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]
