The left-hand derivative of $f(x) = [x] \sin(\pi x)$ at $x = k$,where $k$ is an integer and $[\cdot]$ denotes the greatest integer function,is:

The number of critical points of the function $f(x) = \begin{cases} |\frac{\sin x}{x}|, & x \ne 0 \\ 1, & x = 0 \end{cases}$ in the interval $(-2\pi, 2\pi)$ is equal to:

The value of $m$ for which the function $f(x) = \begin{cases} mx^2, & x \le 1 \\ 2x, & x > 1 \end{cases}$ is differentiable at $x = 1$,is

Let $g: [-2, 2] \rightarrow R$ and $f: [-2, 2] \rightarrow R$ be two functions defined as $g(x) = \begin{cases} -1, & \text{if } -2 \le x < 0 \\ x^2 - 1, & \text{if } 0 \le x \le 2 \end{cases}$ and $f(x) = |g(x)| + g(|x|) + 2$. In the interval $(-2, 2)$, $f$ is not differentiable at $x = $

The function $f(x)=|x^{2}-2 x-3| \cdot e^{|9 x^{2}-12 x+4|}$ is not differentiable at exactly :

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