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

Answer

B

Explanation

Since `f(x) = tan ^(-1) (sin x + cos x)`
`rArr " " f(x) = (1)/(1+(sin x+ cos x)^(2)( cos x - sin x))`
` = (sqrt(2) cos(x+(pi)/(4)))/(1+(sin x+ cos x)^(2))`
For f(x) to be increasing ,
`-(pi)/(2) lt x +(pi)/(4) lt (pi)/(2)`
`rArr -(3pi)/(4) lt x lt (pi)/(4)`
Hence , option (b) is correct , which lies in the above interal .