If $f(x) = \begin{cases} 1, & x < 0 \\ 1 + \sin x, & 0 \le x < \frac{\pi}{2} \end{cases}$,then $f'(0) = $

The set of all points,where the derivative of the function $f(x) = \frac{x}{1+|x|}$ exists,is

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

The number of points at which the function $f(x) = \max \{a-x, a+x, b\}$ for $-\infty < x < \infty$ and $0 < a < b$ is not differentiable, is

Which one of the following functions is continuous everywhere in its domain but has at least one point where it is not differentiable?

Let $S$ be the set of all points in $(-\pi, \pi)$ at which the function $f(x) = \min\{\sin x, \cos x\}$ is non-differentiable. Then $S$ is a subset of which of the following?