Let f: R → R be a twice differentiable function such that f^′ ′(x)>0 for all x ∈ R and f^′( a -1)=0, where a is a real number. Let g (x)=f( tan^2 x-2…

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

Consider the following two statements :

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

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

Then,