Calculus, Pleas show all work

Question

 

5) a) Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that revenue, R(x), and cost, C(x), are in dollars. R(x) =50x – 0.5x2, C(x) = 4x + 10. Hint: first find total profit P(x).

 

 

a) Find C’(x), R’(x), and P’(x)

 

 

 

 

 

 

        b) Find C’ (50), R’ (50), and P’ (50)

Answer 1

Profit = Revenue minus Cost

R(x) =50x – 0.5x2

C(x) = 4x + 10

P(x) = R(x) - C(x) = 50x - 0.5x² - 4x - 10 = 46x - 0.5x² - 10

R'(x) = 50 - x

C'(x) = 4

P'(x) = 46 - x

R'(50) = 0

C'(50) = 4

P'(50) = -4