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

Answer

B

Explanation

Given `f(x)=tan^(-1)(sinx+cosx)`
`f'(x)=1/(1+(sinx+cosx)^(2))(cosx-sinx)`
`=(sqrt(2)cos(x+(pi)/4))/(1+(sinx+cosx)^(2))`
For `f(x)` to be increasing,
`sqrt(2)cos(x+(pi)/4)gt0`
`implies-(pi)/2ltx+(pi)/4lt(pi)/2`
`implies (-3pi)/4lt x ltg (pi)/4`