Let the function `f(x) = tan^(1-) (sin x + cos x)` be defined on `[ 0, 2 pi] Then f(x) is

Options

A. increasing on `[0,pi//4) cup [ 5 pi/4, 2 pi]`
B. decreasing on `(Pi//4,2pi)`
C. increasing on `(0,pi//4,) cup (3 pi//4,2pi)`
D. decreasing on `[pi//4,7 pi//4]`

Answer

A

Explanation

We have
`f(x) = tan^(-1)""(sin x + cos x )`
`rArr f(x)=(1)/(1+(sin x+cos x )^2)xx (cos x - sin x)`
Now
`f(x) gt 0 rArr cos x -sin x gt 0 tArr x in [0,pi//4) cup (5pi //4 ,4,2pi]`
and `f(x) lt 0 rArr cos x sin x lt 0 rArr x in (pi//4,5 pi//4)` decreasing `(pi//4,5 pi//4)`