Find two numbers whose sum is 23 and whose product is a maximum.

Let x = the first numberLet y = the second number x + y = 23                    eq1       y = 23- x 
xy = Product                 eq2

Substitute eq1 into eq2.  Lets get eq2 in terms of x.  

Product = x(23 - x)Product = - x2 + 23x
 
we must find the vertex has coordinate (h, k). h = -b / 2a  where:a = -1b = 23
h= -23  / -2  =  11.5 

 x = 11.5   


Substitute this value of x into eq1 to find y. y = 23 - xy = 23 - 11.5y = 11.5

The two numbers that will give the largest product possible are 11.5 and 11.5.