To encourage buyers to place 100-unit orders, your firm’s sales department applies a continuous discount that makes the unit price a function p(x) of the number of units x ordered. The discount decreases the price at the rate of $0.01 per unit ordered. The price per unit for a 100-unit order is p(100) = $20.09.

a. Find p(x) by solving the following initial value problem:

b. Find the unit price p(10) for a 10-unit order and the unit price 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) = x · p(x), will actually be less for a 100-unit order than, say, for a 90-unit order. Reassure them by showing that r has its maximum value at x = 100.

d. Graph the revenue function r(x) = xp(x) for 0 ≤ x ≤ 200.

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Differential equation: Initial condition: dp dx p(100) = 1 100P = 20.09.