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

If $f(x) = [x]$,where $[x]$ is the greatest integer not greater than $x$,then $f^{\prime}(1^{+}) = \dots$.

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 $f(x) = \begin{cases} a \sin(x + b) & x \ge 0 \\ 6x^7 - x + 1 & x < 0 \end{cases}$ be differentiable for all real $x$. If $a \in \mathbb{R}$ and $b \in [0, 2\pi]$,then the number of ordered pairs $(a, b)$ is:

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

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 $S$