1. What is the maximum of f(x,y) = x2y, given x2 + y2 = 1 ?

 A. (4/27)31/2 B. 31/2/9 C. (2/9)31/2 D. 21/2/3 E. 2/3

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1. Let y = (3x)1/2 for x > 0. What is the x-coordinate of the point on the graph of y that is closedt to the point (5,0)?

 A. 2/7 B. 1 C. 5/2 D. 7/2 E. 13/2
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3.Let f(x) = 5 + 6x + 12x2 - 2x3 - x4 and let g(x) = f(x) for any real number x. At what value of x is g(x) increasing most rapidly?

 A. -2 B. -(1/2) C. 1/2 D. 1 E. 2
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4. A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the xy-plane bounded by the lines y = 0, y = 3x, and y = 30 - 2x. What is the largest possible area of such a rectangle?

 A. 135/8 B. 45 C. 135/2 D. 90 E. 270
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5. A rectangle is to be inscribed in a semicircle of radius 10, with one side lying on the diameter of the semicircle. What is the maximum possible area of the rectangle?

 A. 5(2)1/2 B. 50 C. 100 D. 60(5)1/2 E. 145
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6. A solid rectangular brick is to be made from 1 cubic foot of clay. The brick must be 3 times as long as it is wide. For what value of the ratio of height to width will the brick have minimum surface area?

 A. 1/27 B. 2/9 C. (2/9)1/3 D. 2/3 E. 3/2
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7. Let f(x,y) = 2y2 - 15y + x2y - 2xy. At which of the following critical points

 I. (-3,0) II II. (1,4) III. (5,0)

does f have a relative maximum?

 A. None B. I and II only C. I and III only D. II and III only E. The correct answer is not given by A,B,C, or D

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8. What is the area of the largest rectangle that can be inscribed in the region bounded by y = 3 - x2 and the x-axis?

 A. 1 B. 2 C. 9/4 D. 4 E. 4(3)1/2
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9. If

then at what value of x in the interval is f(x) a maximum?

 A. B. C. D. E.
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10. Let f(x,y) = x3 + 6xy + y3 + 3. What are all the points at which f has a relative maximum?

 A. (0,0) B. (-2,-2) C. (0,0) and (-2,-2) D. (-2,2) and (2,-2) E. (0,0),(-2,2), and (2,-2)
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11. What is the minimum value of f(x,y) = x2y for {(x,y): 4x2 + y2 = 4}?

 A. -2/31/2 B. -4/3(31/2) C. 0 D. 4/3(31/2) E. 2/31/2
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