SOLUTION: What is the maximum product of 2 numbers whose sum is 45? What numbers yield this product?

Let x and y be the numbers; let x be the lesser number and y be the greater number.


Consider the mean value a =  =  = 22.5  and half the distance between x and y,  b = 


Then, obviously,  x = 22.5 - b  and  y = 22.5 + b, so we can write


    xy = (22.5-b)*(22.5+b) = .


From this formula, it is clear that xy is maximum when b= 0, i.e. when the numbers x and y are equal: x = y.


Thus we get that the maximum of the product xy  is achieved when x = y.


Together with x + y = 45 it implies that the maximum is achieved in this problem at x = y = 45/2 = 22.5.


The maximum value of xy is  then   = 506.25.   


ANSWER.  The maximum is achieved when both numbers are equal to half of the sum, i.e. 22.5.

         The maximum value of the product is  506.25  then.