{"id":539970,"date":"2023-04-23T01:34:02","date_gmt":"2023-04-22T20:04:02","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/question\/gxygxgy3xyxy%e2%88%80xy%e2%88%88r-and-g%e2%80%b20%e2%88%924-2\/"},"modified":"2025-06-20T13:28:16","modified_gmt":"2025-06-20T07:58:16","slug":"gxygxgy3xyxy%e2%88%80xy%e2%88%88r-and-g%e2%80%b20%e2%88%924-2","status":"publish","type":"questions","link":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/gxygxgy3xyxyxyrandg04\/","title":{"rendered":"g(x+y)=g(x)+g(y)+3xy(x+y)\u2200x,y\u2208R\u00a0and\u00a0g\u2032(0)=\u22124"},"content":{"rendered":"","protected":false},"author":1,"template":"","meta":{"_yoast_wpseo_focuskw":"","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"g(x+y)=g(x)+g(y)+3xy(x+y)\u2200x,y\u2208R\u00a0and\u00a0g\u2032(0)=\u22124","custom_permalink":"question\/mathematics\/gxygxgy3xyxyxyrandg04\/"},"categories":[],"acf":{"question":"g(x+y)=g(x)+g(y)+3xy(x+y)\u2200x,y\u2208R\u00a0and\u00a0g\u2032(0)=\u22124","options":null,"solution":" <p>Given the functional equation <code>g(x+y) = g(x) + g(y) + 3xy(x+y)<\/code> for all <code>x, y \u2208 \u211d<\/code> and <code>g\u2032(0) = \u22124<\/code>, we find the function <code>g(x)<\/code>.<\/p>\r\n    <ol>\r\n        <li>\r\n            Substitute <code>y = 0<\/code> in the given equation:\r\n            <pre>g(x+0) = g(x) + g(0) + 3x \u22c5 0 \u22c5 (x+0)<\/pre>\r\n            <p>Simplifying, we get:<\/p>\r\n            <pre>g(x) = g(x) + g(0)<\/pre>\r\n            <p>This implies:<\/p>\r\n            <pre>g(0) = 0<\/pre>\r\n        <\/li>\r\n        <li>\r\n            Differentiate both sides of the given functional equation with respect to <code>x<\/code>:\r\n            <pre>\u2202\/\u2202x [g(x+y)] = \u2202\/\u2202x [g(x) + g(y) + 3xy(x+y)]<\/pre>\r\n            <p>Since <code>y<\/code> is treated as a constant:<\/p>\r\n            <pre>g'(x+y) = g'(x) + 3y(x+y) + 3xy<\/pre>\r\n            <p>Substitute <code>x = 0<\/code>:<\/p>\r\n            <pre>g'(y) = g'(0) + 3y^2<\/pre>\r\n            <p>Given <code>g'(0) = -4<\/code>:<\/p>\r\n            <pre>g'(y) = -4 + 3y^2<\/pre>\r\n        <\/li>\r\n        <li>\r\n            Integrate <code>g'(y)<\/code> to find <code>g(y)<\/code>:\r\n            <pre>g(y) = \u222b(-4 + 3y^2) dy = -4y + y^3 + C<\/pre>\r\n            <p>Given <code>g(0) = 0<\/code>, we find <code>C<\/code>:<\/p>\r\n            <pre>g(0) = -4(0) + (0)^3 + C = 0 \u27f9 C = 0<\/pre>\r\n            <p>Therefore:<\/p>\r\n            <pre>g(y) = y^3 - 4y<\/pre>\r\n        <\/li>\r\n    <\/ol>\r\n    <p><strong>Answer:<\/strong> The function <code>g(x)<\/code> is <code>g(x) = x^3 - 4x<\/code>.<\/p>","subject":"Mathematics"},"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>g(x+y)=g(x)+g(y)+3xy(x+y)\u2200x,y\u2208R\u00a0and\u00a0g\u2032(0)=\u22124 - Infinity Learn by Sri Chaitanya<\/title>\n<meta name=\"description\" content=\"g(x+y)=g(x)+g(y)+3xy(x+y)\u2200x,y\u2208R\u00a0and\u00a0g\u2032(0)=\u22124\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/gxygxgy3xyxyxyrandg04\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"g(x+y)=g(x)+g(y)+3xy(x+y)\u2200x,y\u2208R\u00a0and\u00a0g\u2032(0)=\u22124 - 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