Problem 7

Find the maximum of the product: (4--2).

[Problem submitted by Steve Lee, LACC Professor of Mathematics.]

Solution for Problem 7:

According to the arithmetic-geometric mean theorem,

(2) (4--2)≦ ==

∴the maximum of (2) (4--2)=

∴the maximum of (4--2)=

[The arithmetic-geometric mean theorem: If a, b, and c are positive numbers with a fixed sum, then or equivalently abc≦.

The equal sign occurs only when a=b=c.

Proof: If any two factors of abc are not equal, say a≠b, then a=m+d and b=m-d, where m= and d=.

abc=(m+d)(m-d)c=c<c. That is, if we replace a, b, c by m, m, c the sum remains the same but the product is larger. Therefore abc is a maximum if and only if every factor is equal to the mean.]