Problem 1 :
For what value of x does the function
f(x) = x3 - 9x2 - 120x + 6
have a local minimum ?
A) 10 B) 4 C) -4 D) -10
Problem 2 :
Suppose f'(x) = x (x - 2)2 (x + 3). which of the following is (are) true ?
I. f has a local maximum at x = -3
II. f has local minimum at x = 0
III. f has neither a local maximum nor local minimum at x = 2
A) I only B) II only C) III only D) I and II only
E) I, II and III
Problem 3 :
The maximum value of the function
f(x) = x4 - 4x3 + 6 on [1, 4] is
A) 1 B) 0 C) 3 D) 6 E) -27
Problem 4 :
Find all critical numbers of the function g(x) = x4 - 4x2
Problem 5 :
Locate the absolute extrema of the function
f(x) = x3 - 12x
on the closed interval [0, 4]
A) absolute max (2, -16) and absolute min (4, 16)
B) no absolute max, absolute min (4, 16)
C) absolute max (4,16); absolute min (2, -16)
D) Absolute max (4, 16), no absolute min
E) no absolute max or min.
1) x = -4, option C
2) I, II and III, Option E
3) 6, option D
4) x = 0, 2√2 and -2√2
5) Absolute maximum is at (4, 16) and absolute minimum is at (2, -16), option C.
Problem 1 :
If (x + 2y) (dy/dx) = 2x - y, what is the value of d2y/dx2 at the point (3, 0) ?
A) -10/3 B) 0 C) 2 D) 10/3 E) Undefined
Problem 2 :
For t ≥ 0, the position of a particle moving along the x-axis is given by x(t) = sin t - cos t. What is the acceleration of the particle at the point where the velocity is equal to 0 ?
A) -√2 B) -1 C) 0 D) 1 E) √2
Problem 3 :
The graph f', the derivative of the function, f is shown above. Which of the following statements must be true ?
I) f has relative minimum at x = -3
II) The graph of f has a point of inflection at x = -2
III) The graph of f is concave down for 0 < x < 4
A) I only B) II only C) III only D) I and II only
E) I and III only
Problem 4 :
The graph of y = etan x- 2 crosses the x-axis at one point in the interval [0, 1]. What is the slope of the graph at this point ?
A) 0.606 B) 2 C) 2.242 D) 2.961 E) 3.747
Problem 5 :
f'(x) = √(x4+1) + x3 - 3x, then f has a local maximum at x = ?
A) -2.314 B) -1.332 C) 0.350 D) 0.829 E) 1.234
Problem 6 :
For -1.5 < x < 1.5, let f be a function with first derivative given by
Which of the following are all intervals on which graph of f is concave down ?
A) (-0.418, 0.418) B) (-1, 1)
C) (-1.354, -0.409) and (0.409, 1.354)
D) (-1.5, -1) and (0, 1)
E) (-1.5, -1.354) (-0.409, 0) and (1.354, 1.5)
Problem 7 :
The graph of f', the derivative of f is shown above. The function f has local maximum at x = ?
A) -3 B) -1 C) 1 D) 3 E) 4
Problem 8 :
The graph of f'', the second derivative f, is shown above -2 ≤ x ≤ 4. What are all intervals on which the graph of the function f is concave down ?
A) -1 < x < 1 B) 0 < x < 2 C) 1 < x < 3
D) -2 < x < -1 E) -2 < x < -1 and 1 < x < 3
Problem 9 :
Let f be a polynomial with values of f'(x) at selected values of x given in the table above. Which of the following must be true for -2 < x < 6?
A) The graph of f is concave up.
B) The graph of f has at least two points of inflection
C) f is increasing
D) f has no critical points.
E) f has at least two relative extrema
Problem 10 :
The graph of the derivative of a function f is shown in the figure above. The graph has horizontal tangent lines at x = -1, x = 1 and x = 3. At which of the following values of x does f have a relative maximum ?
A) -2 only B) 1 only C) 4 only D) -1 and 3 only
E) -2, 1 and 4.
1) -10/3 , option A
2) -√2, option A
3) I and III only, option E
4) 2.92, option D
5) x = 0.35, option C
6) (-1.5, -1) and (0, 1), option D
7) x = 1, option C
8) On -2 < x < -1 and 1 < x < 3, option E.
9) option B
10) x = 4, option C
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM