MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
For each real parameter k, define the function qk(x)=x3+kx on (−∞,∞). The goal of this problem is to determine for which values of k the function qk possesses a critical point that is not a local extremum.
(a) Compute qk′(x) and find every critical point in terms of k. Distinguish carefully between roots of the derivative and points where the derivative is undefined.
(b) For each critical point obtained in part (a), apply an appropriate test to decide whether the point is a local maximum, a local minimum, or neither. Show all reasoning, and express your conclusions in terms of k.
(c) Determine the complete set of real values of k for which qk has at least one critical point that is not a local extremum. State your answer clearly and provide a rigorous justification drawing upon your earlier analysis.
(d) Reflect on how the parameter k influences the number and type of critical points of qk, and explain why your final result is consistent with the overall shape of the graph of qk.