Question

If \( f(x) > 0 \; \forall x \in \mathbb{R} \), \( f'(3) = 0 \) and \( g(x) = f(\tan^2 x - 2\tan x + 4) \), \( 0 < x < \frac{\pi}{2} \), then \( g(x) \) is increasing in

• A. \(\left(0,\frac{\pi}{4}\right)\)
• B. \(\left(\frac{\pi}{6},\frac{\pi}{3}\right)\)
• C. \(\left(0,\frac{\pi}{3}\right)\)
• D. \(\left(\frac{\pi}{4},\frac{\pi}{2}\right)\)

Hint:

For composite functions, analyze inner function first and use critical point of outer derivative.

Answer 1

The Correct Option is D