If $[t]$ denotes the greatest integer $\leq t$,then the number of points at which the function $f(x) = 4|2x + 3| + 9[x + \frac{1}{2}] - 12[x + 20]$ is not differentiable in the open interval $(-20, 20)$ is:

$A$ function $f$ is defined as follows:
$f(x) = \begin{cases} \sin x & \text{if } x \le c \\ ax + b & \text{if } x > c \end{cases}$
where $c$ is a known quantity. If $f$ is derivable at $x = c$,then the values of $a$ and $b$ are . . . . . . and . . . . . . respectively.

If $f(x) = \begin{cases} x + 2, & -1 < x < 3 \\ 5, & x = 3 \\ 8 - x, & x > 3 \end{cases}$,then at $x = 3$,$f'(x) = $

The function which is continuous for all real values of $x$ and differentiable at $x = 0$ is

Let $[x]$ denote the greatest integer less than or equal to $x$. Then the number of points where the function $y = [x] + |1 - x|$ for $-1 \leq x \leq 3$ is not differentiable,is

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 $........$.