Chapter 11.8, Problem 47E

(a) Find the maximum value of

$f(x_1,x_2,...,x_n)=\sqrt[n]{x_1{x}_2...x_n}$

$f(x_1,x_2,...,x_n)=\sqrt[n]{x_1{x}_2...x_n}$

given that x₁, x₂, . . . , x_n are positive numbers and x₁ + x₂ + . . . + x_n = c, where c is a constant.

(b) Deduce from part (a) that if x₁, x₂, . . . , x_n are positive numbers, then

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

This inequality says that the geometric mean of n numbers is no larger than the arithmetic mean of the numbers. Under what circumstances are these two means equal?