Maximize the product - Math Central

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Question from David, a student:

Hi i have this site call calcchat.com, but i don't understand how they explained this can you take a look? The question is:

Direction: Find two positive numbers that satisfy the given requirements.

The sum is S and the product is a maximum
this is what they did
1) Let x and y be two positive numbers such that x + y = S
2)P = xy
3) = x (S - x)
4) =Sx - x^2
5)...etc. the thing i don't get is how did they go from step 2 to step 3
and also i know this sound dumb but how did they get step 2? =)

Hi David,

You want to maximize the product of x and y which is

P = xy.

You are also told the the sum of x and y is S so x + y = S. Solving this equation for y gives

y = S - x.

Substitute this value for y into the product expression to get

P = xy = x(S - x)

At this point use the distributive law which says for any numbers a, b and c, a × (b + c) = a × b + a × c. Thus

P = x(S - x) = xS - xx = Sx - x²

I hope this helps,
Harley

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