A fast-food restaurant determines the cost and revenue models for its hamburgers: C(x) = 66,000x - x^2, 0 < x < 50,000 and R(x) = 20,000x, 0 < x < 50,000.
(a) Write the profit function for this situation:
P(x) = R(x) - C(x) = 20,000x - (66,000x - x^2) = -x^2 + 86,000x, 0 < x < 50,000.
(b) Determine the intervals on which the profit function is increasing and decreasing (Enter your answers using interval notation.)
increasing: (0, 43,000)
decreasing: (43,000, 50,000)
Determine how many hamburgers the restaurant needs to sell to obtain maximum profit: hamburgers
Explain your reasoning: Because the function changes from decreasing to increasing at this value of x, the maximum profit occurs at this value.