If f(x) = a| x| + be^ |x| + c|x|^3, where a, b, c ∈ R, is differentiable at x = 0, then

JEE Main 2012 Mathematics question

If , where a, b, c ∈ R, is differentiable at x = 0, then

(1)a = 0, b and c are any real numbers
(2)c = 0, a = 0, b is any real number
(3)b = 0, c = 0, a is any real number
(4)a = 0, b = 0, c is any real numberCorrect

Answer: (4)

Solution

Near 0, |sin x| and each have one-sided derivatives ±1 at x = 0 (corners), while is differentiable there with derivative 0. So and ; f is differentiable at 0 exactly

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Question ID VK-103703