Let `f(x) = tan^-1 (g(x))`, where `g (x)` is monotonically increasing for `0 < x < pi/2.`

Options

A. increasing on `(0,pi//2)`
B. decreasing on `(0,pi//2)`
C. increasing on `(0,pi//4)` and decreasing on `(pi//4,pi//2)`
D. none of these

Answer

A

Explanation

We have
`f(x) = tan^(-1) (g(x)) rArr f'(x) = 1/(1+(g+x)^2)xxd/(dx)(g(x))`
For f(x) to be increasing , we must have
`f'(x) gt0 `
`rArr 1/(1+{g(x)}^2)g'(x)gt0`
` rArrg'(x)gt0`
` rArr x in (0,pi//2)`