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

Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be functions satisfying $f(x+y)=f(x)+f(y)+f(x)f(y)$ and $f(x)=x g(x)$ for all $x, y \in R$. If $\lim _{x \rightarrow 0} g(x)=1$,then which of the following statements is/are $TRUE$?
$(A)$ $f$ is differentiable at every $x \in R$
$(B)$ If $g(0)=1$,then $g$ is differentiable at every $x \in R$
$(C)$ The derivative $f^{\prime}(1)$ is equal to $1$
$(D)$ The derivative $f^{\prime}(0)$ is equal to $1$

The function $f(x) = |\cos x|$ is

Number of points where the function $f(x) = \text{maximum}(\sqrt{2x - x^2}, 2 - x)$ is non-differentiable is:

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

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 $x = 0$?