MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Let h be the function
h(x)=x2/3(2−x),−1≤x≤2.
Answer the following questions to analyze the extreme values of h on its domain.
(a) Show that h is continuous on [−1,2].
(b) Compute h′(x) for all x where the derivative exists, and find every interior point of [−1,2] at which h′(x)=0.
(c) Determine all interior points of [−1,2] where h′(x) does not exist and classify the nature of the nondifferentiability (sharp corner, vertical tangent, or cusp).
(d) Produce a table listing h(x) for each endpoint of the interval and for every interior point identified in parts (b) and (c).
(e) Using the information compiled in part (d), state the absolute maximum value and the absolute minimum value of h on [−1,2], giving clear justification for your selections.