Consider the function h(x)= [3] x +|x|

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Consider the function h(x)=x3+∣x∣,h(x)=\sqrt[3]{x}+|x|, which is defined for all real xx. (a) Show that hh is continuous at x=0x=0. (b) For x≠0x\neq 0, compute h′(x)h'(x). Present your work separately for x>0x>0 and x<0x<0 if necessary. (c) Evaluate lim⁡x→0−h′(x)\displaystyle\lim_{x\to 0^-}h'(x) and lim⁡x→0+h′(x)\displaystyle\lim_{x\to 0^+}h'(x). (d) Does hh have a tangent line at x=0x=0? If not, explain which term of hh is responsible for the failure of differentiability and characterize the type of nondifferentiable point observed.

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