MATH 1526: LAGRANGE MULTIPLIERS Locating Relative Extrema Using the Method of Lagrange Multipliers: To locate the candidates for relative extrema of a function f (x,y) subject to the constraint g(x,y) = 0, we solve the following system of equations for x, y, and λ: fx = λgx , f y = λg y, g = 0. The unknown λ is called a Lagrange multiplier. The points (x, y) that occur in the solution are then called the candidates for the relative extrema of f subject to g = 0.

Note: If solving your system of equations for x, y, and λ results in exactly one point, you may assume for purposes of this course that the point you have found is the extreme value that you are seeking. If it should happen that solving your system for x, y, and λ results in more than one point, you need to test the values in your original function, f (x,y), to determine which of the points provides the extreme value that you want. You may assume that the largest of these values is the maximum value of f (x,y), subject to the constraint g(x,y) = 0, and the smallest value is the minimum value of f (x,y), subject to the constraint g(x,y) = 0. Lagrange Multiplier Homework Problems:

1. Using the method of Lagrange multipliers, find the relative minimum of the function f ( x, y ) = 2x2 + y2 subject to the constraint x + y =1.

2. Minimize f ( x, y ) = x2 + y2 subject to 3x + 4y = 25. What is the minimum value?

3. Minimize f (x, y) = 6xy subject to y − x = 6.

4. To fill an order for 100 units of its product, a firm wishes to distribute production between its two plants, plant 1 and plant 2. The total cost function is given by C(x, y) = 0.1x2 + 7x +15y +1000 where x and y are the numbers of units produced at plants 1 and 2, respectively. How should the output be distributed in order to minimize costs?

5. The total weekly profit (in dollars) realized by the Country Workshop in manufacturing and selling its rolltop desks is given by the profit function P(x, y) = −0.2x2 −0.25y2 −0.2xy +100x + 90y −4000 where x stands for the number of finished units and y denotes the number of unfinished units manufactured and sold per week. The company’s management decides to restrict the manufacture of these desks to a total of exactly 200 units/week. How many finished and how many unfinished units should be manufactured per week to maximize the company’s weekly profit?

6. The total weekly profit (in dollars) that the Acrosonic Company realized in producing and selling its bookshelf loudspeaker system is given by the profit function

P(x, y) = −1

4x2 − 3

8y2 − 1

4xy +120x +100y −5000

where x denotes the number of fully assembled units and y denotes the number of kits produced and sold per week. Acrosonic’s management decides that production of these loudspeaker systems should be restricted to a total of exactly 230 units per week. Under this condition, how many fully assembled units and how many kits should be produced per week to maximize Acrosonic’s weekly profit?

7. A farmer wishes to build a fence to enclose a rectangular pasture on a straight riverbank along which no fencing is necessary. Find the dimensions of the pasture that will require the least amount of fencing if the total area to be enclosed is 28,800 square feet. The amount of fencing is given by f (l , w) = 2w + l and the constraint is given by the area required, l ⋅ w = 28800, (l is the length of the pasture parallel to the river and w is the width of the pasture perpendicular to the river, with both distances in feet).

8. The Ross-Simons Company has a monthly advertising budget of $60,000. Their marketing department

estimates that if they spend x dollars on newspaper advertising and y dollars on television advertising, then the monthly sales will be given by S = 90x1/ 4 y3 / 4 dollars. Determine how much money Ross-Simons should spend on newspaper ads and on television ads per month to maximize its monthly sales.

9. Suppose that x units of labor and y units of capital are required to produce f (x, y) =100x3 / 4 y1/ 4 units of a certain product (this is a Cobb-Douglas production function). If each unit of labor costs $200 and each unit of capital costs $300 and a total of $60,000 is available for production, determine how many units of labor and how many units of capital should be used in order to maximize production.

Interpretation of the Lagrange Multiplier, λ: When used in the context of problems such as Cobb-Douglas production functions (like problems 9 and 10), the value of the Lagrange multiplier λ is called the marginal productivity of money. This value is interpreted as the approximate increase in the optimum value of f (x,y) when the value on the constraint is increased by one unit. That is, if one additional dollar is available for production, then approximately λ units of a product can be produced. See additional comment inserted after problem 9 below.

Note: You don’t actually find a value for λ when you solve problem #9 (and many of the other problems). You can, however, use the values of x and y that maximize production to find λ. Simply substitute the x and y value that you found into one of the equations containing λ and solve for λ. In problem #9, you should find that λ ≈ 0.257. Therefore, if the budget is increased from $60,000 to $65,000, then optimum production is increased by approximately (5000)(.257) = 1285 units for a maximum production of approximately 15,448 + 1,285 = 16,733 units.

10. A consulting firm for a manufacturing company arrived at the production function N ( x, y) = 50x0.8 y0.2 for a particular product where x is the number of units of labor and y is the number of units of capital required to produce N(x,y) units of the product. Each unit of labor costs $40 and each unit of capital costs $80. If $400,000 is budgeted for production of the product, determine how this amount should be allocated to maximize production and find the maximum production. Find λ. Use λ to find the new maximum production level if the production budget is increased from $400,000 to $410,000.

Solutions: 1. minimum is 2/3 when x = 1/3; y = 2/3 2. minimum = 25 when x = 3 and y = 4. 3. minimum = -54 at x = -3, y = 3 4. Plant 1 – 40 units; Plant 2 – 60 units. 5. 140 finished and 60 unfinished units. 6. 180 assembled and 50 kit versions of the bookshelf loudspeaker systems 7. l = 240 feet and w = 120 feet. 8. $15,000 on newspaper advertising and $45,000 on TV advertising. 9. Maximum production is approximately 15,448 units when 225 units of labor and 50 units of capital are

used. 10. Max production is approximately 263,902 units when 8,000 units of labor and 1,000 units of capital are

used. λ ≈ 0.660. New max production is approximately 270,502 units.