If $f(x)=|x|+|\sin x|$ for $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$,then its left hand derivative at $x=0$ is

Let $[x]$ denote the greatest integer function and $f(x) = \begin{cases} 4x^2 + [2x]x, & \text{if } x \in [-\frac{1}{2}, 0) \\ ax^2 - bx, & \text{if } x \in [0, \frac{1}{2}) \end{cases}$. Then:

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,

Identify the correct statement about the function $f(x) = \max(x^2 - 1, 7 - x^2, 5)$.

Let $f(x) = \begin{cases} x + 1, & \text{when } x < 2 \\ 2x - 1, & \text{when } x \ge 2 \end{cases}$,then $f'(2) = $

If the function $f: R \rightarrow R$, defined by $f(x) = \begin{cases} 5-3x, & \text{if } x \leq \frac{5}{3} \\ x^2-3x+20, & \text{if } x > \frac{5}{3} \end{cases}$, then $f$ is