Mathematics - Application of Derivative Question with Solution | TestHub

$f^{\text{'}}\left(x\right)=x\left(\pi -{\mathrm{c}\mathrm{o}\mathrm{s}}^{-1}\left(\mathrm{s}\mathrm{i}\mathrm{n}\left|x\right|\right)\right)=x(\pi \left(\frac{\pi }{2}-{\mathrm{s}\mathrm{i}\mathrm{n}}^{-1}\left(\mathrm{s}\mathrm{i}\mathrm{n}\left|x\right|\right)\right)=x\left(\frac{\pi }{2}+\left|x\right|\right)$
$f\left(x\right)=\left\{\begin{array}{cc}x\left(\frac{\pi }{2}+x\right)& x\ge 0\\ x\left(\frac{\pi }{2}-x\right)& x<0\end{array}\right.$
$f^{\text{'}}\left(x\right)=\left\{\begin{array}{cc}\frac{\pi }{2}+2x& x\ge 0\\ \frac{\pi }{2}-2x& x<0\end{array}\right.$
$f^{\text{'}}\left(x\right)$is increasing in$\left(0,\hspace{0.17em}\frac{\pi }{2}\right)$and decreasing in$\left(\frac{-\pi }{2},\hspace{0.17em}0\right)$.