If $f(x) = x(\sqrt{x} - \sqrt{x + 1}),$ then

Let $f(x) = |x-3| + |x+5|$ and $A = \{a \in \mathbb{R} \mid \lim_{x \rightarrow a} \frac{f(x)-f(a)}{x-a} \text{ exists} \}$. Then the number of real numbers which are in $(-\infty, -3) \cup (5, \infty)$ but not in $A$ is

If $f(x) = \begin{cases} x^{\alpha} \sin \left( \frac{1}{x} \right), & x \neq 0 \\ 0, & x = 0 \end{cases}$; Which of the following is true?

If $f$ is a real-valued differentiable function satisfying $|f(x) - f(y)| \le (x - y)^2$ for all $x, y \in R$ and $f(0) = 0$,then $f(1)$ is equal to:

$f(x) = ||x| - 1|$ is not differentiable at

The function $f(x) = e^{-|x|}$ is