Which of the following statements is true?

Let $f:R \to R$ be a function defined by $f(x) = \text{Min}\{x + 1, |x| + 1\}$. Then which of the following is true?

Let $g(x) = x \cdot f(x)$,where $f(x) = \begin{cases} x \sin \frac{1}{x}, & x \ne 0 \\ 0, & x = 0 \end{cases}$. Discuss the differentiability of $g$ at $x = 0$.

Let $S = \{t \in R \mid f(x) = |x - \pi|(e^{|x|} - 1) \sin |x| \text{ is not differentiable at } t\}$. Then $S$ is

Let $f:[-1,1] \rightarrow R$ be a function defined by $f(x)=\begin{cases} x^2 \left| \cos \left(\frac{\pi}{x}\right) \right| & \text{for } x \neq 0 \\ 0 & \text{for } x=0 \end{cases}$. The set of points where $f$ is not differentiable is

If $f(x) = \begin{cases} \frac{1}{2}(b^2 - a^2), & 0 \leq x \leq a \\ \frac{1}{2}b^2 - \frac{x^2}{6} - \frac{a^3}{3x}, & a < x \leq b \\ \frac{1}{3}\left(\frac{b^3 - a^3}{x}\right), & x > b \end{cases}$,then which of the following is true?