Given F^2(x) + G^2(x) + H^2(x) < 9 And U(x) = 3f(x) + 4g(x) + 10h(x ...

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SolveGuidesJoin / LoginUse appLogin0You visited us 0 times! Enjoying our articles? Unlock Full Access!Standard XIIMathematicsQuestionGiven f2(x)+g2(x)+h2(x)≤9 and U(x)=3f(x)+4g(x)+10h(x),where f(x).g(x) and h(x) are continuous ∀x∈R. If maximum value of U(x) is √N. Then find N. Open in AppSolutionVerified by Toppr

Given : U(x)=3f(x)+4g(x)+10h(x)Squaring both sides we getU2(x)=9f2(x)+16g2(x)+100h2(x)+24f(x)g(x)+80g(x)h(x)+60f(x)h(x)we know that,a2+b2>2abHence 2ab≤a2+b2using the above property we can say2.3.4f(x).g(x)≤16f(x)+0g2(x)...[1]2.4.10g(x).h(x)≤16h2(x)+100g2(x)...[2]2.10.3.f(x).h(x)≤100f2(x)+9h2(x)...[3]Using [1] [2] [3] we can writeU2(x)≤9f2(x)+16g2(x)+100h2(x)+16f2+9g2(x)+16h2(x)+100g2(x)+100f2(x)+9h2(x)⇒U2(x)≤125f2(x)+125g2(x)+125h2(x)⇒U2(x)≤125(f2(x)+g2(x)+h2(x)Given that f2(x)+g2(x)+h2(x)≤9∴U2(x)≤125×9⇒U(x)≤√125×9=√1125∴ Maximum value oof U(x)=√1125=√N∴N=11251124957_1031678_ans_65a66c23db4d4a10994c93951292d9c4.jpg

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