MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Consider the cubic polynomial f(x)=x3−3kx2+12x+5, where k is a real parameter.
(i) Obtain f′(x) and show that the locations of its critical points are the real roots of a quadratic equation in x that involves k. Hence write the discriminant of this quadratic in terms of k.
(ii) Determine the range of k for which f(x) possesses two distinct real critical points, and classify each critical point as a local maximum or a local minimum.
(iii) For one admissible value of k, verify your conclusion by sketching a rough graph of f(x) indicating the positions of the turning points and their nature.