Let C be the curve given by y=x3−3x. For each real number k with ∣k∣<2, the horizontal line y=k intersects C at three distinct points, enclosing a lens-shaped region Sk between the curve and the line. Let A(k) denote the area of Sk.
(a) Show that the intersection points satisfy the cubic equation x3−3x−k=0, and explain why the condition ∣k∣<2 guarantees exactly three real roots.
(b) Let a(k)<b(k) be the leftmost and rightmost intersection abscissae. Express A(k) as a single definite integral in terms of a(k), b(k), and k.
(c) Differentiate A(k) with respect to k to obtain A′(k). Find all critical values of k in (−2,2).
(d) Use a second-derivative test or other analytic argument to classify each critical value found in part (c) and determine the value of k that yields the maximum area.