The sum of two numbers is 18. If one of the numbers is x, the other number is 18 - x.
The product of the two numbers is P = x*(18 - x) = 18x - x^2
To determine the maximum value of P, solve P' = 0 for x.
P' = 18 - 2x
P' = 0
=> 18 - 2x = 0
=> x = 9
This gives the other number as 9 too, the product of the two is 9*9 = 81
The maximum value of the product of the numbers is 81.