Chapter 14.8, Problem 49E

(a) Find the maximum value of

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

given that $x_1,x_2,\mathrm{...},x_n$ are positive numbers and $x_1+x_2+\mathrm{...}+x_n=c$, where c is a constant.

(b) Deduce from part (a) that if $x_1,x_2,\mathrm{...},x_n$ are positive numbers, then

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

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?