878 CHAPTER 15 DIFFERENTIATION IN SEVERAL VARIABLES (e) Does P correspond to a minimum or maximum value of f? Re- fer to Figure 12 to justify your answer. Hint: Do the values of f(x, y) 2 increase or decrease as (x, y) moves away from P along the line 8 (x, y) = 0? 8 (x, y) = 0 12 24 36 FIGURE 13 Contou 10-4 constraint g(x, y) 17. Find the poin FIGURE 12 Level curves of f(x, y) = x2 + 2y2 and graph of the least. constraint g(x, y) = 4x - 6y - 25 = 0. 18. Find the rect of the edges is 30 3. Apply the method of Lagrange multipliers to the function f (x, y) = (x2 + 1)y subject to the constraint x2 + yz = 5. Hint: First show that 19. The surface y * 0; then treat the cases x = 0 and x * 0 separately. S = urvr2 + h In Exercises 4-15, find the minimum and maximum values of the function (a) Determine subject to the given constraint. maximum volun (b) What is the 4. f (x, y) = 2x + 3y, x2 + y2 = 4 \\ code clog edi chaste (an) surface area S (c) Does a con 5. f (x, y) = x2+ y2, 2x + 3y = 6 10=1 98 exist? 6. f (x, y) = 4x2 + 9y2, xy = 4 20. In Exam ellipse (x/4)2 7. f (x, y) = xy, 4x2 + 92 =32 grange multi 4 cost, y = 3 8. f ( x, y ) = x2 y + xty, xy = 4 ing single-var 9. f (x, y ) =x2+ y2, x4+ 1 4 = 1 21. Find the 10. f (x, y ) = x2y4, x2 + 2y2 = 6 11. f(x, y, z) = 3x + 2y + 4z, x2 + 2y2 + 622 = 1 with the gre 12. f (x, y, z) = x2 - y - z, x2 - y2+z=0 13. f (x, y, z) = xy + 2z, x2 + y2+ z2 =36 14. f (x, y, z) = x2 + y2 + z2, x + 3y+2z =36 15. f (x, y, z) = xy + xz, x2 + y2 + 22 = 4 16. Let f (x, y ) = x3+xyty, 8(x,y)=x3 -xy+y3 (a) Show that there is a unique point P = (a, b) on g(x, y) = 1 where Vfp = XVgp for some scalar ). 22. Us (b) Refer to Figure 13 to determine whether f( P) is a local minimum or scribed a local maximum of f subject to the constraint. (e) Does Figure 13 suggest that f( P) is a global extremum subject to the constraint?ABLES Re- r, y line 2 -2 FIGURE 13 Contour map of f(x, y) = x3 + xy + y3 and graph of the constraint g(x, y) = x3 - xy + y' = 1. 17. Find the point (a, b) on the graph of y = et where the value ab is the least. 18. Find the rectangular box of maximum volume if the sum of the lengths of the edges is 300 cm. at 19. The surface area of a right-circular cone of radius r and height h is S = urvr2 + h2, and its volume is V = 3xr2h. (a) Determine the ratio h/r for the cone with given surface area S and maximum volume V. (b) What is the ratio h/r for a cone with given volume V and minimum surface area S? (c) Does a cone with given volume V and maximum surface area exist? 20. In Example 1, we found the maximum of f(x, y) = 2x + 5y on the ellipse (x/4)2 + (y/3)2 = 1. Solve this problem again without using La- grange multipliers. First, show that the ellipse is parametrizat 4 cost, y = 3 sint. Then find the maximum value of f (4 cost, 3 sint) us- ing single-variable calculus. Is one method easier than the other? 21. Find the point on the ellipse x2 + 6y2 + 3xy = 40 with the greatest x-coordinate (Figure 14). 4- - 8 FIGURE 14 Graph of x + 6y- + 3xy = 40. 22. Use Lagrange multipliers to find the maximum area of a rectangle in- scribed in the ellipse (Figure 15): + =132. river a (-x, y) (x, y ) should (a) D (-x, -y) (x , - y) Reph sumi FIGURE 15 Rectangle inscribed in the ellipse x2 (b) 62 = 1. (c) lipse the r origin. 23. Find the point (xo, yo) on the line 4x + 9y = 12 that is closest to the (d) min 24. Show that the point (Xo, yo) closest to the origin on the line ax + by = c has coordinates ac XO = bc a2+ 62 ' yo = a2 + 62 25. Find the maximum value of f(x, y) = xay for x 2 0, y 2 0 on the line x + y = 1, where a, b > 0 are constants. 26. Show that the maximum value of f(x, y) = x2y' on the unit circle is 6 25 V uilw 27. Find the maximum value of f(x, y) = xay for x 2 0, y 2 0 on the unit circle, where a, b > 0 are constants. chime ban 28. Find the maximum value of f (x, y, z) = xay z" for x, y, z 2 0 on the unit sphere, where a, b, c > 0 are constants. 29. Show that the minimum distance from the origin to a point on the plane ax + by + cz = d is Vaz + 62 + c 2 ard to oulay and otsluols) (d) 30. Antonio has $5.00 to spend on a lunch consisting of hamburgers ($1.50 each) and french fries ($1.00 per order). Antonio's satisfaction from eating x1 hamburgers and x2 orders of french fries is measured by a func- tion U(x1, X2) = x1X2. How much of each type of food should he pur- chase to maximize his satisfaction? (Assume that fractional amounts of each food can be purchased.) 31. Let Q be the point on an ellipse closest to a given point P outside the ellipse. It was known to the Greek mathematician Apollonius (third century BCE) that P Q is perpendicular to the tangent to the ellipse at Q (Figure 16). Explain in words why this conclusion is a consequence of the method of Lagrange multipliers. Hint: The circles centered at P are level curves of the function to be minimized. wod of agv 4 = Q your sedl of A math obulono ) FIGURE 16acca 880 CHAPTER 15 DIFFERENTIATION IN SEVERAL VARIABLES -xy = bh C = (0, 0, C) L Ay L Wall ( -b , -h ) . p h Ladder B = (0, b, 0) X Fence A = (a, 0, 0) FIGURE 20 FIGURE 19 40. Find the maximum value of f (x, y, z) = xy + xz + yz - xyz subject to the constraint x + y + z = 1, for x 2 0, y 2 0, z 20. 36. With the same set-up as in the previous problem, find the plane 41. Find the minimum of f(x, y, z) = x2 + y2 + zz subject to the two that minimizes V if the plane is constrained to pass through a point constraints x + y + z = 1 and x + 2y + 3z = 6. P = (a, B, y) with a, B, y > 0. 37. Show that the Lagrange equations for f(x, y) = x + y subject to the 42. Find the maximum of f(x, y, z) = z subject to the two constraints constraint g(x, y) = x + 2y = 0 have no solution. What can you conclude x2 + y2 = 1 and x+ y+ z=1. about the minimum and maximum values of f subject to g = 0? Show this directly. 43. Find the point lying on the intersection of the plane x + zy + z = 0 and the sphere x2 + y2 + z2 = 9 with the greatest z-coordinate. 38. Show that the Lagrange equations for f(x, y) = 2x + y subject 44. Find the maximum of f(x, y, z) = x + y + z subject to the two con- to the constraint g(x, y) = x2 - yz = 1 have a solution but that f has no min or max on the constraint curve. Does this contradict Theorem 1? straints x2 + y2 + z2 = 9 and 4x2 + zy2 + 4z2 = 9. 39. Let L be the minimum length of a ladder that can reach over a fence 45. The cylinder x2 + y2 = 1 intersects the plane x + z = 1 in an ellipse. of height h to a wall located a distance b behind the wall. Find the point on such an ellipse that is farthest from the origin. (a) Use Lagrange multipliers to show that L = (12/3 + 62/3)3/2 (Figure 20). Hint: Show that the problem amounts to minimizing f(x, y) = 46. Find the minimum and maximum of f(x, y, z) = y + 2z subject to (x +b)2 + (y + h) subject to y/b = h/x orxy=bh. two constraints, 2x + z = 4 and x2 + y2 = 1. (b) Show that the value of L is also equal to the radius of the circle with 47. Find the minimum value of f (x, y, z) = x2 + y2 + z2 subject to two center (-b, -h) that is tangent to the graph of xy = bh. constraints, x + 2y + z = 3 and x - y = 4