Chapter 14.8, Problem 50E

(a) Maximize ${\Sigma }_{i=1}^n{x}_i{y}_i$ subject to the constraints

${\Sigma }_{i=1}^n{x}_i^2=1$ and ${\Sigma }_{i=1}^n{y}_i^2=1$.

(b) Put

$x_i=\frac{a_i}{\sqrt{\sum a_j^2}}$ and $y_i=\frac{b_i}{\sqrt{\sum b_j^2}}$

to show that

${\displaystyle \sum a_i{b}_i}\le \sqrt{\Sigma a_j^2}\sqrt{\Sigma b_j^2}$

for any numbers $a_1,\mathrm{...},a_n,b_1,\mathrm{...},b_n$. This inequality is known as the Cauchy-Schwarz Inequality.