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The sum of the numbers when they are added is 18. Let one of the numbers be x, the other number is 18 - x. The product of the two numbers is (18 - x)*x = 18x - x^2
The product f(x) = 18x - x^2 has to be minimized. The extreme value of f(x) lies at the points where f'(x) = 0 and this is a minimum if the value of f''(x) at that point is positive.
f'(x) = 18 - 2x
f''(x) = -2
It is seen that the second derivative f''(x) is negative for all values of x. For any two numbers that add up to 18 it is always possible to determine two numbers that have a smaller product no matter what numbers are chosen initially.
This makes it impossible to determine two numbers that add up to 18 such that the product of the numbers is minimum.
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