Chapter 13.10, Problem 44E

Geometric and Arithmetic Means

(a) Use Lagrange multipliers to prove that the product of three positive numbers x, y, and z, whose sum has the constant value S, is a maximum when the three numbers are equal. Use this result to prove that

$\sqrt[3]{xyz}\le \frac{x+y+z}{3}.$

(b) Generalize the result of part (a) to prove that the product $x_1{x}_2{x}_3\dots x_n$ is a maximum when

$x_1=x_2=x_3=\cdots =x_n,{\displaystyle \underset{i=1}{\overset{n}{\sum }}{x}_i=S},\text{and}\text{all}{x}_i\ge 0$

Then prove that

$\sqrt[n]{x_1{x}_2{x}_3\cdots x_n}\le \frac{x_1+x_2+x_3+\cdots +x_n}{n}.$

This shows that the geometric mean is never greater than the arithmetic mean.