If both $f(x)$ and $g(x)$ are differentiable functions at $x = x_0$,then the function defined as $h(x) = \text{Maximum} \{f(x), g(x)\}$:

Given $f(x) = \begin{cases} 1 + x & x < 0 \\ 2 - 3x & x \geq 0 \end{cases}$,find the critical point $x = \dots \dots$.

Let the functions $f, g$ and $h$ be defined as follows:
$f(x) = \begin{cases} x \sin \left( \frac{1}{x} \right) & \text{for } -1 \le x \le 1, x \ne 0 \\ 0 & \text{for } x = 0 \end{cases}$
$g(x) = \begin{cases} x^2 \sin \left( \frac{1}{x} \right) & \text{for } -1 \le x \le 1, x \ne 0 \\ 0 & \text{for } x = 0 \end{cases}$
$h(x) = |x|^3$ for $-1 \le x \le 1$.
Which of these functions are differentiable at $x = 0$?

If the function $g(x) = \begin{cases} ae^x, & x \le 0 \\ b\cos x + x, & x > 0 \end{cases}$ is differentiable,then the value of $a^2 + b^2$ is

The function $y = e^{-|x|}$ is

Let $\alpha, \beta \in R$ be such that the function $f(x) = \begin{cases} 2 \alpha (x^2 - 2) + 2 \beta x, & x < 1 \\ (\alpha + 3) x + (\alpha - \beta), & x \ge 1 \end{cases}$ is differentiable at all $x \in R$. Then $34(\alpha + \beta)$ is equal to