The function $f(x) = a \sin |x| + b e^{|x|}$ is differentiable at $x = 0$ when

Let $[x]$ denote the greatest integer function and $f(x) = \begin{cases} 4x^2 + [2x]x, & \text{if } x \in [-\frac{1}{2}, 0) \\ ax^2 - bx, & \text{if } x \in [0, \frac{1}{2}) \end{cases}$. Then:

Which of the following is differentiable at $x=0$?

Suppose that $f(x)$ is a differentiable function such that $f^{\prime}(x)$ is continuous, $f^{\prime}(0)=1$ and $f^{\prime \prime}(0)$ does not exist. Let $g(x)=x f^{\prime}(x)$. Then,

Let $f(x) = \begin{cases} a \cot^{-1} \left( \frac{b+x}{4} \right), & \frac{-2}{3} < x < 0 \\ 2, & x = 0 \\ \frac{\ln(1-cx)}{x}, & 0 < x < \frac{2}{3} \end{cases}$. If the function $f(x)$ is differentiable at $x = 0$,then find the value of $(b^2 - 2a + c^6)$.

Let $f(x) = \begin{cases} \max \{|x|, x^2\}, & |x| \le 2 \\ 8 - 2|x|, & 2 < |x| \le 4 \end{cases}$. Let $S$ be the set of points in the interval $(-4, 4)$ at which $f$ is not differentiable. Then $S$