(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

(b) Generalize the result of part (a) to prove that the product

is a maximum when

Then prove that

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

Transcribed Image Text:

3/xyz ≤ x + y + z 3