MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Define h:R→R by h(x)=x2−41/3.
(1) Verify that h is continuous at every real number.
(2) Show that h is not differentiable at x=2. Your argument must rely on the limit definition of the derivative and should identify the specific feature (corner, cusp, or vertical tangent) causing non-differentiability.
(3) Determine whether h is differentiable at x=−2 and provide a complete justification.
(4) Identify the set of all real numbers at which h is differentiable, and summarize your reasoning clearly.