If $f(x) = \begin{cases} \frac{1}{|x|} & ; |x| \geq 1 \\ ax^2 + b & ; |x| < 1 \end{cases}$ is differentiable at every point of the domain,then the values of $a$ and $b$ are respectively

Let $f(x)=a_0+a_1|x|+a_2|x|^2+a_3|x|^3$,where $a_0, a_1, a_2, a_3$ are real constants. Then $f(x)$ is differentiable at $x=0$ if and only if:

Let $f: R \rightarrow R$ be defined as $f(x) = \begin{cases} x^{5} \sin \left(\frac{1}{x}\right) + 5x^{2} & , x < 0 \\ 0 & , x = 0 \\ x^{5} \cos \left(\frac{1}{x}\right) + \lambda x^{2} & , x > 0 \end{cases}$. The value of $\lambda$ for which $f''(0)$ exists 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:

Which of the following functions is differentiable at $x = 0$?

The derivative of $y = 1 - |x|$ at $x = 0$ is