Let $f : R \rightarrow R$ be a twice differentiable function such that $f^{\prime\prime}(x) > 0$ for all $x \in R$ and $f^{\prime}(a-1) = 0$, 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 $(0, \frac{\pi}{4})$
$(II)$ $g$ is decreasing in $(\frac{\pi}{4}, \frac{\pi}{2})$
Then,

• A
  Neither $(I)$ nor $(II)$ is True
• B
  Only $(II)$ is True
• C
  Only $(I)$ is True
• D
  Both $(I)$ and $(II)$ are True