If $f(x) = \begin{cases} e^x + ax, & x < 0 \\ b(x - 1)^2, & x \ge 0 \end{cases}$ is differentiable at $x = 0$,then $(a, b)$ is

If the function $f(x) = \begin{cases} \frac{1}{|x|} & , |x| \geq 2 \\ ax^2 + 2b & , |x| < 2 \end{cases}$ is differentiable on $\mathbb{R}$,then $48(a+b)$ is equal to . . . . . . .

Let $g: R \rightarrow R$ be a differentiable function with $g(0)=0, g^{\prime}(0)=0$ and $g^{\prime}(1) \neq 0$. Let $f(x)=\begin{cases} \frac{x}{|x|} g(x), & x \neq 0 \\ 0, & x=0 \end{cases}$ and $h(x)=e^{|x|}$ for all $x \in R$. Let $(f \circ h)(x)$ denote $f(h(x))$ and $(h \circ f)(x)$ denote $h(f(x))$. Then which of the following is (are) true?
$(A)$ $f$ is differentiable at $x=0$
$(B)$ $h$ is differentiable at $x=0$
$(C)$ $f \circ h$ is differentiable at $x=0$
$(D)$ $h \circ f$ is differentiable at $x=0$

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

Let $f$ be a differentiable function from $R$ to $R$ such that $|f(x) - f(y)| \le 2|x - y|^{\frac{3}{2}}$ for all $x, y \in R$. If $f(0) = 1$,then $\int_{0}^{1} f^2(x) dx$ is equal to

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