MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
For a real parameter k, define the function
Hk(x)=x4+kx2
for all real x.
(1) Find all critical points of Hk in terms of k. Express your answers clearly, indicating any conditions on k that change the number or location of critical points.
(2) Compute the second derivative Hk′′(x) and evaluate it at each critical point found in part (a).
(3) Determine all real values of k for which Hk possesses exactly two distinct relative extrema. Show all reasoning leading to your conclusion.
(4) For each value of k identified in part (c), classify every critical point as a relative maximum or a relative minimum and give the coordinates of each extremum point.
(5) Describe how the local extremum structure of Hk changes as k moves through the intervals k<0, k=0, and k>0. Support your description with analytical justification.