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