868 CHAPTER 15 DIFFERENTIATION IN SEVERAL VARIABLES 5. Let f (x, y) = 12x - yx2 + xy. (c) be ch (a) Show that the critical points (x, y) satisfy the equations max for t y(y - 2x + 1) =0, x ( 2y - x + 1)=0 27. (b) Show that f has three critical points where x = 0 or y = 0 (or both) mati and one critical point where x and y are nonzero. (c) Use the Second Derivative Test to determine the nature of the critical points. App mini 6. Show that f(x, y) = vx2 + y2 has one critical point P and that f is nondifferentiable at P. Does f have a minimum, maximum, or saddle point at P? In Exercises 7-23, find the critical points of the function. Then use the Second Derivative Test to determine whether they are local minima, local maxima, or saddle points (or state that the test fails). 7 . f ( x, y ) = x2+ 12 - xy+ x 8. f (x, y ) =x3 -xy+ y' 9. f (x, y ) = x3 + 2xy - 2y2 - 10x 10. f (x, y ) = x3y + 12x2 - 8y 11. f (x, y ) = 4x - 3x3 - 2xy2 12 . f (x , y ) = x3 + y4 - 6.x - 2y2 13. f (x, y ) = x4+4- 4xy 14. f (x, y ) = ex - 12 +4y 15 . f (x , y ) = xye - x2 -y2 16. f(x, y) = ex - xey 17. f(x, y) = sin(x + y) - cosx 18. f(x, y) = x In(x + y) 19. f(x, y) = Inx + 2lny - x - 4y 20. f (x, y) = (x + y) In(x2+2) 21. f (x, y) = x - y2 - In(x + y) 22. f (x, y) = (x - y)ex2->2 23. f ( x, y) = (x+ 3y)ey-x2 24. Show that f(x, y) = x2 has infinitely many critical points (as a func- tion of two variables) and that the Second Derivative Test fails for all of them. What is the minimum value of f? Does f(x, y) have any local maxima? 18 25. Prove that the function f(x, y) = 3x3 + 3y3/2 - xy satisfies f(x, y) 2 0 for x 2 0 and y 2 0. (a) First, verify that the set of critical points of f is the parabola y = x2 and that the Second Derivative Test fails for these points. (b) Show that for fixed b, the function g(x) = f(x, b) is concave up for x > 0 with a critical point at x = 61/2. (c) Conclude that f(a, b) > f(b1/2, b) = 0 for all a, b 2 0. 26. Let f (x. y) = (x2+ yz )e-$2-82. (a) Where does f take on its minimum value? Do not use calculus to answer this question. (b) Verify that the set of critical points of f consists of the origin (0,0) and the unit circle x2 + y? = 1.VARIABLES (c) The Second Derivative Test fails for points on the unit circle (this can be checked by some lengthy e lengthy algebra). Prove, however, that f takes on its maximum value on the unit circle by analyz y analyzing the function g(1) = re- for t > 0. 0 (or both) 27. CAS Use a computer algebra system to find a numerical approxi- mation to the critical point of he critical f (x, y ) = (1-*+3)@12 + (1 - >+12)0*2 Apply the Second Derivative Test to confirm that it corresponds to a local nd that f minimum as in Figure 21. or saddle i use the na, local FIGURE 21 Plot of f (x, y) = (1 -x+x2)ex- + (1-y+>2)ex2. 28. Which of the following domains are closed and which are bounded? (a) {(x, y) ER2 : x2+ 12 =1] (b) { ( x, y) ER2 : x2 + 12 0, y > 0} (e) { (x, y) E R2 : 1 0,x2+ 12 = 10} In Exercises 29-32, determine the global extreme values of the func- C- tion on the given set without using calculus. all al 29. f ( x, y ) = x+y, O2, x2+2 =1 A linear function f(x, y) = ax + by + c has no critical points. There- fore, the global minimum and maximum values of f(x, y) on a closed and bounded domain must occur on the boundary of the domain. Furthermore, it is not difficult to see that if the domain is a polygon, as in Figure 22, then the global minimum and maximum values of f must occur at a vertex of the polygon. In Exercises 33-36, find the global minimum and maxi- mum values of f(x, y) on the specified polygon, and indicate where on the polygon they occur. 33. f(x, y) = 2x - by + 4 on the polygon in Figure 22(A). 34. f(x, y) = lly - 7x + 7 on the polygon in Figure 22(B)