Chapter 7 Section 3 Optimizing Functions of Two Variables 533 (b) Read about the principle of diminishing returns in an economics tet. Then write a paragraph discussing the economic factors that might account for this phenomenon. 53. It is estimated that the weekl output at a certain plant is given b Q(, ) 1,175 483 3.1 2 1.2 3 2.7 2 units, where is the number of skilled workers and is the number of unskilled workers emploed at the plant. Currentl the workforce consists of 37 skilled and 71 unskilled workers. (a) Store the output function as 1,175X 483Y 3.1(X^2)*Y 1.2(X^3) 2.7(Y^2) Store 37 as X and 71 as Y and evaluate to obtain Q(37, 71). Repeat for Q(38, 71) and Q(37, 72). (b) Store the partial derivative Q (, ) in our calculator and evaluate Q (37, 71). Use the result to estimate the change in output resulting when the workforce is increased from 37 skilled workers to 38 and the unskilled workforce stas fied at 71. Then compare with the actual change in output, given b the difference Q(38, 71) Q(37, 71). (c) Use the partial derivative Q (, ) to estimate the change in output that results when the number of unskilled workers is increased from 71 to 72 while the number of skilled workers stas at 37. Compare with Q(37, 72) Q(37, 71). 54. Repeat Problem 53 with the output function Q(, ) 1,731 925 2 2.7 2 1.3 3/2 and initial emploment levels of 43 and 85. 3 Optimizing Functions of Two Variables Suppose a manufacturer produces two VCR models, the delue and the standard, and that the total cost of producing units of the delue and units of the standard is given b the function C(, ). How would ou find the level of production a and b that results in minimal cost? Or perhaps the output of a certain production process is given b Q(K, L), where K and L measure capital and labor ependiture, respectivel. What levels of ependiture K 0 and L 0 result in maimum output? In Section 4 of Chapter 3, ou learned how to use the derivative f () to find the largest and smallest values of a function of a single variable f(), and the goal of this section is to etend those methods to functions of two variables f(, ). We begin with a definition.
534 Chapter 7 Calculus of Several Variables Relative Etrema The function f(, ) is said to have a relative maimum at the point P(a, b) in the domain of f if f(a, b) f(, ) for all points sufficientl close to P. Likewise, f(, ) has a relative minimum at Q(c, d) if f(c, d) f(, ) for all points (, ) sufficientl close to Q. In geometric terms, there is a relative maimum of f(, ) at P(a, b) if the surface z f(, ) has a peak at the point (a, b, f(a, b)); that is, if (a, b, f(a, b)) is at least as high as an nearb point on the surface. Similarl, a relative minimum of f(, ) occurs at Q(c, d) if the point (c, d, f(c, d)) is at the bottom of a valle, so (c, d, f(c, d)) is at least as low as an nearb point on the surface. For eample, in Figure 7.11, the function f(, ) has a relative maimum at P(a, b) and a relative minimum at Q(c, d). z Relative maimum (a, b, f (a, b)) Surface z = f (, ) (c, d, f (c, d)) Relative minimum P(a, b) Q(c, d) FIGURE 7.11 Relative etrema of the function f(, ). CRITICAL POINTS The points (a, b) in the domain of f(, ) for which both f (a, b) 0 and f (a, b) 0 are said to be critical points of f. Like the critical points for functions of one variable, these critical points pla an important role in the stud of relative maima and minima. To see the connection between critical points and relative etrema, suppose f(, ) has a relative maimum at (a, b). Then the curve formed b intersecting the surface z f(, ) with the vertical plane b has a relative maimum and hence a horizontal tangent when a (see Figure 7.12a). Since the partial derivative f (a, b) is the slope of this tangent, it follows that f (a, b) 0. Similarl, the curve formed b intersecting the surface z f(, ) with the plane a has a relative maimum when b (see Figure 7.12b), and so f (a, b) 0. This shows that a point at which a function of two variables has a relative maimum must be a critical point. A similar argument shows that a point at which a function of two variables has a relative minimum must also be a critical point.
Chapter 7 Section 3 Optimizing Functions of Two Variables 535 z z Horizontal tangent (a, b, f(a, b)) (a, b, f(a, b)) Horizontal tangent (a, b) (a, b) (a) (b) FIGURE 7.12 The partial derivatives are zero at a relative etremum. Here is a more precise statement of the situation. z Saddle point FIGURE 7.13 The surface z 2 2. THE SECOND PARTIALS TEST Critical Points and Relative Etrema A point (a, b) in the domain of f(, ) for which the partial derivatives f and f both eist is called a critical point of f if both f (a, b) 0 and f (a, b) 0 If the first-order partial derivatives of f eist at all points in some region R in the plane, then the relative etrema of f in R can occur onl at critical points. Although all the relative etrema of a function must occur at critical points, not ever critical point of a function is necessaril a relative etremum. Consider, for eample, the function f(, ) 2 2, whose graph, which resembles a saddle, is sketched in Figure 7.13. In this case, f (0, 0) 0 because the surface has a relative maimum (and hence a horizontal tangent) in the direction, and f (0, 0) 0 because the surface has a relative minimum (and hence a horizontal tangent) in the direction. Hence (0, 0) is a critical point of f, but it is not a relative etremum. For a critical point to be a relative etremum, the nature of the etremum must be the same in all directions. A critical point that is neither a relative maimum nor a relative minimum is called a saddle point. Here is a procedure involving second-order partial derivatives that ou can use to decide whether a given critical point is a relative maimum, a relative minimum, or a saddle point. This procedure is the two-variable version of the second derivative test for functions of a single variable that ou saw in Chapter 3, Section 2.
536 Chapter 7 Calculus of Several Variables The Second Partials Test Suppose that (a, b) is a critical point of the function f(, ). Let D f (a, b)f (a, b) [ f (a, b)] 2 If D 0, then f has a saddle point at (a, b). If D 0 and f (a, b) 0, then f has a relative maimum at (a, b). If D 0 and f (a, b) 0, then f has a relative minimum at (a, b). If D 0, the test is inconclusive and f ma have either a relative etremum or a saddle point at (a, b). Notice that there is a saddle point at the critical point (a, b) onl when the quantit D in the second partials test is negative. If D is positive, there is either a relative maimum or a relative minimum in all directions. To decide which, ou can restrict our attention to an one direction (sa, the direction) and use the sign of the second partial derivative f in eactl the same wa as the single variable second derivative was used in the second derivative test given in Chapter 3; namel, a relative minimum if f (a, b) 0 a relative maimum if f (a, b) 0 You ma find the following tabular summar a convenient wa of remembering the conclusions of the second partials test: Sign of D Sign of f Behavior at (a, b) Relative minimum Relative maimum Saddle point The proof of the second partials test involves ideas beond the scope of this tet and is omitted. The following eamples illustrate how the test can be used. Find all critical points for the function f(, ) 2 2 and classif each as a relative maimum, a relative minimum, or a saddle point. Solution Since EXAMPLE 3.1 f 2 and f 2
Chapter 7 Section 3 Optimizing Functions of Two Variables 537 the onl critical point of f is (0, 0). To test this point, use the second-order partial derivatives f 2 f 2 and f 0 to get D(, ) f f ( f ) 2 2(2) 0 4 That is, D(, ) 4 for all points (, ) and, in particular, D(0, 0) 4 0 Hence, f has a relative etremum at (0, 0). Moreover, since f (0, 0) 2 0 it follows that the relative etremum at (0, 0) is a relative minimum. For reference, the graph of f is sketched in Figure 7.14. z Eplore! Refer to Eample 3.2. Store f(, ) 2 2 as Y1 L1 2 X 2, where L1 { 2, 1, 0, 0.8, 1.5}. Graph these curves using the decimal window [ 4.7, 4.7]1 b [ 3.1, 3.1]1, paing close attention to the order in which the shapes appear. Describe what ou observe. Relative minimum FIGURE 7.14 The surface z 2 2 with a relative minimum at (0, 0). EXAMPLE 3.2 Find all critical points for the function f(, ) 2 2 and classif each as a relative maimum, a relative minimum, or a saddle point. Solution Since f 2 and f 2
538 Chapter 7 Calculus of Several Variables z Saddle point FIGURE 7.15 The surface z 2 2 with a saddle point at (0, 0). the onl critical point of f is (0, 0). To test this point, compute the second-order partial derivatives f 2 f 2 and f 0 to get D(, ) f f ( f ) 2 2(2) 0 4 That is, D(, ) 4 for all points (, ) and, in particular, D(0, 0) 4 0 It follows that f must have a saddle point at (0, 0). The graph of f is shown in Figure 7.15. Solving the equations f 0 and f 0 simultaneousl to find the critical points of a function of two variables is rarel as simple as in Eamples 3.1 and 3.2. The algebra in the net eample is more tpical. Before proceeding, ou ma wish to refer to the Algebra Review at the back of the book, in which techniques for solving sstems of two equations in two unknowns are discussed. EXAMPLE 3.3 Find all critical points for the function f(, ) 3 3 6 and classif each as a relative maimum, a relative minimum, or a saddle point. Solution Since f 3 2 6 and f 3 2 6 ou find the critical points of f b solving simultaneousl the two equations 3 2 6 0 and 3 2 6 0 2 From the first equation, ou get which ou can substitute into the second 2 equation to find 3 4 6 0 4 or ( 3 8) 0 The solutions of this equation are 0 and 2. These are the coordinates of the critical points of f. To get the corresponding coordinates, substitute these values 2 of into the equation (or into either one of the two original equations). 2 You will find that 0 when 0 and 2 when 2. It follows that the critical points of f are (0, 0) and (2, 2).
Chapter 7 Section 3 Optimizing Functions of Two Variables 539 The second-order partial derivatives of f are f 6 f 6 and f 6 Hence, D(, ) f f ( f ) 2 36 36 36( 1) Since D(0, 0) 36[0(0) 1] 36 0 it follows that f has a saddle point at (0, 0). Since D(2, 2) 36[2( 2) 1] 108 0 and f (2, 2) 6(2) 12 0 ou see that f has a relative minimum at (2, 2). These results are summarized in the following table: Critical Point (a, b) D(a, b) f (a, b) Behavior at (a, b) (0, 0) Saddle point (2, 2) Relative minimum PRACTICAL OPTIMIZATION PROBLEMS In the net eample, ou will see how to appl the theor of relative etrema to solve an optimization problem from economics. Actuall, ou will be tring to find the absolute maimum of a certain function. It turns out, however, that the absolute and relative maima of this function coincide. In fact, in the majorit of two-variable optimization problems in the social sciences, the relative etrema and absolute etrema coincide. For this reason, the theor of absolute etrema for functions of two variables will not be developed in this tet, and ou ma assume that the relative etremum ou find as the solution to a practical optimization problem is actuall the absolute etremum. EXAMPLE 3.4 The onl grocer store in a small rural communit carries two brands of frozen apple juice, a local brand that it obtains at the cost of 30 cents per can and a well-known national brand that it obtains at the cost of 40 cents per can. The grocer estimates that if the local brand is sold for cents per can and the national brand for cents per can, approimatel 70 5 4 cans of the local brand and 80 6 7 cans of the national brand will be sold each da. How should the grocer price each brand to maimize the profit from the sale of the juice?
540 Chapter 7 Calculus of Several Variables Solution Since Total profit profit from the sale of the local brand profit from the sale of the national brand A(1, 5) it follows that the total dail profit from the sale of the juice is given b the function f(, ) ( 30)(70 5 4) ( 40)(80 6 7) 5 2 10 20 7 2 240 5,300 Compute the partial derivatives f 10 10 20 and f 10 14 240 and set them equal to zero to get 10 10 20 0 and 10 14 240 0 or 2 and 5 7 120 Then solve these equations simultaneousl to get 53 and 55 It follows that (53, 55) is the onl critical point of f. Net appl the second partials test. Since f 10 f 14 and f 10 ou get D(, ) f f ( f ) 2 10( 14) (10) 2 40 and since D(53, 55) 40 0 and f (53, 55) 10 0 it follows that f has a (relative) maimum when 53 and 55. That is, the grocer can maimize profit b selling the local brand of juice for 53 cents per can and the national brand for 55 cents per can. B(0, 0) W(, ) C(8, 0) FIGURE 7.16 Locations of businesses A, B, and C and warehouse W. EXAMPLE 3.5 A planner for Acme Corporation plots a grid on a map and determines that Acme s three most important customers are located at A(1, 5), B(0, 0), and C(8, 0), where units are in miles. At what point W(, ) should a warehouse be located in order to minimize the sum of the distances from P to A, B, and C (see Figure 7.16).
Chapter 7 Section 3 Optimizing Functions of Two Variables 541 Solution The point W(, ) where the sum of the distances is minimized is the same point that minimizes the sum of the squares of the distances; namel, S(, ) [( 1) 2 ( 5) 2 ] ( 2 2 ) [( 8) 2 2 ] agfdddefbgddddffc afdbfddc aedddgbeddgdc W to A W to B W to C (B working with the squares of distances, we eliminate the square roots and make the calculations easier.) To minimize S, begin b computing the partial derivatives S 2( 1) 2 2( 8) 6 18 S 2( 5) 2 2 6 10 Then S 0 and S 0 when 6 18 0 6 10 0 5 or 3 and. Since S 6, S 0, and S 6, ou get 3 D S S S 2 (6)(6) 0 2 36 0 and 3, 5 3 S 6 0 3, 5 3 Thus, the sum of squares is minimized at the map point W. THE METHOD OF LEAST SQUARES Suppose in the process of analzing a particular phenomenon we gather the set of data plotted in Figure 7.17a. The data points seem to lie roughl in a straight line, but what line? In other words, given a collection of points ( 1, 1 ), ( 2, 2 ),..., ( n, n ), what line m b best fits the data?
542 Chapter 7 Calculus of Several Variables m b ( 1, 1 ) d 3 (a) A collection of data points d 1 ( 3, 3 ) d2 ( 2, 2 ) 1 2 3 (b) The least-squares criterion FIGURE 7.17 Least-squares approimation of a set of data: (a) a collection of data points and (b) the least-squares criterion. One of the most frequentl used procedures for determining the best-fitting line is to compute the sum of squares S of the vertical distances from the data points to the line m b (Figure 7.17b). The sum S will be a function of the two variables m (slope) and b ( intercept), and we obtain the best-fitting line b using the optimization procedures of this section to minimize the function S(m, b). Here is an eample. Eplore! Refer to Eample 3.6. Store the points (1, 1), (2, 3), and (4, 3) into the data list feature of our graphing calculator, with coordinates in L1 and in L2. Find the equation of the leastsquares regression line and graph it along with the data points. Now place the line 0.6 1 in Y2 of the equation editor. Which line appears to fit the data better and wh? EXAMPLE 3.6 Use the least-squares criterion to find the equation of the line that is closest to the three points (1, 1), (2, 3), and (4, 3). Solution As indicated in Figure 7.18, the sum of the squares of the vertical distances from the three given points to the line m b is d 2 1 d 2 2 d 2 3 (m b 1) 2 (2m b 3) 2 (4m b 3) 2 This sum depends on the coefficients m and b that define the line, so the sum can be thought of as a function S(m, b) of the two variables m and b. The goal, therefore, is to find the values of m and b that minimize the function S(m, b) (m b 1) 2 (2m b 3) 2 (4m b 3) 2 S S You do this b setting the partial derivatives and equal to zero to get m b S 2(m b 1) 4(2m b 3) 8(4m b 3) m 42m 14b 38 0
Chapter 7 Section 3 Optimizing Functions of Two Variables 543 4 3 (2, 3) (4, 4m + b) d 3 d 2 (4, 3) = m + b 2 1 (1, m + b) d 1 (1, 1) (2, 2m + b) 1 2 3 4 FIGURE 7.18 Minimize the sum d 2 1 d 2 2 d 2 3. S and 2(m b 1) 2(2m b 3) 2(4m b 3) b 14m 6b 14 0 and solving the resulting simplified equations 21m 7b 19 7m 3b 7 simultaneousl for m and b to conclude that 4 m and b 1 7 Since S mm 42 S mb 14 S bb 6 we have D S mm S bb S 2 mb (42)(6) (14) 2 56 So D 0 and S mm 0, and the second partials test tells us that the critical point 4 corresponds to a relative minimum. Thus, the line that best fits the given three 7, 1 4 points has the equation 1. 7 The procedure illustrated in Eample 3.6 can be generalized to find the best-fitting line m b for an arbitrar set of data points ( 1, 1 ), ( 2, 2 ),..., ( n, n ).
544 Chapter 7 Calculus of Several Variables Specificall, ou would minimize the sum of squares function S(m, b) (m 1 b 1 ) 2... (m n b n ) 2 and It can be shown that the minimum occurs when where, for simplicit, we have dropped the indices in the sums. For instance, 2 n m j 1 n n 2 ( ) 2 b 2 n 2 ( ) 2 2 j 2 1 2 2... 2 n The derivation of these formulas involves several complicated algebraic steps and is omitted. For practice, appl the formulas to the data in Eample 3.6 to assure ourself that the do indeed ield the result ou found directl. We close with an applied eample that illustrates how data can be efficientl organized for substitution into the formulas to obtain a least-squares line. EXAMPLE 3.7 A college admissions officer has compiled the following data relating students highschool and college grade-point averages: High-school GPA 2.0 2.5 3.0 3.0 3.5 3.5 4.0 4.0 College GPA 1.5 2.0 2.5 3.5 2.5 3.0 3.0 3.5 Find the equation of the least-squares line for these data and use it to predict the college GPA of a student whose high-school GPA is 3.75. Solution Let denote the high-school GPA and the college GPA and arrange the calculations as follows: 2 2.0 1.5 3.0 4.0 2.5 2.0 5.0 6.25 3.0 2.5 7.5 9.0 continued on net page
Chapter 7 Section 3 Optimizing Functions of Two Variables 545 2 3.0 3.5 10.5 9.0 3.5 2.5 8.75 12.25 3.5 3.0 10.5 12.25 4.0 3.0 12.0 16.0 4.0 3.5 14.0 16.0 25.5 21.5 71.25 2 84.75 Use the least-squares formula with n 8 to get 8(71.25) 25.5(21.5) m 8(84.75) (25.5) 2 0.78 84.75(21.5) 25.5(71.25) and b 8(84.75) (25.5) 2 0.19 The equation of the least-squares line is therefore 0.78 0.19 To predict the college GPA of a student whose high-school GPA is 3.75, substitute 3.75 into the equation of the least-squares line. This gives 0.78(3.75) 0.19 3.12 which suggests that the student s college GPA might be about 3.1. (college GPA) 4 3 Least-squares line: = 0.78 + 0.19 2 1 1 2 3 4 (high-school GPA) FIGURE 7.19 The least-squares line for high-school and college GPAs.
546 Chapter 7 Calculus of Several Variables A graph of the original data and of the corresponding least-squares line is shown in Figure 7.19. Actuall, in practice, it is a good idea to plot the data before proceeding with the calculations. B looking at the graph ou will usuall be able to tell whether approimation b a straight line is appropriate or whether a curve of some other shape should be used instead. P. R. O. B. L. E. M. S 7.3 P. R. O. B. L. E. M. S 7.3 In Problems 1 through 20, find the critical points of the given function and classif each as a relative maimum, a relative minimum, or a saddle point. 1. f(, ) 5 2 2 2. f(, ) 2 2 3 2 3. f(, ) 4. f(, ) 2 2 2 14 1 8 8 5. f(, ) 6. f(, ) 1 1 7. f(, ) 2 3 3 3 2 3 12 4 8. f(, ) ( 1) 2 3 3 2 9 5 9. f(, ) 3 2 6 9 5 2 10. f(, ) 4 32 3 12 7 11. f(, ) ( 2 2 2 ) e 1 2 2 12. f(, ) ( 4) ln () 13. f(, ) 3 4 3 14. f(, ) 2 2 4 15. f(, ) e (2 2 6) 16. f(, ) 2 4 2 2 3 2 2 5 1 17. f(, ) 18. f(, ) e 162 9 2 288 2 2 3 2 1 RETAIL SALES 2 19. f(, ) ln 3 2 20. f(, ) 4 2 4 2 4 2 21. A T-shirt shop carries two competing shirts, one endorsed b Michael Jordan and the other b Shaq O Neal. The owner of the store can obtain both tpes at a cost of $2 per shirt and estimates that if Jordan shirts are sold for dollars apiece and O Neal shirts for dollars apiece, consumers will bu approimatel 40 50 40 Jordan shirts and 20 60 70 O Neal shirts each da. How should the owner price the shirts in order to generate the largest possible profit?
Chapter 7 Section 3 Optimizing Functions of Two Variables 547 PRICING CONSTRUCTION CONSTRUCTION RETAIL SALES RETAIL SALES RESPONSE TO STIMULI SOCIAL CHOICES 22. The telephone compan is planning to introduce two new tpes of eecutive communications sstems that it hopes to sell to its largest commercial customers. It is estimated that if the first tpe of sstem is priced at hundred dollars per sstem and the second tpe at hundred dollars per sstem, approimatel 40 8 5 consumers will bu the first tpe and 50 9 7 will bu the second tpe. If the cost of manufacturing the first tpe is $1,000 per sstem and the cost of manufacturing the second tpe is $3,000 per sstem, how should the telephone compan price the sstems to generate the largest possible profit? 23. Suppose ou wish to construct a rectangular bo with a volume of 32 ft 3. Three different materials will be used in the construction. The material for the sides costs $1 per square foot, the material for the bottom costs $3 per square foot, and the material for the top costs $5 per square foot. What are the dimensions of the least epensive such bo? 24. A farmer wishes to fence off a rectangular pasture along the bank of a river. The area of the pasture is to be 6,400 d 2, and no fencing is needed along the river bank. Find the dimensions of the pasture that will require the least amount of fencing. 25. A dair produces whole milk and skim milk in quantities and gallons, respectivel. Suppose that the price of whole milk is p() 100 and that of skim milk is q() 100 and assume that C(, ) 2 2 is the joint-cost function of the commodities. What should and be to maimize profit? 26. Repeat Problem 25 for the case where p() 20 5, q() 4 2, and C(, ) 2 4. 27. Consider an eperiment in which a subject performs a task while being eposed to two different stimuli (for eample, sound and light). For low levels of the stimuli, the subject s performance might actuall improve, but as the stimuli increase, the eventuall become a distraction and the performance begins to deteriorate. Suppose in a certain eperiment in which units of stimulus A and units of stimulus B are applied, the performance of a subject is measured b the function f(, ) C where C is a positive constant. How man units of each stimulus result in maimum performance? 28. The social desirabilit of an enterprise often involves making a choice between the commercial advantage of the enterprise and the social or ecological loss that ma result. For instance, the lumber industr provides paper products to societ and income to man workers and entrepreneurs, but the gain ma be offset b the destruction of habitable territor for spotted owls and other endangered species. Suppose the social desirabilit of a particular enterprise is measured b the function D(, ) (16 6) ( 2 4 40) 0, 0 where measures commercial advantage (profit and jobs) and measures e 1 2 2
548 Chapter 7 Calculus of Several Variables ecological disadvantage (species displacement, as a percentage). The enterprise is deemed desirable if D 0 and undesirable if D 0. (a) What values of and will maimize social desirabilit? Interpret our result. Is it possible for this enterprise to be desirable? (b) The function given in part (a) is artificial, but the ideas are not. Research the topic of ethics in industr and write a paragraph on how ou feel these choices should be made.* PARTICLE PHYSICS ALLOCATION OF FUNDS PROFIT UNDER MONOPOLY PROFIT UNDER MONOPOLY 29. A particle of mass m in a rectangular bo with dimensions,, z has ground state energ E(,, z) 8m k2 1 2 1 2 1 z 2 where k is a phsical constant. If the volume of the bo satisfies z V 0 for constant V 0, find the values of,, and z that minimize the ground state energ. 30. A manufacturer is planning to sell a new product at the price of $150 per unit and estimates that if thousand dollars is spent on development and thousand dollars 320 is spent on promotion, consumers will bu approimatel units of 2 160 4 the product. If manufacturing costs for this product are $50 per unit, how much should the manufacturer spend on development and how much on promotion to generate the largest possible profit from the sale of this product? [Hint: Profit (number of units)(price per unit cost per unit) total amount spent on development and promotion.] 31. A manufacturer with eclusive rights to a sophisticated new industrial machine is planning to sell a limited number of the machines to both foreign and domestic firms. The price the manufacturer can epect to receive for the machines will depend on the number of machines made available. (For eample, if onl a few of the machines are placed on the market, competitive bidding among prospective purchasers will tend to drive the price up.) It is estimated that if the manufacturer supplies machines to the domestic market and machines to the foreign market, the machines will sell for 60 thousand dollars apiece at home and for 5 20 50 thousand dollars apiece abroad. If the manufacturer can produce the 10 20 machines at the cost of $10,000 apiece, how man should be supplied to each market to generate the largest possible profit? 32. A manufacturer with eclusive rights to a new industrial machine is planning to sell a limited number of them and estimates that if machines are supplied to the domestic market and to the foreign market, the machines will sell for 150 6 * A good place to start is the article b K. R. Stoller, Environmental Controls in Etractive Industries, Land Economics, Vol. 61, 1985, page 169.
Chapter 7 Section 3 Optimizing Functions of Two Variables 549 thousand dollars apiece at home and for 100 thousand dollars apiece abroad. 20 (a) How man machines should the manufacturer suppl to the domestic market to generate the largest possible profit at home? (b) How man machines should the manufacturer suppl to the foreign market to generate the largest possible profit abroad? (c) How man machines should the manufacturer suppl to each market to generate the largest possible total profit? (d) Is the relationship between the answers in parts (a), (b), and (c) accidental? Eplain. Does a similar relationship hold in Problem 31? What accounts for the difference between these two problems in this respect? CITY PLANNING MAINTENANCE SALES 33. Four small towns in a rural area wish to pool their resources to build a television station. If the towns are located at the points ( 5, 0), (1, 7), (9, 0), and (0, 8) on a rectangular map grid, where units are in miles, at what point S(a, b) should the station be located to minimize the sum of the distances from the towns? 34. In relation to a rectangular map grid, four oil rigs are located at the points ( 300, 0), ( 100, 500), (0, 0), and (400, 300) where units are in feet. Where should a maintenance shed M(a, b) be located to minimize the sum of the distances from the rigs? In Problems 35 through 38 plot the given points and use the method of Eample 3.6 to find the corresponding least-squares line. 35. (0, 1), (2, 3), (4, 2) 36. (1, 1), (2, 2), (6, 0) 37. (1, 2), (2, 4), (4, 4), (5, 2) 38. (1, 5), (2, 4), (3, 2), (6, 0) In Problems 39 through 42, find the indicated least-squares line m b. You ma use the formulas given just before Eample 3.7. 39. A compan s annual sales (in units of 1 billion dollars) for its first 5 ears of operation are shown in the following table: Year 1 2 3 4 5 Sales 0.9 1.5 1.9 2.4 3.0 (a) Plot these data on a graph. (b) Find the equation of the least-squares line. (c) Use the least-squares line to predict the compan s sith-ear sales. DRUG ABUSE 40. The following table gives the percentage of high-school seniors in four different ears who had tried cocaine at least once in their lives.* * L. Hoffman, S. Paris, and E. Hall, Developmental Pscholog Toda, McGraw-Hill, Inc., New York, 1994, page 405.
550 Chapter 7 Calculus of Several Variables Year 1975 1980 1985 1990 Percentage Using Cocaine 9.0 15.7 17.3 9.4 (a) Plot these data on a graph. (b) Find the equation of the least-squares line. (c) Use the least-squares line to predict the percentage of high school seniors in the ear 2000 who used cocaine at least once. VOTER TURNOUT 41. On election da, the polls in a certain state open at 8:00 A.M. Ever 2 hours after that, an election official determines what percentage of the registered voters have alread cast their ballots. The data through 6:00 P.M. are shown below: Time 10:00 12:00 2:00 4:00 6:00 Percentage Turnout 12 19 24 30 37 (a) Plot these data on a graph. (b) Find the equation of the least-squares line. (Let denote the number of hours after 8:00 A.M.) (c) Use the least-squares line to predict what percentage of the registered voters will have cast their ballots b the time the polls close at 8:00 P.M. SPREAD OF AIDS 42. Recall that in Problem 40 of Section 1, Chapter 4, we gave the following table for the number of reported cases of AIDS for the period 1984 1991: Year 1984 1985 1986 1987 1988 1989 1990 1991 Cases 4,445 8,249 12,932 12,070 31,001 33,722 41,595 43,672 (a) Plot these data on a graph with time t (ears after 1984) on the ais. Find the equation for the least-squares line for the given data. (b) Use the least-squares line to predict the number of cases of AIDS reported in the ear 2000. 43. Let f(, ) 2 2 4. Show that f does not have a relative minimum at its critical point (0, 0), even though it does have a relative minimum at (0, 0) in both the and directions. [Hint: Consider the direction defined b the line. That is, substitute for in the formula for f and analze the resulting function of.] In Problems 44 through 47 find the partial derivatives f and f, and then use our graphing utilit to determine the critical points of each function. 44. f(, ) ( 2 3 5)e 2 2 2
Chapter 7 Section 4 Constrained Optimization: The Method of Lagrange Multipliers 551 LEVEL CURVES 2 7 2 45. f(, ) ln 46. f(, ) 6 2 12 4 16 3 47. f(, ) 2 4 4 2 (11 18) 48. Sometimes ou can classif the critical points of a function b inspecting its level curves. In each case shown in the figure, determine the nature of the critical point of f at (0, 0). f = 1 f = 2 f = 3 f = 3 f = 2 f = 1 2 f = 1 f = 1 f = 1 3 f = 3 f = 2 f = 3 f = 2 (a) (b) f = 1 f = 1 4 Constrained Optimization: The Method of Lagrange Multipliers In man applied problems, a function of two variables is to be optimized subject to a restriction or constraint on the variables. For eample, an editor, constrained to sta within a fied budget of $60,000, ma wish to decide how to divide this mone between development and promotion in order to maimize the future sales of a new book. If denotes the amount of mone allocated to development, the amount allocated to promotion, and f(, ) the corresponding number of books that will be sold, the editor would like to maimize the sales function f(, ) subject to the budgetar constraint that 60,000. For a geometric interpretation of the process of optimizing a function of two variables subject to a constraint, think of the function itself as a surface in three-dimensional space and of the constraint (which is an equation involving and ) as a curve in the plane. When ou find the maimum or minimum of the function subject to the given constraint, ou are restricting our attention to the portion of the surface that lies directl above the constraint curve. The highest point on this portion of the surface is the constrained maimum, and the lowest point is the constrained minimum. The situation is illustrated in Figure 7.20.