A Simple Differential Proof of the Inequality Between the Arithmetic and Geometric Means

Abstract

We prove that if x₁, x₂, ..., x_n are positive numbers not all equal, then

$$ {\left[ {\left( {{x_1} + \cdots + {x_n}} \right)/n} \right]^n} - {x_1} \cdots {x_n} > 0. $$