Mathematics - Application of Derivative Question with Solution | TestHub

$g^{\prime }(x)=\left(f^{\prime }\left((\tan x-1{)}^2+3\right)\right)2(\tan x-1){\sec }^{2}x$ since $f^{\prime \prime }(x)>0\Rightarrow f^{\prime }(x)$ is increasing

So, $f^{\prime }\left((\tan x-1{)}^2+3\right)>f^{\prime }(3)=0\forall x\in \left(0,\frac{\pi }{4}\right)\cup \left(\frac{\pi }{4},\frac{\pi }{2}\right)$

Also, $(\tan x-1)>0$ for $x\in \left(\frac{\pi }{4},\frac{\pi }{2}\right)$. So, $g(x)$ in increasing in $\left(\frac{\pi }{4},\frac{\pi }{2}\right)$