For the function $f(x)=\mathrm{e}^{\sin |x|}-|x|, x \in \mathbf{R}$, consider the following statements: Statement I : $f$ is differentiable for all $x \in \mathbf{R}$. Statement II: $\quad f$ is increasing in $\left(-\pi,-\frac{\pi}{2}\right)$. In the light of the above statements, choose the correct answer from the options given below :