JEE-Advanced 2015 (16/40)

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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