1982 AHSME Problems/Problem 29
Problem
Let , and
be three positive real numbers whose sum is
. If no one of these numbers is more than twice any other,
then the minimum possible value of the product
is
Solution
Suppose that the product is minimized at
Without the loss of generality, let
and fix
To minimize we minimize
Note that
By a corollary of the AM-GM Inequality (If two nonnegative numbers have a constant sum, then their product is minimized when they are as far as possible.), we get
It follows that
Recall that so
This problem is equivalent to finding the minimum value of
in the interval
The graph of
is shown below:
............DIAGRAM GOES HERE ............
Since
has a relative minimum at
and cubic functions have at most one relative minimum, we conclude that the minimum value of
in
is at either
or
As
the minimum value of
in
is
~MRENTHUSIASM
See Also
1982 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 28 |
Followed by Problem 30 | |
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