The sum of two numbers is 45. If one of the numbers is x, the other number is 45 - x. The product of the two numbers is P = (45 - x)*x = 45x - x^2
To determine the minimum value of the product the minimum value of P has to be determined. This is done by solving P' = 0 for x.
P' = 45 - 2x
45 - 2x = 0
=> x = 45/2 = 22.5
But P'' = -2
As a result the value of P for x = 22.5 is the point of maximum. It is not possible to determine a minimum value of the product, only the maximum value can be determined. For appropriate values of the numbers the sum can be 45 and the product can tend to `-oo` .
The minimum product of the numbers cannot be determined.