Mathematics - Application of Derivative Question with Solution | TestHub

$f^{\prime }(x)=\frac{-g^{\prime }(x)}{(g(x){)}^2}$ where

$g(x)=\{\begin{array}{ll}2{\sin }^{-1}x& ifx\in \left[-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right]\setminus \{0\}\\ \pi -2{\sin }^{-1}x& ifx\in \left[\frac{1}{\sqrt{2}},1\right)\\ -\pi -2{\sin }^{-1}x& ifx\in \left(-1,-\frac{1}{\sqrt{2}}\right]\end{array}$.

$g$ is not differentiable at $x=\pm \frac{1}{\sqrt{2}}$.

$g^{\prime }(x)$ is never zero in $(-1,1)\setminus \{0\}$.

Therefore, $f$ has two critical points, namely $x=\pm \frac{1}{\sqrt{2}}$.