By considering the function

MATHEMATICS · Subjective · AoPS · Mathematics · More Inequalities · More Inequalities

Problem

y = (a_1x - b_1)^2 + (a_2x - b_2)^2

, use quadratic theory to prove the Cauchy-Schwarz inequality

|a_1b_1 + a_2b_2| \le \sqrt{a_1^2 + a_2^2} \sqrt{b_1^2 + b_2^2}