Exercise 50

  1. Maximize \(\sum_{i=1}^n x_i y_i\) subject to the constraints \(\sum_{i=1}^n x_i^2 = 1\) and \(\sum_{i=1}^n y_i^2 = 1\).
  2. 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 \(\sum a_i b_i \le \sqrt{\sum a_i^2} \sqrt{\sum b_i^2}\) for any numbers \(a_1, ..., a_n, b_1, ..., b_n\). This inequality is known as the Cauchy-Schwarz Inequality.