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Calculus

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#Calculus
Let f(x)= |x|^3/2, x€R then
A.f is uniformly continuous
B.f is Continous but not differentiable ar x=0
C. f is differentiable and f' is continuous
D. f is differentiable but f' is discontinuous at x=0
What is the answer and how to solve this kind of questions?

My Answer is option D ,

I want to confirm if my reasoning to this question is correct as im learning calculus now. 

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