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

solution

 

  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

    solution

     

    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

    solution