To encourage buyers to place 100-unit orders, your firm’s sales department applies a continuous discount that makes the unit price as a function P(x) of the number of unites ordered. The discount decreases the prices at the rate EUR 0.01 per unit ordered. The price per unit for a 100-unit order is P(100) = 20.09 EUR.
(a) Find P(x) by solving the equation P′(x) = − 1/100 P(x), P(100) = 20.09.
(b) Find the unit price P(10) for a 10-unit order and P(90) for a 90-unit order.
(c) The sales department has asked you to find out if it is discounting so much that the firm’s revenue, r(x) = xP(x), will actually be less for a 100-unit order than, say, for a 90-unit order. Reassure them by showing that r(x) has its maximum value at x = 100.
(d) Graph the revenue function r(x) = xP(x) for 0 ≤ x ≤ 200.
