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...

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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**.