If x1 and x2 are the number of items of two goods bought, a customers utility is U(x1

In Problems 110, use Lagrange multipliers to find the maximum or minimum values of f(x, y) subject to the constraint f(x, y) = x + y, x2 + y2 = 1

In Problems 110, use Lagrange multipliers to find the maximum or minimum values of f(x, y) subject to the constraint f(x, y) = x2 + 4xy, x + y = 100

In Problems 110, use Lagrange multipliers to find the maximum or minimum values of f(x, y) subject to the constraint f(x, y) = xy, 5x +2y = 100

In Problems 110, use Lagrange multipliers to find the maximum or minimum values of f(x, y) subject to the constraint f(x, y) = x2 + 3y2 + 100, 8x + 6y = 88

In Problems 110, use Lagrange multipliers to find the maximum or minimum values of f(x, y) subject to the constraint f(x, y) = 5xy, x+ 3y = 24

In Problems 110, use Lagrange multipliers to find the maximum or minimum values of f(x, y) subject to the constraint f(x, y) = 3x 2y, x2 +2y2 = 44

In Problems 110, use Lagrange multipliers to find the maximum or minimum values of f(x, y) subject to the constraint f(x, y) = x2 + y, x2 y2 = 1

In Problems 110, use Lagrange multipliers to find the maximum or minimum values of f(x, y) subject to the constraint f(x, y) = xy, 4x2 + y2 = 8

In Problems 110, use Lagrange multipliers to find the maximum or minimum values of f(x, y) subject to the constraint f(x, y) = x2 + y2, 4x 2y = 15

In Problems 110, use Lagrange multipliers to find the maximum or minimum values of f(x, y) subject to the constraint f(x, y) = x2 + y2, x4 + y4 = 2

Figure 8.50 shows contours labeled with values of f(x, y) and a constraint g(x, y) = c. Mark the approximate points at which: (a) f has a maximum (b) f has a maximum on the constraint g = c. 9 10 11 12 13 14 g = c Figure 8.50

Figure 8.51 shows contours of f(x, y) and the constraint g(x, y) = c. Approximatelywhat values of x and y maximize f(x, y) subject to the constraint? What is the approximate value of f at this maximum?

The quantity, Q, of a good produced depends on the quantities x1 and x2 of two raw materials used: Q = x0.3 1 x0.7 2 . A unit of x1 costs $10, and a unit of x2 costs $25. We want to maximize production with a budget of $50 thousand for raw materials. (a) What is the objective function? (b) What is the constraint?

The quantity, Q, of a certain product manufactured depends on the quantity of labor, L, and of capital, K, used according to the function Q = 900L1/2K2/3. Labor costs $100 per unit and capital costs $200 per unit. What combination of labor and capital should be used to produce 36,000 units of the goods at minimum cost? What is that minimum cost?

The Cobb-Douglas production function for a product is P = 5L0.8K0.2, where P is the quantity produced, L is the size of the labor force, and K is the amount of total equipment. Each unit of labor costs $300, each unit of equipment costs $100, and the total budget is $15,000. (a) Make a table of L and K values which exhaust the budget. Find the production level, P, for each. (b) Use the method of Lagrange multipliers to find the optimal way to spend the budget.

A firm manufactures a commodity at two different factories. The total cost of manufacturing depends on the quantities, q1 and q2, supplied by each factory, and is expressed by the joint cost function, C = f(q1, q2) = 2q2 1 + q1q2 + q2 2 + 500. The companys objective is to produce 200 units, while minimizing production costs. How many units should be supplied by each factory?

The quantity, Q, of a product manufactured by a company is given by Q = aK0.6L0.4, where a is a positive constant, K is the quantity of capital and L is the quantity of labor used. Capital costs are $20 per unit, labor costs are $10 per unit, and the company wants costs for capital and labor combined to be no higher than $150. Suppose you are asked to consult for the company, and learn that 5 units each of capital and labor are being used. (a) What do you advise? Should the company use more or less labor? More or less capital? If so, by how much? (b) Write a one-sentence summary that could be used to sell your advice to the board of directors.

For a cost function, f(x, y), the minimum cost for a production of 50 is given by f(33, 87) = 1200, with = 15. Estimate the cost if the production quota is: (a) Raisedto51 (b) Loweredto49

A company has the production function P(x, y), which gives the number of units that can be produced for given values of x and y; the cost function C(x, y) gives the cost of production for given values of x and y. (a) If the company wishes to maximize production at a cost of $50,000, what is the objective function f? What is the constraint equation? What is the meaning of in this situation? (b) If instead the company wishes to minimize the costs at a fixed production level of 2000 units, what is the objective function f? What is the constraint equation? What is the meaning of in this situation?

You have set aside 20 hours to work on two class projects. You want to maximize your grade (measured in points), which depends on how you divide your time between the two projects. (a) What is the objective function for this optimization problem and what are its units? (b) What is the constraint? (c) Suppose you solve the problem by the method of Lagrange multipliers. What are the units for ? (d) What is the practical meaning of the statement = 5?

A steel manufacturer can produce P(K,L) tons of steel using K units of capital and L units of labor, with production costs C(K,L) dollars. With a budget of $600,000, the maximum production is 2,500,000 tons, using $400,000 of capital and $200,000 of labor. The Lagrange multiplier is = 3.17. (a) What is the objective function? (b) What is the constraint? (c) What are the units for ? (d) What is the practical meaning of the statement = 3.17?

The quantity, q, of a product manufactured depends on the number of workers, W, and the amount of capital invested, K, and is given by the Cobb-Douglas function q = 6W3/4K1/4. In addition, labor costs are $10 per worker and capital costs are $20 per unit and the budget is $3000. (a) What are the optimum number of workers and the optimum number of units of capital? (b) Recompute the optimum values of W and K when the budget is increased by $1. Check that increasing the budget by $1 allows the production of extra units of the product, where is the Lagrange multiplier.

The terminal velocity (meters/second) that a two-stage rocket achieves is a function of the amount of fuel x1 and x2 (measured in liters) loaded into the two stages. You wish to minimize the total quantity of fuel required to achieve a specified terminal velocity, v0. (a) What is the objective function and what are its units? (b) What is the constraint? (c) Suppose you solve the problem by the method of Lagrange multipliers. What are the units for ? (d) What is the practical meaning of the statement = 8 when the terminal velocity of the rocket is 50 meters/second?

Each person tries to balance his or her time between leisure and work. The tradeoff is that as you work less your income falls. Therefore each person has indifference curves which connect the number of hours of leisure, l, and income, s. If, for example, you are indifferent between 0 hours of leisure and an income of $1125 a week on the one hand, and 10 hours of leisure and an income of $750 a week on the other hand, then the points l = 0, s = 1125, and l = 10, s = 750 both lie on the same indifference curve. Table 8.11 gives information on three indifference curves, I, II, and III. Table 8.11 Weekly income Weekly leisure hours I II III I II III 1125 1250 1375 0 20 40 750 875 1000 10 30 50 500 625 750 20 40 60 375 500 625 30 50 70 250 375 500 50 70 90 (a) Graph the three indifference curves. (b) You have 100 hours a week available for work and leisure combined, and you earn $10/hour. Write an equation in terms of l and s which represents this constraint. (c) On the same axes, graph this constraint. (d) Estimate from the graph what combination of leisure hours and income you would choose under these circumstances. Give the corresponding number of hours per week you would work.

If x1 and x2 are the number of items of two goods bought, a customers utility is U(x1, x2) = 2x1x2 + 3x1. The unit cost is $1 for the first good and $3 for the second. Use Lagrange multipliers to find the maximum value of U if the consumers disposable income is $100. Estimate the new optimal utility if the consumers disposable income increases by $6.