Let set x and y be the numbers.
Taking note that the sum of the numbers is 800.
Our first equation will be x + y = 800.
Solve for x in terms of y, subtract both sides by y.
x = 800 - y.
Let set P = the product of the two numbers.
So, we will have: P = xy.
Plug-in x = 800 - y.
P = (800 - y)(y)
Use Distributive property.
P = 800y - y^2
Take the derivative.
P' = 800 - 2y.
Equate the zero for the critical number.
800 - 2y = 0
Add 2y on both sides.
800 = 2y
Divide both sides by 2.
y = 400
Take the second derivative P'' = -2. hence, we have a mximum at y = 400.
Solve for x = 800 - 400 = 400.
Therefore the largest possible number that will create the largest product are 400 and 400.