Let`f'(sin x)lt0 and f''(sin x) gt0 forall x in (0,(pi)/(2))` and g(x) =f(sinx)+f(cosx)
which of the following is true?

Options

A. g(x) is decreasing in `((pi)/(4),(pi)/(2))`
B. g(x) increasing in `(0,(pi)/(4))`
C. g(x) is nonotonically increasing in `(0,(pi)/(2))`
D. none of these

Answer

4

Explanation

`g(x)=f(sinx)cosx-f(cosx)sinx`
or `g(x)=f(sinx)sinx+cos^(2)xf(sinx)`
`f(cos x)sin^(2)x-f(cosxgt0forall x in (0,pi//2)`
[as it is given `f(sinx)=f(cos x (pi//2-x))lt0`
Thus g(x) is increasing in `(0,pi//2)`. Also` g(pi//4)=0`
or `g(x)gt0forallx in ((pi)/(4),(pi)/(2))`
and `g(x) ltforall x in (0,pi//4)`
Thus g(X) is decreasing in `(0,pi//4)`