Question 1129224: Find two positive real numbers whose product is a maximum. The sum is 978.
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! the product of numbers whose sum is a constant is half the sum^2, so here it would be 489^2 or 239,121.
(you can do this by finding the maximum when the sum is 100. It occurs when each number is 50 and the product is 2500. Furthermore, the distance from the maximum squared subtracted from the sum is the product, so 45*55 is 2475.)
x+y=978
maximize xy
y=978-x
so xy=978-x^2
The maximum occurs when x is -b/2a, the vertex, or -978/-2 or 489
The two numbers are both 489.
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