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 ab, 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.]