SOLUTION: find two positive real numbers whose product is a maximum if the sum of the first and twice the second is 32.

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Question 421822: find two positive real numbers whose product is a maximum if the sum of the first and twice the second is 32.
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
Suppose the numbers are x and 2y, with x + 2y = 32. We wish to maximize the product xy.

Using the AM-GM inequality, %28x+%2B+2y%29%2F2+%3E=+sqrt%282xy%29. Replace x+2y with 32 to obtain

16+%3E=+sqrt%282xy%29

256+%3E=+2xy

xy+%3C=+128, hence the maximum product is 128. The maximum product occurs when all the terms are equal, i.e. x = 2y, which implies x = 16 and 2y = 16, y = 8. So our numbers are 16 and 8.