S = x1y1 + x2y2 +· · · + xnyn, where x1, x2, . . . , xn and y1, y2, . . . , yn are orderings of two different sequences of positive real numbers, each containing n elements.
a) Show that S takes its maximum value over all orderings of the two sequences when both sequences are sorted (so that the elements in each sequence are in nondecreasing order).
b) Show that S takes its minimum value over all orderings of the two sequences when one sequence is sorted into nondecreasing order and the other is sorted into nonincreasing order.