{"id":5666,"date":"2021-09-27T19:41:56","date_gmt":"2021-09-27T14:11:56","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5666"},"modified":"2021-11-13T17:33:49","modified_gmt":"2021-11-13T12:03:49","slug":"monotonic-function-definition-and-examples","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/monotonic-function-definition-and-examples\/","title":{"rendered":"Monotonic Function – Definition and Examples"},"content":{"rendered":"

Here you will learn definition of monotonic function and condition for monotonicity with examples.<\/p>\n

Let’s begin –<\/p>\n

Monotonic Function<\/h2>\n

The function f(x) is said to be monotonic on an interval (a, b) if it is either increasing or decreasing on (a, b).<\/p>\n

A function f(x) is said to be increasing (decreasing) at a point \\(x_0\\), if there is an interval (\\(x_0 – h, x_0 + h\\)) containing \\(x_0\\) such that f(x) is increasing (decreasing) on (\\(x_0 – h, x_0 + h\\)).<\/p>\n

A function f(x) is said to be increasing (decreasing) on [a, b] if it is increasing (decreasing) on (a, b) and it is also increasing (decreasing) at x = a and x = b. <\/p>\n

Necessary and Sufficient Conditions for Monotonicity<\/h2>\n

Let f be a differentiable real function defined on an open interval (a, b).<\/p>\n

\n

(i) If f'(x) > 0 for all x \\(\\in\\) (a, b), then f(x) is increasing on (a, b).<\/p>\n

(ii) If f'(x) < 0 for all x \\(\\in\\) (a, b), then f(x) is decreasing on (a, b).<\/p>\n<\/blockquote>\n

Corollary : <\/strong><\/p>\n

Let f(x) be a function defined on (a, b).<\/p>\n

\n

(i) If f'(x) > 0 for all x \\(\\in\\) (a,b) except for a finite number of points, where f'(x) = 0, then f(x) is increasing on (a, b).<\/p>\n

(ii) If f'(x) < 0 for all x \\(\\in\\) (a,b) except for a finite number of points, where f'(x) = 0, then f(x) is decreasing on (a, b).<\/p>\n<\/blockquote>\n

Algorithm<\/strong><\/p>\n

\n

1 : Obtain the function and put it equal to f(x).<\/p>\n

2 : find f'(x)<\/p>\n

3 : Put f'(x) > 0 and solve this inequation.<\/p>\n

for the values of x obtained in step 3 f(x) is increasing and for the remaining points in its domain it is decreasing.<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : find the interval in which f(x) = \\(-x^2 – 2x + 15\\) is increasing or decreasing.<\/p>\n

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

f(x) = \\(-x^2 – 2x + 15\\)<\/p>\n

\\(\\implies\\) f'(x) = -2x – 2 = -2(x + 1)<\/p>\n

for f(x) to be increasing, we must have<\/p>\n

f'(x) > 0<\/p>\n

-2(x + 1) > 0<\/p>\n

\\(\\implies\\) x + 1 < 0<\/p>\n

\\(\\implies\\) x < -1 \\(\\implies\\) x \\(\\in\\) \\((-\\infty, -1)\\).<\/p>\n

Thus f(x) is increasing on the interval \\((-\\infty, -1)\\).<\/p>\n

for f(x) to be decreasing, we must have<\/p>\n

f'(x) > 0<\/p>\n

-2(x + 1) < 0<\/p>\n

\\(\\implies\\) x + 1 > 0<\/p>\n

\\(\\implies\\) x > -1 \\(\\implies\\) x \\(\\in\\) \\((-1, \\infty)\\).<\/p>\n

Thus f(x) is decreasing on the interval \\((-1, \\infty)\\).<\/p>\n

\n

Related Questions<\/h3>\n

Prove that \\(f(\\theta)\\) = \\({4sin \\theta\\over 2 + cos\\theta} \u2013 \\theta\\) is an increasing function of \\(\\theta\\) in \\([0, {\\pi\\over 2}]\\).<\/a><\/p>\n

Separate \\([0, {\\pi\\over 2}]\\) into subintervals in which f(x) = sin 3x is increasing or decreasing.<\/a><\/p>\n

Prove that the function f(x) = \\(x^3 \u2013 3x^2 + 3x \u2013 100\\) is increasing on R<\/a><\/p>\n\n\n

\n
Next – Maxima and Minima Class 12<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Increasing and Decreasing Function<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn definition of monotonic function and condition for monotonicity with examples. Let’s begin – Monotonic Function The function f(x) is said to be monotonic on an interval (a, b) if it is either increasing or decreasing on (a, b). A function f(x) is said to be increasing (decreasing) at a point \\(x_0\\), …<\/p>\n

Monotonic Function – Definition and Examples<\/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":[76,85,88,86,87],"yoast_head":"\nMonotonic Function - Definition and Examples - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn definition of monotonic function and condition for monotonicity 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\/monotonic-function-definition-and-examples\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Monotonic Function - 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