MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A polynomial family is defined by fk(x)=x3+kx, where k is a real parameter.
(a) Determine all critical points of fk in terms of k.
(b) Explain why fk possesses no relative extremum when k≥0. Your answer should reference the behavior of fk′(x) on the real line.
(c) For k<0, describe which critical point corresponds to a relative maximum and which corresponds to a relative minimum. Justify your conclusion by analyzing the sign of fk′(x) on the intervals determined by the critical points.