Let $f(x) = \begin{cases} 1, & \forall x < 0 \\ 1 + \sin x, & \forall 0 \le x \le \pi/2 \end{cases}$,then what is the value of $f'(x)$ at $x = 0$?

If $f(x) = a|\sin x| + be^{|x|} + c|x|^3$,where $a, b, c \in \mathbb{R}$,is differentiable at $x = 0$,then:

The function $f(x) = x^2 \sin \frac{1}{x}$ for $x \ne 0$ and $f(0) = 0$ at $x = 0$ is:

Let $R$ denote the set of all real numbers. Define the function $f: R \rightarrow R$ by $f(x) = \begin{cases} 2-2x^2-x^2 \sin \frac{1}{x} & \text{if } x \neq 0 \\ 2 & \text{if } x=0 \end{cases}$. Then which one of the following statements is True?

Let $a \in Z$ and $[t]$ be the greatest integer $\leq t$. Then the number of points,where the function $f(x) = [a + 13 \sin x], x \in (0, \pi)$ is not differentiable,is $........$.

Let $S = \{(\lambda, \mu) \in R \times R : f(t) = (\|\lambda\|e^{\|t\|} - \mu) \sin(2\|t\|), t \in R\}$ be a differentiable function. Then $S$ is a subset of?