Extrema of functions of several variables are important in numerous applications in economics and business. Particularly important variables are profit, revenue, and cost. Their rates of change (i.e., derivatives) with respect to the number of units produced or sold are referred to as marginal profit, revenue, and cost. These are central to many applications involving extrema. For instance, to find the maximum profit, the profit function, P = R - C, is analyzed. In the formula, R = xp is the total revenue from selling x units, where p is the price per unit, and C is the total cost of producing x units. The differentiation of the profit function is carried out with respect to x. Assuming the result of the differentiation is equal to zero, one can see that the maximum profit occurs when the marginal revenue (that is, dR/dx) is equal to the marginal cost (that is, dC/dx).

Example:

The profit obtained by producing x units of product A and y units of product B is approximated by the model

P(x,y) = 8x + 10y - (0.001)(x² + xy + y²) - 10,000

To find the production level that produces a maximum profit, the partial derivatives of the profit function are set equal to 0, and the resulting system of two equations is solved with respect to x and y; this gives x = 2,000 and y = 4,000. The Second Derivatives Test shows that P_xx < 0 and P_xx P_yy - P_xy² > 0, which means the obtained production level (x = 2,000 units, y = 4,000 units) indeed yields a maximum profit.

Practical profit problems usually involve several models of one type of product, with prices per unit and profits per unit varying from model to model, and demand for each model which is a function of the price of the other models as well as its own price, etc.

Question:

a. To reduce shipping distances between the manufacturing facilities and a major consumer, a Korean computer brand, Intel Corp. intends to start production of a new controlling chip for Pentium III microprocessors at their two Asian plants. The cost of producing _x1 chips at Chiangmai (Thailand) is:

C1 = 0.002_x1² + 4_x1 + 500,

and the cost of producing _x2 chips at Kuala-Lumpur (Malaysia) is

C2 = 0.005_x2² + 4_x2 + 275

The Korean computer manufacturer buys them for $150 per chip. Find the quantity that should be produced at each Asian location to maximize the profit if, in accordance with Intel's marketing department, it is described by the expression:

P(_x1, _x2) = 150(_x1 + _x2) - C1 - C2

b. Repeat the same process as in part a, however, this time we are going to use numbers which are a bit more realistic. We also know that the maximum number of chips that the second plant can ship to the manufacturing plant in Korea is 11,000 chips.

C1 = 0.001998_x1² + 3.813_x1 + 531.6

C2 = 0.005698_x2² + 4.045_x2 + 349.6

P(_x1, _x2) = 148.6(_x1 + _x2) - C1 - C2