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If x^3f(x)=sqrt(1+cos2x)+|f(x)|, (-3pi)/...

If `x^3f(x)=sqrt(1+cos2x)+|f(x)|, (-3pi)/4ltxlt(-pi) /2` and `f(x)=(alphacosx)/(1+x^3)` , then the value of `alpha` is

A

2

B

`-sqrt(2)`

C

`sqrt(2)`

D

1

Text Solution

Verified by Experts

The correct Answer is:
B
`x^(3) f(x)=sqrt(1+cos 2x)+|f(x)|=sqrt(2)|cos x|+|f(x)|`
Since R.H.S is positive, L.H.S must be positive.
So, `x^(3)f(x) gt 0`
`implies f(x) lt 0`
`implies f(x) lt 0 " "( :' (-3pi)/(4) lt x lt (-pi)/(2) implies x^(3) lt 0)`
` :. x^(3) f(x) = -sqrt(2) cos x-f(x)`
` implies f(x)= (-sqrt(2) cos x)/(1+x^(3))`
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