Problems 11, 13, 18, 22, 33 please
94 CHAPTER 1 . FUNCTIONS OF SEVERAL VARIABLES SECTION 1.8 EXERCISES Review Questions 1. Describe the appearance of a smooth surface with a local maxi- 30. f(x, y) = x - 1 mum at a point. x2 + ya 2. Describe the usual appearance of a smooth surface at a saddle point. 31. f(x, y) = x4 + 4x7 (y - 2) + 8( -1)3 3. What are the conditions for a critical point of a function f? 32. f ( x, y) = xe 'sin y, for | x | $ 2, 0 s y s T 4. 33. f (x, y ) = ye' - e If f.(a, b) = f,(a, b) = 0, does it follow that f has a local maxi- mum or local minimum at (a, b)? Explain. 34. f(x, y) = sin (2Tx) cos (try), for |x| s ; and ly| s 4. 5. Consider the function z = f(x, y). What is the discriminant of f, 35. Shipping regulations A shipping company handles rectangular and how do you compute it? boxes provided the sum of the height and the girth of the box does not exceed 96 in. (The girth is the perimeter of the smallest side of 6. Explain how the Second Derivative Test is used. the box.) Find the dimensions of the box that meets this condition 7. What is an absolute minimum value of a function f on a set R and has the largest volume. in R?? 36. Cardboard boxes A lidless box is to be made using 2 m of 8. What is the procedure for locating absolute maximum and cardboard. Find the dimensions of the box with the largest minimum values on a closed bounded domain? possible volume. Basic Skills 37. Cardboard boxes A lidless cardboard box is to be made with a volume of 4 m'. Find the dimensions of the box that requires the 9-18. Critical points Find all critical points of the following functions. least amount of cardboard. 9. f (x, y ) = 1+x2+ 2 38. Optimal box Find the dimensions of the largest rectangular 10. f(x, y ) = x2 - 6x + 12 + 8y box in the first octant of the xyz-coordinate system that has one vertex at the origin and the opposite vertex on the plane 11. f(x, y) = (3x - 2)2 + (x - 4)? x + 2y + 3z = 6. 12. f(x, y ) = 3.x2 - 4y? 39-42. Inconclusive tests Show that the Second Derivative Test is inconclusive when applied to the following functions at (0, 0). 13. f(x, y ) = x4 + y* - 16ry Describe the behavior of the function at the critical point. 14 . f(x. y ) = x3/3 -1/3+ 3xy 39. f(x, y) = 4 + x4+ 3y 40. f (x, y ) = xy - 3 15. f(x,y) = x4 - 2x2 + y2 - 4y + 5 41. f(x, y) = xy2 42. f( x, y ) = sin (x 3y? ) 16. f(x, y ) = x2 + xy - 2x - y + 1 43-52. Absolute maxima and minima Find the absolute maximum 17. f(x, y ) = x2 + 6x + 2 + 8 and minimum values of the following functions on the given region R. 18. f(x. y ) = 12-andty 43. f( x, y ) = x2 + y2 - 2y + 1; R = {(x,y):x3 + y's 4) 19-34. Analyzing critical points Find the critical points of the 44. f ( x, y) = 2x2 + y?; R = ( (my ):x + y= 16] following functions. Use the Second Derivative Test to determine (if 45. f(x, y ) = 4 + 2x2 + yz; possible) whether each critical point corresponds to a local maximum, R = {(x,y): -Isx s l, -Isys1} local minimum, or saddle point. Confirm your results using a graphing 46. f(x, y) = 6 - x2 - 4y2; utility. R = { (x, y): - 2 5 x 5 2, - 1 sys1} 19. f(x, y ) = 4 + 2x2 + 3y2 47. f(x, y) = 2x2 - 4x + 3y2 + 2; 20. f(x, y) = (4x - 1)2 + (2y + 4)2 + 1 R = { (x,y ): (x - 1)3 + y's1} 21. A(x, y) = -4x2 + 8y2 -3 48. f(x, y) = x2 + y2 - 2x - 2y; R is the closed region bounded by the triangle with vertices (0, 0). (2, 0), and (0, 2). 22. f(x, y ) = x* + 34 - 4x - 32y + 10 49. f(x, y) = -2x2 + 4x - 3y2 - 6y - 1; 23. f(x, y ) = x4 + 2y2 - 4xy R = {(x,y): (x - 1)2 + ( + 1)? $1 ) 24. f (x, y ) = xe *-" 50. f(x, y) = Vx2 + y? - 2x + 2; R = { (x,y):x2+ y's4, 25. f(x, y ) = Vx2 + y2 - 4x + 5 yz0} 2y2 - x2 26. f(x, y) = tan xy 51. f(x, y) = 2 + 2x2,2; R is the closed region bounded by the lines 27. f (x, y ) = 2rye -r y = x, y = 2x, and y = 2. 28. f(x, y) = x2 + xy? - 2 + 1 52. f(x, y) = Vx2 + y; R is the closed region bounded by the ellipse =+ yz = 1. 29. f( x, y) =1+ x2+
