Mathematics Question from JEE Main 2026 · 21 January, Shift 2

Let $f: \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f''(x) > 0$ for all $x \in \mathbb{R}$ and $f'(a-1) = 0$, where $a$ is a real number. Let $g(x) = f(\tan^2 x - 2\tan x + a)$, $0 < x < \frac{\pi}{2}$.

Consider the following two statements :

(I) $g$ is increasing in $\left(0, \frac{\pi}{4}\right)$

(II) $g$ is decreasing in $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$

Then,