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.]