Consider the polynomial function defined for all real numbers.
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Consider the polynomial function f(x)=x3 defined for all real numbers.
(a) Determine all critical points of f on (−∞,∞). Show all analytical work that leads to your conclusion.
(b) For each critical point you identified in part (a), decide whether it is a local maximum, a local minimum, or neither. Provide a clear justification based on an appropriate derivative test.
(c) Describe the behavior of f in a neighborhood of the critical point(s) by referencing both the sign of f′(x) and the concavity indicated by f′′(x). Explain how this behavior supports your conclusion in part (b).