Let $a, b \in R$ and $f: R \rightarrow R$ be defined by $f(x)=a \cos (|x^3-x|)+b|x| \sin (|x^3+x|)$. Then $f$ is

Define a function $f: R \rightarrow R$ by $f(x) = \begin{cases} \frac{\sin x^2}{x}, & \text{for } x < 0 \\ x^2 + ax + b, & \text{for } x \geq 0 \end{cases}$. Suppose $f(x)$ is differentiable on $R$. Then,

If $f(x)=\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$,then $f(x)$ is differentiable on

Let $f : R \to R$ be a function such that $|f(x)| \leq x^2$,for all $x \in R$. Then,at $x = 0$,$f$ is

$\text{The domain of the derivative of the function } f(x) = \begin{cases} \tan^{-1} x, & \text{if } |x| \le 1 \\ \frac{1}{2}(|x|-1), & \text{if } |x| > 1 \end{cases} \text{ is given by:}$

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