Let g: R R be a differentiable function with g(0)=0, g'(0)=0 and g'(1) 0. Let f(x)= cases x |x| g(x), & x 0 0, & x=0 cases and h(x)=e^ |x| for all x R . Let (f h)(x) denote f(h(x)) and (h f)(x) denote h(f(x)). Then which of the following is (are) true?

JEE Advanced 2015 Mathematics question

Let be a differentiable function with , and . Let

f(x)=

and for all . Let denote and denote . Then which of the following is (are) true?

(A) is differentiable at Correct
(B) is differentiable at
(C) is differentiable at
(D) is differentiable at Correct

Answer: (A),(D)

Solution

(A) ; since , the limit is 0 from both sides, so is differentiable at 0. (B) has right derivative and left derivative at 0: not differentiable. (C) , so $(f°

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Question ID VK-026419