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CBSE Class 11-science Answered

If f(x)=2(7cosx+24sinx)(7sinx-24cosx), then find the maximum value of (f(x))^1/4
Asked by aahnik.mohanty | 22 May, 2019, 09:59: PM
answered-by-expert Expert Answer
f(x)=2(7cosx+24sinx)(7sinx-24cosx)
Multiply and divide f(x) by 25, we get
 
begin mathsize 16px style straight f left parenthesis straight x right parenthesis space equals space 2 space cross times space 25 space cross times 25 space cross times open parentheses 7 over 25 cosx space plus space 24 over 25 sinx close parentheses open parentheses 7 over 25 sinx space minus space 24 over 25 cosx close parentheses
Let space 7 over 25 space equals space cosα space and space fraction numerator 24 over denominator space 25 end fraction space equals space sinα
rightwards double arrow space space straight f left parenthesis straight x right parenthesis space equals 2 space cross times space 25 space cross times 25 space cross times open parentheses cosα cosx space plus sinα space sinx close parentheses open parentheses cosα sinx space minus space sinα cosx close parentheses
rightwards double arrow space space straight f left parenthesis straight x right parenthesis space equals 2 space cross times space 25 space cross times 25 open square brackets cos open parentheses straight x space minus space straight alpha space close parentheses sin open parentheses straight x space minus space straight alpha space close parentheses close square brackets
rightwards double arrow space space straight f left parenthesis straight x right parenthesis space equals 25 space cross times space 25 space cross times 2 cos open parentheses straight x space minus space straight alpha space close parentheses sin open parentheses straight x space minus space straight alpha space close parentheses
rightwards double arrow space space straight f left parenthesis straight x right parenthesis space equals 25 space cross times space 25 space cross times sin open square brackets 2 open parentheses straight x space minus space straight alpha space close parentheses close square brackets
rightwards double arrow space space straight f left parenthesis straight x right parenthesis space equals 25 space cross times space 25 space cross times sinθ
We space know space that space sin space function space has space the space maximum space value space as space 1.
Therefore space the space maximum space value space of space straight f left parenthesis straight x right parenthesis space is space
rightwards double arrow space space straight f left parenthesis straight x right parenthesis space equals 25 space cross times space 25 space cross times 1 space equals space 625
rightwards double arrow space The space maximum space value space of space space open square brackets straight f left parenthesis straight x right parenthesis close square brackets to the power of 1 fourth end exponent space equals space open parentheses 625 close parentheses to the power of 1 fourth end exponent space space space equals space 5 space end style
Answered by Yasmeen Khan | 23 May, 2019, 11:08: AM
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