Calculus Level 4

Let $g:\mathbb{R} \to \mathbb{R}$ be a differentiable function with $g(0)=0, g'(0)=0$ and $g'(1) \neq 0$. Let $f(x)=\begin{cases} \frac{x}{|x|} g(x) \ \ , \quad x \neq 0 \\ 0 \ \ \ , \ \quad x=0 \end{cases}$ and $h(x)=e^{|x|}$ for all $x \in \mathbb{R}$. Let $(foh)(x)$ denotes $f(h(x))$ and $(hof)(x)$ denotes $h(f(x))$. Then which of the following is (are) true ?
$\begin{array}{} (1) \, f \ \text{is differentiable at} \ x=0 \quad \quad \quad \quad & (2) \, h \ \text{is differentiable at} \ x=0 \\ (3) \, foh \ \text{is differentiable at} \ x=0 & (4) \, hof \ \text{is differentiable at} \ x=0 \end{array}$
Note :

• Submit your answer as the increasing order of the serial numbers of all the correct options.

• For eg, if your answer is $(1),(2)$, then submit 12 as the correct answer, if your answer is $(2),(3),(4)$, then submit 234 as the correct answer.

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