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

Let $f(x) = \begin{cases} |4x^2 - 8x + 5|, & \text{if } 8x^2 - 6x + 1 \geq 0 \\ [4x^2 - 8x + 5], & \text{if } 8x^2 - 6x + 1 < 0 \end{cases}$,where $[\alpha]$ denotes the greatest integer less than or equal to $\alpha$. Then the number of points in $\mathbb{R}$ where $f$ is not differentiable is $.......$

Which of the following statements is true?

Let the function $f: R \rightarrow R$ be defined by $f(x)=x-x^2+(x-1) \sin x$ and let $g: R \rightarrow R$ be an arbitrary function. Let $f g: R \rightarrow R$ be the product function defined by $(f g)(x)=f(x) g(x)$. Then which of the following statements is/are $TRUE$?
$(A)$ If $g$ is continuous at $x=1$,then $f g$ is differentiable at $x=1$
$(B)$ If $fg$ is differentiable at $x=1$,then $g$ is continuous at $x=1$
$(C)$ If $g$ is differentiable at $x=1$,then $f g$ is differentiable at $x=1$
$(D)$ If $fg$ is differentiable at $x=1$,then $g$ is differentiable at $x=1$

Let $f(x) = |2x^2 + 5|x| - 3|$,$x \in R$. If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively,then $m + n$ is equal to:

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