868 CHAPTER 15 DIFFERENTIATION IN SEVERAL VARIABLES 5. Let f (x, y) = 12x - yx2 +xy. (c) be c (a) Show that the critical points (x, y) satisfy the equations max for 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) ma and one critical point where x and y are nonzero. (c) Use the Second Derivative Test to determine the nature of the critical points. Ap 6. Show that f(x, y) = vx2 + y2 has one critical point P and that f mi 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 + y2 - 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 - 6x - 2y2 of et owned at garn wor too oil 13. f (x, y ) = x4+ y4 - 4xy 14. f (x, y ) = ex2 -12+4y 15. f (x, y ) = xye -x2-yz 16. f(x, y) = ex - xey 17. f(x, y) = sin(x + y) - cos.x 18. f(x, y) = x In(x + y) 19. f(x, y) = Inx +21ny -x -4y 20. f (x, y) = (x + y) In(x2 + >2) 21. f (x, y ) = x - y2 - In(x + x) 22. f ( x, y ) = (x - y )ex2-12 23. f (x, y) = (x+ 3y)ex-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? 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) 2 f(b1/2, b) = 0 for all a, b 2 0. 26. Let f (x, y) = (x2+ 13 )e-x2-yz (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 + y2 = 1