Re-Post: The total cost of producing x books is C(x)=50,000+2x dollars, and the total revenue generated by selling x books for R(x)=10x-0.0001x^2 dollars.
1) Find R'(x)
2)Find C'(x)
3)Find d/dx (r(x)-C(x))
4) When is the derivative positive, negative?
5) To maximize total profits, how many books should be produced? What is the maximum total profit? Explain reason.
***How would you get 4) and 5) without a graph?***
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
Sorry I'm just confused.
+129849 It might be easier to answer 5, first, before answering 4
5. The profit function is -0.0001x^2 + 8x - 50000 which is an "upside-down" parabola.....it's form is -ax^2 + bx - c.......this type of parabola will always have a maximum........the x co-ordinate of the max is found by :
x = -b / -2a where a = 0.0001 and b = 8....so we have
-8 / [ -2(0.0001)] = 40000
So.....this means that 40,000 books sold will maximize the profit
To find what that profit is just put 40000 back into -0.0001x^2 + 8x - 50000 and you should get 110,000 [dollars] as the max profit
4. We can find the roots of -0.0001x^2 + 8x - 50000........by setting it to 0
Thus -0.0001x^2 + 8x - 50000 = 0
I used a computer algebra system to solve this and obtained ≈ 6833 and ≈ 73166 as roots
This means that from x ≈ 6733 to x ≈ 40,000, the curve will be "rising" and the derivative will be positive [the profit will be increasing until it hits the max value of x = 40000]
Likewise.....from x ≈ 40000 to x ≈ 73166, the curve will be "falling" and the dervative will be negative [the profit will be decreasing until it falls to 0 at 73166 books produced]
