Consider the function defined for all real numbers by h(x)=|x^ 3 -3x|
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Consider the function h defined for all real numbers by
h(x)=∣x3−3x∣.
The absolute value introduces sharp corners into the graph of h.
Analyze the first derivative h′ as follows.
(a) Determine all real numbers c for which h is not differentiable at x=c. Give a complete justification based on the definition of differentiability.
(b) For x=c, write a simplified algebraic expression for h′(x) as a piecewise-defined function. Show all steps in your derivation.
(c) Create a number line sign chart for h′(x), indicating precisely where the derivative is positive, negative, or zero.
(d) Using your work in parts (a)–(c), sketch an accurate graph of y=h′(x) on the interval [−3,3]. Label intercepts, discontinuities, and any horizontal asymptotes if they occur.
(e) Discuss the relationship between the zeros and discontinuities of h′ and the critical points and sharp corners of h. Support your discussion with specific references to the graph of h′ you sketched.