Inequalities

The AM-GM inequality states that for non-negative real numbers $a_1,a_2,\dots ,a_n$, the arithmetic mean is at least the geometric mean: $\frac{a_1+a_2+\cdots +a_n}{n}\ge \sqrt[n]{a_1{a}_2\dots a_n}$. Equality holds exactly when all terms are equal. For two positive numbers $a$ and $b$, this becomes $\frac{a+b}{2}\ge \sqrt{ab}$. This principle is used to prove inequalities and find minimum values of expressions where variables are positive.