The function `f(x)=tan^(-1)(sinx+cosx)` is an increasing function in `(-pi/2,pi/4)` (b) `(0,pi/2)` `(-pi/2,pi/2)` (d) `(pi/4,pi/2)`

Options

A. `(0,(pi)/2)`
B. `-((pi)/2,(pi)/2)`
C. `((pi)/4,(pi)/2)`
D. `-((pi)/2,(pi)/4)`

Answer

D

Explanation

We have
`f(x)=tan^(-1)(sinx+cosx)`
`rArrf'(x)=1/(1+(sinx+cosx)^2)xx(cosx-sinx)`
`rArrf'(x)=cosx/(1+(sinx+cosx)^2)xx(1-tanx)`
Clearly ,f'(x) `gt 0 for all x in(-pi//2,pi//4)`
Hence f(x) is increasing in `(-pi//2,pi//4)`