JEE-Advanced 2015 (16/40)

Calculus Level 4

Let g:RRg:\mathbb{R} \to \mathbb{R} be a differentiable function with g(0)=0,g(0)=0g(0)=0, g'(0)=0 and g(1)0g'(1) \neq 0. Let f(x)={xxg(x)  ,x00   , x=0f(x)=\begin{cases} \frac{x}{|x|} g(x) \ \ , \quad x \neq 0 \\ 0 \ \ \ , \ \quad x=0 \end{cases} and h(x)=exh(x)=e^{|x|} for all xRx \in \mathbb{R}. Let (foh)(x)(foh)(x) denotes f(h(x))f(h(x)) and (hof)(x)(hof)(x) denotes h(f(x))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)(1),(2), then submit 12 as the correct answer, if your answer is (2),(3),(4)(2),(3),(4), then submit 234 as the correct answer.

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