Chapter 12.9, Problem 60E

Geometric and arithmetic means Given positive numbers x₁, …, x_n, prove that the geometric mean (x₁x₂ … x_n)^1/n is no greater than the arithmetic mean (x₁+ … + x_n)/n in the following cases.

a.    Find the maximum value of xyz, subject to x + y + z = k, where k is a real number and x > 0, y > 0, and z > 0. Use the result to prove that

     ${(xyz)}^{1/3}\le \frac{x+y+z}{3}.$

b.    Generalize part (a) and show that

     ${(x_1{x}_2\cdots x_n)}^{1/n}\le \frac{x_1+\cdots +x_n}{n}.$