A company looks to maximize its profits.
(a) The demand function relates the price of a product and the quantity sold. Suppose that the company sells 1000 units when the price is 10 dollars, and sells 20% fewer units every time the price is increased by one dollar (selling the corresponding amount more due to price decreases). Find a formula for x = f (p), the number of goods sold when the price is p dollars.
(b) The revenue is the money the company receives from sales. Find a formula for R = g(p), the revenue the company receives when the price is p dollars. (Hint: revenue is the price times the quantity sold.)
(c) The company can manufacture x goods for a cost of C = 1000 + 2x dollars. Find a formula for C = h(p), the cost of producing the number of goods sold when the price is p dollars.
(d) The profit is revenue minus cost. Find a formula for P = j(p), the profit earned by the company when the price is p dollars. What price is a critical number for profit? (Consider all positive prices p > 0 as your domain.)
(e) Classify the above critical number as corresponding to a local maximum by using a calculator to test nearby values. Then, argue why this must be the absolute maximum on the domain p > 0.