$\text{ Let, then }\mathrm{f}\left(\mathrm{x}\right)={\mathrm{xcos}}^{-1}(-\sin |\mathrm{x}\left|\right),\mathrm{x}\in \left[-\frac{\pi }{2},\frac{\pi }{2}\right]\text{ which of the following is true? }$

$\begin{array}{l}f\left(x\right)=x\left(\pi -{\cos }^{-1}\left(\sin \right|x\left|\right)\right)=x\left(\pi -\left(\frac{\pi }{2}-{\sin }^{-1}\left(\sin \right|x\left|\right)\right)\right\}=x\left(\frac{\pi }{2}+\left|x\right|\right)\\ f\left(x\right)=\left\{\begin{array}{ll}x\left(\frac{\pi }{2}+x\right)& x\ge 0\\ x\left(\frac{\pi }{2}-x\right)& x<0\end{array}\Rightarrow f^{\text{'}}\left(x\right)=\left\{\begin{array}{ll}\left(\frac{\pi }{2}+2x\right)& x\ge 0\\ \left(\frac{\pi }{2}-2x\right)& x<0\end{array}\right.\right.\\ f^{\text{'}}\left(x\right)\text{ is increasing in }\left(0,\frac{\pi }{2}\right)\text{ and decreasing in }\left(\frac{-\pi }{2},0\right)\end{array}$