5.1 MVT
Let f be a polynomial function with a degree greater than 2. If a does not equal b and f(a) = f(b) = 1, which of the following must be true for at least one value of x between a and b?
I f(x) = 0
II f'(x) = 0
III f''(x) = 0
a) None b) I only c) II only d) I and II only e) I, II, and III
c) II only
Quiz AP classroom #10
Calculator Question:
Let f be the function defined by f(x) = ln(x)/x. What is the absolute maximum value of f?
a) 1 b) 1/e c) 0 d) -e e) f does not have an absolute maximum value
b) 1/e
Quiz AP classroom #13
The function given by f(x) = x^3+12x -24 is
a) increasing for x<-2, decreasing for -2<x<2, increasing for x>2 b) decreasing fr x<0, increasing for x>0 c) increasing for all x d) decreasing for all x e) decreasing for x<-2, increasing for -2<x<2, decreasing for x>2
c) increasing for all x
Quiz AP classroom #22
The function given by d(x) = 2x^3-3x^2-12x has a relative minimum at x =
a) -1 b) 0 c) 2 d) (3-105^(1/2))/4 e) (3+105^(1/2))/4
c) 2
Quiz AP classroom #43
Let f be the function given by f(x) =x^3-6x^2-15x. What is the maximum value of f on the interval [0,6]?
a) 0 b) 5 c) 6 d) 8
a) 0
Quiz AP classroom #46
Which of the following does not satisfy the Mean Value Theorem? [-1.5,1.5]
a) tan(x) b) ln(x+2) c) 4/(x^2-x-2) d) x^(4/3)
c) 4/(x^2-x-2)
AP video
Calculator Question
The function f given by f(x) = 9x^(2/3)+3x -6 has a relative minimum at x =
a) -8 b) -2^1/3 c) -1 d) -1/8 e) 0
e) 0
Quiz AP classroom #32
Let f be the function with derivative given by f'(x) = x^2-2/x. om which of the following intervals is f decreasing?
a) (-infinity,-1] only b) (-infinity,0) c) [-1,0) only d) (0, 2^1/3) e) [2^1/3,infinity)
d) (0, 2^1/3)
Quiz AP classroom #23
Let g be the function defined by g(x) = x^4 + 4x^3. How many relative extrema does g have?
a) 0 b) 1 c) 2 d) 3
B) 1
Quiz AP classroom #38
The absolute maximum values of f(x) = x^3-3x^2+12 on the closed interval [-2,4] occurs at x =
a) 4 b) 2 c) 1 d) 0 e) -2
a) 4
Quiz AP classroom #45
The function f is given by f(x) = x^3. The application of the Mean Value Theorem to f on the interval [1,3] guarantees a point in the interval (1,3) at which the slope of the line tangent to the graph of f is equal to which of the following
a) 0 b) 12 c) 13 d) 24
c) 13
Quiz AP classroom #4
Calculator Question:
Let f be the function defined by f(x) = x^3 -3x^2 -9x +11. At which of the following values of x does f attain a local minimum?
a) -3 b) 1 c) -1 d) -3
a) 3
Quiz AP classroom #33
If f(x) = x+1/x, then the set of values where f increases is a) (-infinity, -1] U [1,infinity) b) [-1,1] c) (-infinity, infinity) d) (0,infinity) e) (-infinity, 0) U (0,infinity)
a) (-infinity, -1] U [1,infinity)
Quiz AP classroom #25
For what value of k will x+ k/x have a relative maximum at x=-2?
a) -4 b) -2 c) 2 d) 4 e) None of these
d) 4
Quiz AP classroom #40
The function g is given by g(x) = 4x^3 +3x^2 -6x +1. What is the absolute minimum value of g on the closed interval [-2,1]?
a) -7 b) -3/4 c) 0 d) 2 e) 6
a) -7
Quiz AP classroom #47
Let f be the function given f(x) = x^3 - 3x^2. What are all the values of c that satisfy the conclusion of the Mean Value Theorem of differential calculus on the closed interval [0,3]?
a) 0 only b) 2 only c) 3 only d) 0 and 3 e) 2 and 3
b) 2 only
Quiz AP classroom #9
Calculator Question:
The function f is defined by f(x) = e^-x(x^2+2x) At what values does f have a relative maximum?
a) -2+2^1/2 b) -2^1/2 only c) -2 and x = 0 d) 2^1/2 only
d) 2^1/2 only
Quiz AP classroom #34
Let f be the functioned defined by f(x) = (sin(x))e^-x on the interval [-pi/2,pi/2]. On which of the following open intervals is f increasing?
a) [-pi/4,pi/2] b) [0,pi/2] only c) [pi/4,pi/2] only d) [-pi/2,pi/4]
d) [-pi/2,pi/4]
Quiz AP classroom #27
Calculator Question
For what value of x does the function f(x) = (x-2)(x-3)^2 have a relative maximum.
a) -3 b) -7/3 c) -5/2 d) 7/3 e) 5/2
d) 7/3
Quiz AP classroom #42
What is the absolute minimum value of y = -cos(x) - sin(x) on the closed interval [0, pi/2]?
a) -2*2^1/2 b) -2 c) -2^1/2 d) -1
c) -2^1/2
Quiz AP classroom #44
x | 0 | 2 | 5 | 7 | 11 |
f(x) | 13 | 5 | 17 | 28 | 41 |
Selected values of a differentiable function are given in the table above. what is the fewest possible number of values c in the interval for which the mean value theorem guarantees the f'(c) = 4?
a) 0 b) 1 c) 2 d) 3
c) 2
Quiz AP classroom #8
Calculator Question:
Which of the following functions of x is guaranteed by the Extreme value theorem to have an absolute maximum on the interval [0,4]
a) y = tan(x) b) y = arctan(x) c) y = (x^2-16)/(x^2+x-20)
d) y = 1/(e^x - 1)
b) arctan(x)
Quiz AP classroom #15
Let f be the function given by f(x) = (e^x)*sin(x) on the closed interval [0,2pi]. On which of the following intervals is f increasing?
a) [pi/2,3pi/2] b) [3pi/4,7pi/4] c) [0,pi/2] and [3pi/2,2pi] d) [0,3pi/4] and [7pi/4,2pi]
d) [0,3pi/4] and [7pi/4,2pi]
Quiz AP classroom #26
What is the minimum value of f(x) = xln(x)?
a) -e b) 0-1 c) -1/e d) 0 e) has no minimum value
-1/e
Quiz AP classroom #39
Let g be the function defined by g(x) = |x|-3|x+1|. What is the absolute maximum value of g on the closed interval [-2,2]?
a) 1 b) -1 c) -3 d) -7
a) 1
Quiz AP classroom #48