Analyze profit functions

Question

For a certain company the cost of function for producing c items is C(x)=40x+100 and the revenue function for selling x items is R(x)=-0.5(x-100)^2+5,000. The maximum capacity of the company is 150 items.

assuming the company sells all that it produces what is the profit function? P(x)=

what is the domain of P(x)?

the company can choose to produce either 60 or 70 items. What is their profit for each case and which level production should they choose?

Answer 1

P(x) = R(x) - C(x) = -x^2/2 + 100x - 40x - 100 = -x^2/2 +60x -100

P'(x) = -x + 60 = 0

x =60 is the profit maximizing output level

P(60) = -(60)^2/2 +60(60) -100 = 1700 = maximum profit

P(70) = -(70)^2/2 + 60(70) -100 = 1650

domain is 0 < x < 150

P(0) = -100 = P(120)

P(150) = a greater loss = -2,350

doman for non-negative profits is about 1.5 to 118.5

go below or above that, and you suffer losses, with no profits