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: 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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