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) ` sqrt2` (C) ` -sqrt2` (D) ` 1`

Options

A. 2
B. `-sqrt(2)`
C. `sqrt(2)`
D. 1

Answer

c

Explanation

`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))`