{"id":8021,"date":"2021-11-13T15:09:20","date_gmt":"2021-11-13T09:39:20","guid":{"rendered":"https:\/\/mathemerize.com\/?p=8021"},"modified":"2021-11-14T01:00:17","modified_gmt":"2021-11-13T19:30:17","slug":"prove-that-ftheta-4sin-thetaover-2-costheta-theta-is-an-increasing-function-of-theta-in-0-piover-2","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/prove-that-ftheta-4sin-thetaover-2-costheta-theta-is-an-increasing-function-of-theta-in-0-piover-2\/","title":{"rendered":"Prove that \\(f(\\theta)\\) = \\({4sin \\theta\\over 2 + cos\\theta} – \\theta\\) is an increasing function of \\(\\theta\\) in \\([0, {\\pi\\over 2}]\\)."},"content":{"rendered":"

Solution :<\/h2>\n

We have, \\(f(\\theta)\\) = \\({4sin \\theta\\over 2 + cos\\theta} – \\theta\\)<\/p>\n

\\(\\implies\\) \\(f'(\\theta)\\) = \\((2 + cos\\theta)(4 cos\\theta) + 4 sin^2\\theta\\over (2 + cos\\theta)^2\\) – 1<\/p>\n

\\(\\implies\\) \\(f'(\\theta)\\)\u00a0 = \\(8 cos\\theta + 4\\over (2 + cos\\theta)^2\\) – 1<\/p>\n

\\(\\implies\\) \\(f'(\\theta)\\) = \\(4\\cos\\theta – cos^2\\theta\\over (2 + cos\\theta)^2\\)<\/p>\n

\\(\\implies\\) \\(f'(\\theta)\\) = \\(cos\\theta(4 – cos\\theta)\\over (2 + cos\\theta)^2\\) > 0\u00a0 for all \\(\\theta\\) \\(\\in\\) \\((0, {\\pi\\over 2})\\).<\/p>\n

[ \\(\\because\\)\u00a0 \u00a0\\(cos\\theta\\) > 0 , 4 – \\(cos\\theta\\) > 0 and 2 + \\(cos\\theta\\) > 0 ]<\/p>\n

Hence, \\(f(\\theta)\\) is increasing<\/a> on \\([0, {\\pi\\over 2}]\\).<\/p>\n

\n

Similar Questions<\/h3>\n

Prove that the function f(x) = \\(x^3 \u2013 3x^2 + 3x \u2013 100\\) is increasing on R<\/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(\\theta)\\) = \\({4sin \\theta\\over 2 + cos\\theta} – \\theta\\) \\(\\implies\\) \\(f'(\\theta)\\) = \\((2 + cos\\theta)(4 cos\\theta) + 4 sin^2\\theta\\over (2 + cos\\theta)^2\\) – 1 \\(\\implies\\) \\(f'(\\theta)\\)\u00a0 = \\(8 cos\\theta + 4\\over (2 + cos\\theta)^2\\) – 1 \\(\\implies\\) \\(f'(\\theta)\\) = \\(4\\cos\\theta – cos^2\\theta\\over (2 + cos\\theta)^2\\) \\(\\implies\\) \\(f'(\\theta)\\) = \\(cos\\theta(4 – cos\\theta)\\over …<\/p>\n

Prove that \\(f(\\theta)\\) = \\({4sin \\theta\\over 2 + cos\\theta} – \\theta\\) is an increasing function of \\(\\theta\\) in \\([0, {\\pi\\over 2}]\\).<\/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 \\(f(\\theta)\\) = \\({4sin \\theta\\over 2 + cos\\theta} - \\theta\\) is an increasing function of \\(\\theta\\) in \\([0, {\\pi\\over 2}]\\).<\/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-ftheta-4sin-thetaover-2-costheta-theta-is-an-increasing-function-of-theta-in-0-piover-2\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Prove that \\(f(\\theta)\\) = \\({4sin \\theta\\over 2 + cos\\theta} - \\theta\\) is an increasing function of \\(\\theta\\) in \\([0, {\\pi\\over 2}]\\).\" \/>\n<meta property=\"og:description\" content=\"Solution : We have, (f(theta)) = ({4sin thetaover 2 + costheta} – theta) (implies) (f'(theta)) = ((2 + costheta)(4 costheta) + 4 sin^2thetaover (2 + costheta)^2) – 1 (implies) (f'(theta))\u00a0 = (8 costheta + 4over (2 + costheta)^2) – 1 (implies) (f'(theta)) = (4costheta – cos^2thetaover (2 + costheta)^2) (implies) (f'(theta)) = (costheta(4 – costheta)over … Prove that (f(theta)) = ({4sin thetaover 2 + costheta} – theta) is an increasing function of (theta) in ([0, {piover 2}]). 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