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102 CHAPTER 1 . FUNCTIONS OF SEVERAL VARIABLES SECTION 1.9 EXERCISES Review Questions 25-34. Applications of Lagrange multipliers Use Lagrange multipli- 1. Explain why, at a point that maximizes or minimizes f subject to ers in the following problems. When the domain of the objective func- a constraint g(x, y) = 0, the gradient of f is parallel to the gradi- tion is unbounded or open, explain why you have found an absolute ent of g. Use a diagram. maximum or minimum value. 2. If f(x, y) = x2 + y? and g(x, y) = 2x + 3y - 4 = 0, write the 25. Shipping regulations A shipping company requires that the sum Lagrange multiplier conditions that must be satisfied by a point of length plus girth of rectangular boxes must not exceed 108 in. that maximizes or minimizes f subject to g(x, y) = 0. Find the dimensions of the box with maximum volume that meets this condition. (The girth is the perimeter of the smallest side of 3. If f ( x, y, z) = x2 + y2 + 23 and 8(x, y, z) = the box.) 2x + 3y - 5z + 4 = 0, write the Lagrange multiplier conditions that must be satisfied by a point that maximizes or minimizes f 26. Box with minimum surface area Find the rectangular box with a subject to g (x, y, z) = 0. volume of 16 ft that has minimum surface area. 4. Sketch several level curves of f(x, y) = x2 + y and sketch the 27. Extreme distances to an ellipse Find the minimum and maxi- constraint line g(x, y) = 2x + 3y - 4 = 0. Describe the extrema mum distances between the ellipse x + xy + 2yz = 1 and the (if any) that f attains on the constraint line. origin. Basic Skills 28. Maximum area rectangle in an ellipse Find the dimensions of the rectangle of maximum area with sides parallel to the coordi- 5-14. Lagrange multipliers in two variables Use Lagrange multipli- nate axes that can be inscribed in the ellipse 4x2 + 16y? = 16. ers to find the maximum and minimum values of f (when they exist) subject to the given constraint. 29. Maximum perimeter rectangle in an ellipse Find the dimen- sions of the rectangle of maximum perimeter with sides paral- 5. f (x, y) = x + 2y subject to x2 + >2 = 4 lel to the coordinate axes that can be inscribed in the ellipse 6. f(x, y) = xy2 subject to x2 + y2 = 1 2x2 + 4y2 = 3. 7. f(x, y) = x + y subject to x2 - xy + y/2 = 1 30. Minimum distance to a plane Find the point on the plane 2x + 3y + 62 - 10 = 0 closest to the point (-2, 5, 1). 8. f ( x, y ) = x2 + y2 subject to 2x2 + 3xy + 2y2 = 7 31. Minimum distance to a surface Find the point on the surface 9. f(x, y ) = xy subject to x 2 + y2 - xy = 9 4x + y - 1 = 0 closest to the point (1, 2, -3). 10. f(x, y) = x - y subject to x2 + y2 - 3xy = 20 32. Minimum distance to a cone Find the points on the cone 11. f(x, y) = e2ty subject to x2 + y2 = 16 z? = x2 + y2 closest to the point (1, 2, 0). 12. f(x, y) = x2 + y2 subject to x6 + yo = 1 33. Extreme distances to a sphere Find the minimum and maximum distances between the sphere x2 + y' + z? = 9 and the point 13. f(x, y) = y2 - 4x2 subject to x2 + 2y2 = 4 2, 3, 4 ) . 14. f (x, y ) = xy + x + y subject to x 3/2 = 4 34. Maximum volume cylinder in a sphere Find the dimensions of a right circular cylinder of maximum volume that can be inscribed 15-24. Lagrange multipliers in three variables Use Lagrange multi- in a sphere of radius 16. pliers to find the maximum and minimum values of f (when they exist) subject to the given constraint. 35-38. Maximizing utility functions Find the values of & and g with e 2 0 and g 2 0 that maximize the following utility functions subject 15. f(x, y, z) = x + 3y - z subject to x2 + y2 + z? = 4 to the given constraints. Give the value of the utility function at the 16. f(x, y, z) = xyz subject to x2 + 2y2 + 472 = 9 optimal point. 17. f(x, y, z) = x subject to x2 + y2 + 22 - z = 1 35. U = f((, 8) = 10el/2g1/2 subject to 36 + 68 = 18 18. f (x, y, z) = x - z subject to x2 + y2 + z2 - y = 2 36. U = f(l, 8) = 3202/381/3 subject to 40 + 28 = 12 19. f(x, y, z) = x2 + y2 + z2 subject to x2 + y2 + 2? - 4xy = 1 37. U = f(l, 8) = 84/581/5 subject to 101 + 88 = 40 20. f ( x, y, z) = x + y + z subject to x2 + y2 + 22 - 2x - 2y = 1 38. U = f(l, 8) = (1/685/6 subject to 40 + 58 = 20 21. f(x, y, z) = 2x + z2 subject to x2 + y2 + 2z? = 25 Further Explorations 22. f (x, y, z) = x2 + y2 - z subject to z = 2x2y2 + 1 39. Explain why or why not Determine whether the following state- ments are true and give an explanation or counterexample. 23. f (x, y, z) = x2 + y2 + z subject to xyz = 4 a. Suppose you are standing at the center of a sphere looking at a 24. f ( x, y, z) = (xyz) 1/2 subject to x + y + z = 1 with x 2 0, point P on the surface of the sphere. Your line of sight to P is y 20,z20 orthogonal to the plane tangent to the sphere at P. b. At a point that maximizes f on the curve g(x, y) = 0, the dot product Vf . Vg is zero