For the function $$f(x) = e^{\sin|x|} - |x|$$, $$x \in \mathbf{R}$$, consider the following statements :
Statement I : $$f$$ is differentiable for all $$x \in \mathbf{R}$$.
Statement II : $$f$$ 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 :

A

Both Statement I and Statement II are true

B

Both Statement I and Statement II are false

C

Statement I is true but Statement II is false

D

Statement I is false but Statement II is true