{"id":8019,"date":"2021-11-13T14:59:50","date_gmt":"2021-11-13T09:29:50","guid":{"rendered":"https:\/\/mathemerize.com\/?p=8019"},"modified":"2021-11-13T16:24:06","modified_gmt":"2021-11-13T10:54:06","slug":"prove-that-the-function-fx-x3-3x2-3x-100-is-increasing-on-r","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/prove-that-the-function-fx-x3-3x2-3x-100-is-increasing-on-r\/","title":{"rendered":"Prove that the function f(x) = \\(x^3 – 3x^2 + 3x – 100\\) is increasing on R"},"content":{"rendered":"

Solution :<\/h2>\n

We have, f(x) = \\(x^3 – 3x^2 + 3x – 100\\)<\/p>\n

\\(\\implies\\)\u00a0 f'(x) = \\(3x^2 – 6x + 3\\) = \\(3(x – 1)^2\\)<\/p>\n

Now, x \\(\\in\\) R \\(\\implies\\)\u00a0 \\((x – 1)^2\\)\u00a0 \\(\\ge\\)\u00a0 0\u00a0 \\(\\implies\\)\u00a0 f'(x)\u00a0 \\(\\ge\\) 0.<\/p>\n

Thus, f'(x) \\(\\ge\\) 0 for all x \\(\\in\\) R.<\/p>\n

Hence, f(x) is increasing on R.<\/p>\n

\n

Similar 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

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

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

Find the inflection point of f(x) = \\(3x^4 \u2013 4x^3\\).<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Solution : We have, f(x) = \\(x^3 – 3x^2 + 3x – 100\\) \\(\\implies\\)\u00a0 f'(x) = \\(3x^2 – 6x + 3\\) = \\(3(x – 1)^2\\) Now, x \\(\\in\\) R \\(\\implies\\)\u00a0 \\((x – 1)^2\\)\u00a0 \\(\\ge\\)\u00a0 0\u00a0 \\(\\implies\\)\u00a0 f'(x)\u00a0 \\(\\ge\\) 0. Thus, f'(x) \\(\\ge\\) 0 for all x \\(\\in\\) R. Hence, f(x) is increasing on R. Similar …<\/p>\n

Prove that the function f(x) = \\(x^3 – 3x^2 + 3x – 100\\) is increasing on R<\/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":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[66,43],"tags":[],"yoast_head":"\nProve that the function f(x) = \\(x^3 - 3x^2 + 3x - 100\\) is increasing on R<\/title>\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\/prove-that-the-function-fx-x3-3x2-3x-100-is-increasing-on-r\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Prove that the function f(x) = \\(x^3 - 3x^2 + 3x - 100\\) is increasing on R\" \/>\n<meta property=\"og:description\" content=\"Solution : We have, f(x) = (x^3 – 3x^2 + 3x – 100) (implies)\u00a0 f'(x) = (3x^2 – 6x + 3) = (3(x – 1)^2) Now, x (in) R (implies)\u00a0 ((x – 1)^2)\u00a0 (ge)\u00a0 0\u00a0 (implies)\u00a0 f'(x)\u00a0 (ge) 0. 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