The sum of two positive numbers is 120. The product of those numbers will be maximum if both numbers are equal to 60 and the maximum value is 3600.
Suppose we have a function f(x). At the extreme point (maximum/minimum), it holds:
f ' (x) = 0
In the given problem, let the two numbers be a and b. Then,
a + b = 120 or
a = 120 - b (equation 1)
f(b) = a x b = (120 - b) x b
f(b) = -b² + 120b
At the maximum point, the derivative with respect to b is zero,
f ' (b) = 0
-2b + 120 = 0
b = 120 / 2 = 60
Substitute b = 60 to equation 1,
a = 120 - 60 = 60
Hence, the two numbers that give maximal value of the product is 60 and 60. The product is 60 x 60 = 3600.
We can conclude that the maximum value of the product of those two numbers is 3600.
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