{"id":6848,"date":"2021-10-21T19:53:39","date_gmt":"2021-10-21T14:23:39","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6848"},"modified":"2021-10-22T17:32:32","modified_gmt":"2021-10-22T12:02:32","slug":"if-fx-y-fx-fy-2xy-1-for-all-x-and-y-if-f0-exists-and-f0-sinalpha-then-find-ff0","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/if-fx-y-fx-fy-2xy-1-for-all-x-and-y-if-f0-exists-and-f0-sinalpha-then-find-ff0\/","title":{"rendered":"If f(x + y) = f(x) + f(y) &#8211; 2xy &#8211; 1 for all x and y. If f'(0) exists and f'(0) = -sin\\(\\alpha\\), then find f{f'(0)}."},"content":{"rendered":"<h2>Solution :<\/h2>\n<p>f'(x) = \\(\\displaystyle{\\lim_{h \\to 0}}\\) \\(f(x+h) &#8211; f(x)\\over h\\)<\/p>\n<p>= \\(\\displaystyle{\\lim_{h \\to 0}}\\) \\({f(x)+f(h)-2xh-1} &#8211; f(x)\\over h\\)<\/p>\n<p>= \\(\\displaystyle{\\lim_{h \\to 0}}\\) -2x + \\(\\displaystyle{\\lim_{h \\to 0}}\\) \\(f(h)-1\\over h\\) = -2x + \\(\\displaystyle{\\lim_{h \\to 0}}\\) \\(f(h) &#8211; f(0)\\over h\\)<\/p>\n<p>[Putting x = 0 = y in the given relation we find f(0) = f(0) + f(0) + 0 &#8211; 1 \\(\\implies\\) f(0) = 1]<\/p>\n<p>\\(\\therefore\\) f'(x) = -2x + f'(0) = -2x &#8211; sin\\(\\alpha\\)<\/p>\n<p>\\(\\implies\\)&nbsp; f(x) = -\\(x^2\\) &#8211; (sin\\(\\alpha\\)).x + c<\/p>\n<p>f(0) = &#8211; 0 &#8211; 0 + c \\(\\implies\\) c = 1<\/p>\n<p>\\(\\therefore\\)&nbsp; f(x) = -\\(x^2\\) &#8211; (sin\\(\\alpha\\)).x + 1<\/p>\n<p>so, f{f'(0)} = f(-sin\\(\\alpha\\)) = -\\(sin^2\\alpha\\) + \\(sin^2\\alpha\\) + 1 = 1<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solution : f'(x) = \\(\\displaystyle{\\lim_{h \\to 0}}\\) \\(f(x+h) &#8211; f(x)\\over h\\) = \\(\\displaystyle{\\lim_{h \\to 0}}\\) \\({f(x)+f(h)-2xh-1} &#8211; f(x)\\over h\\) = \\(\\displaystyle{\\lim_{h \\to 0}}\\) -2x + \\(\\displaystyle{\\lim_{h \\to 0}}\\) \\(f(h)-1\\over h\\) = -2x + \\(\\displaystyle{\\lim_{h \\to 0}}\\) \\(f(h) &#8211; f(0)\\over h\\) [Putting x = 0 = y in the given relation we find f(0) = f(0) &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathemerize.com\/if-fx-y-fx-fy-2xy-1-for-all-x-and-y-if-f0-exists-and-f0-sinalpha-then-find-ff0\/\"> <span class=\"screen-reader-text\">If f(x + y) = f(x) + f(y) &#8211; 2xy &#8211; 1 for all x and y. If f'(0) exists and f'(0) = -sin\\(\\alpha\\), then find f{f'(0)}.<\/span> Read More &raquo;<\/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":[43],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.12 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>If f(x + y) = f(x) + f(y) - 2xy - 1 for all x and y. If f&#039;(0) exists and f&#039;(0) = -sin\\(\\alpha\\), then find f{f&#039;(0)}.<\/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\/if-fx-y-fx-fy-2xy-1-for-all-x-and-y-if-f0-exists-and-f0-sinalpha-then-find-ff0\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"If f(x + y) = f(x) + f(y) - 2xy - 1 for all x and y. If f&#039;(0) exists and f&#039;(0) = -sin\\(\\alpha\\), then find f{f&#039;(0)}.\" \/>\n<meta property=\"og:description\" content=\"Solution : f&#039;(x) = (displaystyle{lim_{h to 0}}) (f(x+h) &#8211; f(x)over h) = (displaystyle{lim_{h to 0}}) ({f(x)+f(h)-2xh-1} &#8211; f(x)over h) = (displaystyle{lim_{h to 0}}) -2x + (displaystyle{lim_{h to 0}}) (f(h)-1over h) = -2x + (displaystyle{lim_{h to 0}}) (f(h) &#8211; f(0)over h) [Putting x = 0 = y in the given relation we find f(0) = f(0) &hellip; If f(x + y) = f(x) + f(y) &#8211; 2xy &#8211; 1 for all x and y. If f&#039;(0) exists and f&#039;(0) = -sin(alpha), then find f{f&#039;(0)}. 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