Applications of Derivatives · Monotonicity of composite functions

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 real number. Let $g(x) = f(\tan^2 x - 2\tan x + a)$, $0 < x < \dfrac{\pi}{2}$.

Consider the following two statements:

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

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

Then,