What can be the minimum product of the two numbers if the sum of two numbers is 45?
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.