\/p>\n

Solution<\/strong><\/span> : We have,\u00a0<\/p>\n

f(x) = \\(2x^3 – 21x^2 + 36x – 20\\)<\/p>\n

\\(\\implies\\) f'(x) = \\(6x^2 – 42x +36\\)<\/p>\n

The critical points of f(x) are given by f'(x) = 0.<\/p>\n

Now, f'(x) = 0 \\(\\implies\\) \\(6x^2 – 42x +36\\)\u00a0<\/p>\n

\\(\\implies\\) (x – 1)(x – 6) = 0 \\(\\implies\\) x = 1, 6.<\/p>\n

Thus, x = 1 and x =6 are the possible points of local maxima and minima.<\/p>\n

Now, we test the function at each of thes points.<\/p>\n

We have, f”(x) = 12x – 42<\/p>\n

At x = 1 : we have,<\/p>\n

f”(1) = 12 – 42 = -30 < 0<\/p>\n

So, x = 1 is a point of local maximum.<\/p>\n

The local maximum value is f(1) = 2 – 21 + 36 – 20 = -3<\/p>\n

At x = 6, We have,<\/p>\n

f”(6) = 12(6) – 42 = 30 > 0<\/p>\n

So, x = 6 is a point of local minimum.<\/p>\n

The local minimum value is f(6) = -128<\/p>\n\n\n

\n
Next – What is the Point of Inflection<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – First Derivative Test for Maxima and Minima<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn second derivative test for maxima and minima with examples. Let’s begin – Second Derivative Test for Maxima and Minima If f(x) is continuous and differentiable at x = a where f'(a) = 0 and f”(a) also exists then for ascertaining maxima\/minima at x = a, 2nd dervative can be used (i) …<\/p>\n

Second 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":[76,77,78,72,74,75,73],"yoast_head":"\nSecond Derivative Test for Maxima and Minima - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn second 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\/second-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=\"Second Derivative Test for Maxima and Minima - 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