Chapter 1.8, Problem 39E

Let $S=x_1{y}_1+x_2{y}_2+\cdot \cdot \cdot +x_n{y}_n$ , where $x_1,x_2\mathrm{...},x_n$ and $y_1,y_2\mathrm{..},y_n$ are orderings of two different sequences of positive real numbers, each containing n elements.

a) Show thatStakes its maximum value over all orderings of the two sequences when both sequences are sorted (so that the elements in each sequence are in non-decreasing order).

b) Show that S takes its minimum value over all orderings of the two sequences when one sequence is sorted into non-decreasing order and the other is sorted into non-increasing order.