A. (4/27)3^{1/2} |
B. 3^{1/2}/9 |
C. (2/9)3^{1/2} |
D. 2^{1/2}/3 |
E. 2/3 |
A. 2/7 |
B. 1 |
C. 5/2 |
D. 7/2 |
E. 13/2 |
3.Let f(x) = 5 + 6x + 12x^{2} - 2x^{3} - x^{4} 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 |
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 |
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 |
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 |
7. Let f(x,y) = 2y^{2} - 15y + x^{2}y - 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 |
8. What is the area of the largest rectangle that can be inscribed in the region bounded by y = 3 - x^{2} and the x-axis?
A. 1 |
B. 2 |
C. 9/4 |
D. 4 |
E. 4(3)^{1/2} |
9. If
then at what value of x in the interval is f(x) a maximum?
A. |
B. |
C. |
D. |
E. |
10. Let f(x,y) = x^{3} + 6xy + y^{3} + 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) |
11. What is the minimum value of f(x,y) = x^{2}y for {(x,y): 4x^{2} + y^{2} = 4}?
A. -2/3^{1/2} |
B. -4/3(3^{1/2}) |
C. 0 |
D. 4/3(3^{1/2}) |
E. 2/3^{1/2} |