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) increase or decrease as (x, y) moves away from P along the line g(x, y) = 0? 4 g(x, y)=0 0- 12 24 36 -4 -4 0 8 2- 0 -2 FIGURE 13 Contou constraint g(x, y) FIGURE 12 Level curves of f(x, y) = x+2y2 and graph of the constraint g(x, y) = 4x - 6y - 25 = 0. 3. Apply the method of Lagrange multipliers to the function f(x, y) = (x + 1)y subject to the constraint x + y = 5. Hint: First show that y = 0; then treat the cases x = 0 and x 0 separately. In Exercises 4-15, find the minimum and maximum values of the function subject to the given constraint. 17. Find the poin least. 18. Find the recta of the edges is 30 19. The surface S = rr+h (a) Determine maximum volun E 4. f(x, y) = 2x + 3y, x + y = 4 now amniog (0) (b) What is the surface area S? 5. f(x, y) = x + y, 2x + 3y = 6 0: (c) Does a com exist? 6. f(x, y) = 4x+9y, xy=4 7. f(x, y) = xy, 4x2 +9y = 32 8. f(x, y) = xy+x+y, xy=4 9. f(x, y) = x + y, x + y = 1 20. In Examp ellipse (x/4)2 grange multip 4 cost, y = 3 ing single-va 21. Find the 10. f(x, y) = xy4, x+2y = 6 11. f(x, y, z) = 3x+2y+4z, x2 + 2y+6z = 1 12. f(x, y, z) = x-y-z, x- y+z = 0 13. f(x, y, z) = xy +2z, x + y+z = 36 14. f(x, y, z) = x + y+z, x+3y+2z = 36 15. f(x, y, z) = xy+xz, x + y+z = 4 with the gre 16. Let f(x, y) = x + xy + y, g(x, y) = x xy + y (a) Show that there is a unique point P = (a,b) on g(x, y) = 1 where Vfp=Vgp for some scalar >.. (b) Refer to Figure 13 to determine whether f(P) is a local minimum or a local maximum of f subject to the constraint. 22. Use scribed (e) Does Figure 13 suggest that f(P) is a global extremum subject to the constraint?