If `F(x) and G(x)` are even and odd extensions of the functions `f(x) = x|x|+ sin|x|+ xe^x`, where `x in (0, 1), g(x) = cos|x| + x^2-x`, is where `x in (0, 1)` respectively to the ars interval `(-1, 0)` then `F(x)+G(x) `in `(-1,0)` is

Options

A. `sinx +cosx +xe^(-x)`
B. `-(sin x +cos x +xe^(-x))`
C. `-(sin x +cos x +x + xe^(-x))`
D. `-(sin x +cos x +x^(2)+xe^(-x))`

Answer

C

Explanation

`f(x)=f(-x)` where `f(x)=x|x|+sin|x|+xe^(x)`,
`therefore" "F(x)=x^(2)-sinx-xe^(-x)" (i)"`
Also, `g(x)=-g(-x)`, where `g(x)=cosx+x^(2)-x`
`therefore" "G(x)=-(cosx+x^(2)+x)=-cosx-x^(2)-x" (ii)"`
`therefore" "F(x)+G(x)=-sinx-xe^(-x)-cosx-x`
`=-(sinx+cosx+x+x+ex^(-x))`