We already know (1) and (2)
3) R(x) - C( x) =
10x - 0.0001x^2 - [ 50000 + 2x ] =
-0.0001x^2 + 8x - 50000 this the "profit" function
The derivative of this is
-0.0002x + 8
4) Set the derivative to 0 and solve
-0.0002x + 8 = 0
8 = 0.0002x divide both sides by 0.0002
x = 40000
.Here's the profit function graph : https://www.desmos.com/calculator/ejbztbmci4
The derivative will be positive from about ( 6834, 40000) and negative from about (40000, 73,166)
5) 40000 books maximizes the profit......and, as the graph shows, the max profit is $110,000
The maximum profit in this type of problem will always occur at the x value which makes the derivative = 0
