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

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justaguide | College Teacher | (Level 2) Distinguished Educator

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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.

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