**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_{}.]