{"id":5674,"date":"2021-09-27T19:47:38","date_gmt":"2021-09-27T14:17:38","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5674"},"modified":"2021-11-13T16:55:49","modified_gmt":"2021-11-13T11:25:49","slug":"what-is-the-point-of-inflection","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/what-is-the-point-of-inflection\/","title":{"rendered":"What is the Point of Inflection ?"},"content":{"rendered":"

Here you will learn what is the point of inflection and properties of maxima and minima.<\/p>\n

Let’s begin –<\/p>\n

What is the Point of Inflection ?<\/h2>\n

A point of inflection is a point at which a curve is changing concave upward to concave downward, or vice versa.<\/p>\n

\"point<\/p>\n

\n

A curve y = f(x) has one of its points x = c as an inflection point, if<\/p>\n

(i) f”(c) = 0 or is not defined and<\/p>\n

(ii) f”(x) changes sign as x increases through x = c.<\/p>\n

The later conditions may be replaced by f”'(c) \\(\\ne\\) 0 when f”'(c) exists.<\/p>\n

Thus, x = c is a point of inflection if f”(c) = 0 and f”'(c) \\(\\ne\\) 0.<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : The point of inflection for the curve f(x) =\\(x^{5\/3}\\) is –<\/p>\n

(A) (1, 1) <\/p>\n

(B) (0, 0)<\/p>\n

Solution<\/span><\/strong> : Here, f”(x) = \\(10\\over 9x^{1\/3}\\)<\/p>\n

from the given points we find that (0, 0) is the point of the curve where<\/p>\n

f”(x) does not exist but sign of f”(x) changes about this point.<\/p>\n

Hence (0, 0) is the required point.<\/p>\n

Properties of Maxima and Minima <\/h2>\n

(i) If f(x) is continous function in its domain, then atleast one maximum or one minimum must lie between two equal values of f(x).<\/p>\n

(ii) Maxima and minima occur alternately, that is, between two maxima there  is one minimum and vice-versa.<\/p>\n

(iii) If f(x) \\(\\rightarrow\\) \\(\\infty\\) as x \\(\\rightarrow\\) a or b and f'(x) = 0 only for one value of x (say c) between a and b, then f(c) is necessarily the minimum and the least value.<\/p>\n

If f(x) \\(\\rightarrow\\) -\\(\\infty\\) as x \\(\\rightarrow\\) a or b, then f(c) is necessarily the maximum and the greatest value.<\/p>\n

\n

Related Questions<\/h3>\n

Find the point of inflection for the curve y = \\(x^3 \u2013 6x^2 + 12x + 5\\).<\/a><\/p>\n

Find the inflexion point of f(x) = \\(3x^4 \u2013 4x^3\\).<\/a><\/p>\n

Find the point of inflexion for f(x) = \\(x^4\\over 12\\) \u2013 \\(5x^3\\over 6\\) + \\(3x^2\\) + 7.<\/a><\/p>\n

<\/p>\n\n\n

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

Here you will learn what is the point of inflection and properties of maxima and minima. Let’s begin – What is the Point of Inflection ? A point of inflection is a point at which a curve is changing concave upward to concave downward, or vice versa. A curve y = f(x) has one of …<\/p>\n

What is the Point of Inflection ?<\/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":[70,68,71,69,67],"yoast_head":"\nWhat is the Point of Inflection ? - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is the point of inflection and properties of maxima and minima.\" \/>\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\/what-is-the-point-of-inflection\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What is the Point of Inflection ? 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