It was obtained by Cauchy, a great mathematician, when he studied the residue problem in mathematical analysis. But from a historical point of view, this inequality should be called Cauch-Buniakowsky-Schwarz inequality, because it is the latter two mathematicians who independently popularized this inequality in integral calculus and applied it to an almost perfect degree.

Cauchy inequality is a very important inequality. Flexible and ingenious application can solve some difficult problems. It can be used to prove inequalities, solve triangle-related problems, find the maximum function and solve equations.

Proof of Cauchy inequality

Cauchy inequality is generally proved as follows:

The formal writing of Cauchy inequality is: If the number of two columns is AI and BI, there is (∑ AI 2) * (∑ BI 2) ≥ (∑ AI * BI) 2.

Let f (x) = ∑ (ai+x * bi) 2 = (∑ bi 2) * x 2+2 * (∑ ai * bi) * x+(∑ ai 2).

Then we know that there is always f(x) ≥ 0.

Using the condition that the quadratic function has no real root or only one real root, there are δ = 4 * (∑ AI * Bi) 2-4 * (∑ AI 2) * (∑ Bi 2) ≤ 0.

So move the item to the conclusion.