Consider the cubic polynomial g(x)=x3+ax2+bx+c, where a, b, and c are real constants. Assume that g has a relative maximum at x=−2 and a relative minimum at x=1.
(a) Derive the system of equations that a and b must satisfy based on the given information about the locations of the extrema.
(b) Solve the system obtained in part (a) to determine the exact values of a and b.
(c) Confirm that x=−2 is a relative maximum and x=1 is a relative minimum by analyzing the sign of g′(x) on the intervals determined by the critical points.