Question 994285
The total profit $P, generated from the production and marketing of n items of a certain product is given by 
P = −10800n − 4 n^3 + 600n^2 −122
How many items should be made for maximum profit? What is the maximum profit? 

Enter your answers as a list [in square brackets] of the form: [ n, p] 
for some number of items n and profit p

This is what I have

P = −10800n − 4 n^3 + 600n^2 −122
P' = -10800 -12n^2 + 1200n

-10800 -12n^2 + 1200n = 0
n = 10, n = 90

But I am not sure what to do from there?
Do I sub them in?

-10800 -12(10)^2 + 1200(10) = 0
-10800 -12(90)^2 + 1200(90) = 0

I'm lost!

Thank you!!
<pre>You're correct up to this point: {{{P = - 10800 - 12n^2 + 1200n}}} -------> {{{P(n) = - 12n^2 + 1200n - 10800}}}
Now, maximum profit (P) is realized at: {{{n = - b/(2a)}}}
With b being 1200, and a being - 12, {{{n = - b/(2a)}}} becomes: {{{n = - 1200/(2 * - 12)}}}, or {{{n = 1200/24}}}, or 50
Number of units at which maximum profit occurs is: {{{highlight_green(n = 50)}}}

As mentioned before, this means that maximum profit is realized at n = 50 (I presume this should be in 000s, or millions) units

{{{P = - 12(50)^2 + 1200(50) - 10800}}} -------- Substituting 50 for n
{{{P = - 12(2500) + 60000 - 10800}}}
P, or maximum profit = - 30,000 + 60,000 - 10,800, or {{{highlight_green("$"19200)}}}
This results in: [n, P], or [{{{50}}}{{{","}}}{{{19200}}}]