AP Calculus AB
4.0 Worksheet
Name: ___________________________________
Hour: _________
1. Multiply by the conjugate to rationalize the numerator to find the limit of f(x):
f ( x)  lim
x 0
1  x 1
x
2. Use L’Hopital’s rule to find the limit of f(x): f ( x)  lim
x 0
1  x 1
x
Use any appropriate method to evaluate each limit (substitution, EBM, factor/simplify, rationalize the
numerator, L’Hopital’s rule, etc.)
sin x
x 0 x  4
3. lim
6. lim
x2
2 x 2
x2
3x 2  5 x
x 0 4 x 2  7 x
2sin x cos x
x 0
x
5. lim
( x  4) 2  16
x 0
x
8. lim
4. lim
7. lim
1
(2 x  3) 2  9
x 0
4x
1  sin x
 1  cos(2 x )
x
sin(5 x)
x 0 sin(7 x)
10. lim
x  5x  4
x  4 3 x 2  11x  4
1
1

13. lim x  7 7
x 0
x
9. lim
2
12. lim
15. lim
x 0
2sin x
x2  5x
2
2 x sin x
x 0 x 2  5 x
16. lim
2
x2  5x  4
x 0 3 x 2  11x  4
11. lim
ln( x  3)
x  4 x 2  16
14. lim
3x3  x
x 0 5 x 2  x
17. lim
5x4  4 x2 1
18. lim
x 1 10  x  9 x 3
2sin( x)  sin(2 x)
19. lim
x 0
x  sin( x)
e x  e x
20. lim
x 0
x
21. Find the limit using the alternate definition of the derivative and L’Hopital’s rule:
lim
x
22. What is the 1st method you should try when evaluating any limit problem?
3

3
tan( x)  3
x

3
AP Calculus AB
4.1A Worksheet
Name: _____________________________________________________________
Hr: ___
1) The extreme value theorem says that when a _________________________ function is taken
over a _________________ interval, the function has both an absolute _________________ and
an absolute _____________________.
In questions 2-4, explain whether or not the extreme value theorem applies. If possible, list the absolute
maximum value and where it occurs and list the absolute minimum value and where it occurs.
2)
(a) Does the extreme value theorem apply? Explain.
(b) List any absolute extreme value and where it occurs.
3)
(a) Does the extreme value theorem apply? Explain.
(b) List any absolute extreme value and where it occurs.
4)
(a) Does the extreme value theorem apply? Explain.
(b) List any absolute extreme value and where it occurs.
4
In questions 5-6, find the absolute maximum and absolute minimum values of the function listed over the
interval given.
5) Suppose f ( x)  x 4/3 ,  2  x  4.
6) Suppose f ( x)  2sin  x  , 0  x 
3
.
4
7) Suppose f ( x)  2 x 2  8 x  9.
(a) Does the Extreme Value Theorem apply to f(x)? Why or why not?
(b) Find the extreme values of the function and where they occur.
5
AP Calculus AB
4.1B Worksheet
Find the extreme values of the function and where they occur. [Note: because the extreme value theorem
doesn’t apply without a closed interval, we aren’t going to state anything about absolute extrema. We are
only going to comment on maximum and minimum values, without specifying whether they are local or
absolute.]
1)
f ( x)  e  x
2) f ( x)  x3  x 2  8 x  5
3) f ( x)   x 2  6 x  11
4) f ( x)  xe x
6
AP Calculus AB
4.2A Worksheet
1) For the mean value theorem to apply to a function from x = a to x = b, the function must be
_________________ over the interval ___________________ and the function must also be
___________________ over the interval ___________________.
For questions 2-5, (a) state whether or not the function satisfies the hypotheses of the Mean Value Theorem
on the give interval, and (b) if it does, find each value of c in the interval (a, b) that satisfies the equation
f '(c) 
2)
f (b)  f (a)
.
ba
f ( x)  x , 1  x  1.
4) f ( x)  sin x, 0  x 
3) f ( x)  x 2  3x,  1  x  2.

5) f ( x) 
3
7
x  4, 0  x  5
6) Consider the function f ( x)  x 
1
, 0.5  x  2.
x
(a) Find the equation of the secant line from x = 0.5 to x = 2.
(b) Find the equation of the tangent line to f(x) that is parallel to the secant line you found in part
(a).
In questions 7 and 8, use the mean value theorem to answer the questions described.
7) A trucker handed in a ticket at a toll booth showing in 2 hours he had driven 159 miles on a toll
with speed limit 65 miles per hour. The trucker was cited for speeding. Why?
road
8) It took 10 seconds for the temperature of water to change from 5 degrees Celsius to 80 degrees Celsius.
Explain why at some moment the temperature of water was changing at exactly 7.5 degrees Celsius per
second.
8
AP Calculus AB
4.2B Worksheet
For all questions, use analytic methods to find (a) the local extrema, (b) the intervals on which the function
is increasing and (c) the intervals on which the function is decreasing.
1)
f ( x)  6 x  x 2
2) f ( x)  xe x
9
3) f ( x)  x 4  10 x 2  9
4) f ( x)  x 9  x
10
AP Calculus AB
4.3A Worksheet
In questions 1 and 2, use the First Derivative Test (a.k.a. sign charts) to determine the local extreme
values of the function.
1)
2) y   2 x  3 e
y  2 x3  6 x 2  3
x
In questions 3 and 4, use the Concavity Test to determine the intervals on which the graph of the function
is (a) concave up and (b) concave down.
3) y  4 x3  21x 2  36 x  20
4) y  3  x 2/3
11
In questions 5 and 6, find all points of inflection of the function.
5)
y  xe x
6) y   x 4  4 x3
Question #7 – Suppose f ( x)  x3  5x 2  3x  7.
(a) When is f ( x ) increasing?
(b) When is f ( x ) decreasing?
(c) List any maximum values of f ( x ) and when they occur.
(d) List any minimum values of f ( x ) and when they occur.
(e) When is f ( x ) concave up?
(f) When is f ( x ) concave down?
(g) Where does f ( x ) have inflection points?
12
AP Calculus AB
4.3B Worksheet
In questions 1-2, use the second derivative test to find the local extrema for the function. Give the values
of x for which these values occur.
1)
2) y  x 2e x
y  x 3  3x 2  2
3) Suppose f '( x)   x  1
2
 x  2  x  4  .
(a) Determine the x-coordinate(s) where a local maximum value of f ( x ) occurs.
(b) Determine the x-coordinate(s) where a local minimum value of f ( x ) occurs.
(c) It is impossible to compute the actual maximum and minimum value for this function. Why?
(d) Find where f(x) is increasing.
(e) Find where f(x) is decreasing.
13
4) Suppose f ( x ) is continuous on [0, 3] and satisfies the following criteria:
0  x 1 1 x  2 2  x  3
x
0
1
2
3
x
f
0
2
0
2
f



f' 3
0
f'



f"



dne 3
f " 0 1 dne
0
(a) Sketch a graph of f(x).
(b) Find the absolute extrema of f and when they occur.
(c) Find any points of inflection.
(d) Over the interval [0, 3], must there exist a value “c” such that f "'(c)  0? Explain why or why
not.
(e) Over the interval [1, 3], must there exist a value “c” such that f (c)  1? Explain why or why
not.
(f) According to the extreme value theorem, must f(x) have an absolute maximum and absolute
minimum over the interval [0, 3]? Explain why or why not.
14
5) True or False: If f "(c)  0, then (c, f(c)) is a point of inflection. Justify your answer.
6) True or False: If f '(c )  0 and f "(c)  0, then f (c ) is a local maximum. Justify your answer.
7) If a  0, the graph of y  ax3  3x 2  4 x  5 is concave up on _____________________.


1
a
(A)  ,  
1 
,
a 
(D) 


1
a
(B)  , 
(E)
 1
 a


(C)   ,  
 , 1
8) If f (0)  f '(0)  f "(0)  0, which of the following must be true?
(A) There is a local maximum at the origin.
(B) There is a local minimum at the origin.
(C) There is an absolute minimum value at the origin.
(D) There is a point of inflection at the origin.
(E) There is a horizontal tangent at the origin.
9) The x-coordinates of the points of inflection of the graph of y  x5  5x 4  3x  7 are:
(A) 0 only
(B) 1 only
(C) 3 only
(D) 0 and 3
(E) 0 and 1
15
10) Consider a function f with a domain of all real numbers. Which of the following conditions would
enable you to conclude that the graph of f has a point of inflection when x = c?
(A) There is a local maximum of f ' at x = c.
(B) f "(c)  0.
(C) f "(c) does not exist.
(D) The sign of f ' changes at x = c.
(E) f is a cubic polynomial and c = 0.
11) Suppose f is twice-differentiable for all real numbers. Selected values of f '( x) & f "( x) are given
in the table below. Determine whether each statement listed must be true or false. Justify your answer.
x
0
1
2
3
f '( x)
1 0 2
3
f "( x)
7
4
6
0
(A) f ( x ) must have an inflection point when x = 2.
(B) f ( x ) must have a local minimum at x = 1.
(C) There exists a value c, 0 < c < 3, such that f '''(c)  
(D) There exists a value c, 2 < c < 3, such that f '(c)  0.
16
11
.
3
AP Calculus AB
4.3C Worksheet
1) Use the graph of f(x) below to answer the questions that follow. Assume that f is continuous and
twice-differentiable for all real numbers.
a) When is f '( x)  0? Justify your answer.
b) When is f '( x)  0? Justify your answer.
c) When is f '( x)  0? Justify your answer.
d) When is f "( x)  0? Justify your answer.
e) When is f "( x)  0? Justify your answer.
17
Question #1 is continued below.
f) Place the following in order from least to greatest: f (0), f '(0), f "(0) [Justify your answer]
g) Place the following in order from least to greatest: f '(0), f '(1), f '(2) [Justify your answer]
h) Place the following in order from least to greatest: f "(2), f "(0), f "(2) [Justify your
answer]
18
2) Use the graph of f '( x) below to answer the questions that follow. Assume that f is continuous
and twice-differentiable for all real numbers.
a) When is f increasing? Justify your answer.
b) When is f decrasing? Justify your answer.
c) Give the x-coordinates of any local extrema for f. State whether these correspond to
maximum or minimum values. Justify your answer.
d) When is f concave up? Justify your answer.
e) When is f concave down? Justify your answer.
f) When does f have an inflection point? Justify your answer.
g) Suppose f (3)  8. Write the equation of the line tangent to f at x = 3.
19
3) Use the graph of f "( x ) below to answer the questions that follow. Assume that f is continuous
and twice-differentiable for all real numbers.
a) When is f ( x ) concave up? Justify your answer.
b) When is f ( x ) concave down? Justify your answer.
c) List the x-coordinates of any inflection points for f. Justify your answer.
d) Suppose f '(1)  0. Is x = 1 a relative maximum, minimum, or neither? Justify your answer.
20
4) The graph below represents the position of a particle (in meters) at time t from t = 0 seconds to t =
5 seconds. The graph below has horizontal tangents at t = 2 and t = 4 and an inflection point
when t = 3. Answer the questions that follow.
a) What is the position of the particle at t = 0?
b) What is the displacement of the particle from t = 0 to t = 4 seconds?
c) When is the particle moving to the right? Justify your answer.
d) When does the particle have positive acceleration? Justify your answer.
e) When is the particle’s velocity increasing? Justify your answer.
f) Is the particle speeding up or slowing down at t = 3.5?
21
5) The graph below represents the velocity of a particle (in meters/second) at time t from t = 0 seconds
to t = 5 seconds. The graph below has horizontal tangents at t = 2 and t = 4 and an inflection point
when t = 3. Answer the questions that follow. (If a question cannot be answered, state this.)
a) What is the position of the particle at t = 0?
b) What is the displacement of the particle from t = 0 to t = 4 seconds?
c) When is the particle moving to the right? Justify your answer.
d) When does the particle have positive acceleration? Justify your answer.
e) When is the particle’s velocity increasing? Justify your answer.
f) Is the particle speeding up or slowing down at t = 3.5?
22
AP Calculus AB
4.1-4.3 Worksheet
Name: _____________________________________________
Hr: ________
1) Find the absolute extrema for the function, f ( x)  3x 2  11x  5, over the interval [0, 5].
Absolute Maximum Value is ___________, and it occurs at x = __________.
Absolute Minimum Value is ___________, and it occurs at x = __________.
  
2) Find the absolute extrema for the function, f ( x)  sin  3x  , over the interval  ,  .
6 2
Absolute Maximum Value is ___________, and it occurs at x = __________.
Absolute Minimum Value is ___________, and it occurs at x = __________.
23
3) Find the value of c that satisfies the mean value theorem for f ( x)  x 
4)
48
on the interval [1, 4].
x

4
Find the value of c that satisfies the mean value theorem for f ( x)  sin 

x  on the interval

[4,8].
5) Why does the function f ( x)  x satisfy the mean value theorem over the interval [3, 5], but not
over the interval [-2, 2]?
24
6) Find the local extrema for the function, f ( x)  x3  10 x 2  7 x  4, determine where the function
is increasing and decreasing, determine any inflection points and determine when the function is
concave up and concave down.
Maximum: ______________________
Minimum: _____________________________
Increasing: _______________________
Decreasing: ____________________________
Concave Up: _____________________
Concave Down: _________________________
Inflection Points: ____________________


7) Find the local extrema for the function, f ( x)  2 x 2  3 x e x . Determine where the function is
increasing and decreasing, determine any inflection points and determine when the function is
concave up and concave down.
Maximum: _______________________
Minimum: _____________________________
Increasing: _______________________
Decreasing: ____________________________
Concave Up: _____________________
Concave Down: _________________________
Inflection Points: _____________________
25
8) Let f be a function defined on the closed interval 3  x  4 with f(0) = 3. The graph of the derivative
of f, f ' , consists of one line segment and a semicircle as shown above.
a. On what intervals, if any, is f increasing? Justify your answer.
b. Find the x-coordinate of each point of inflection of the graph of f on the open interval (-3,
4). Justify your answer.
c. Find an equation for the line tangent to the graph of f at the point (0, 3).
26
9) The graph of the second derivative of f, f '' , is given above. Assume f is twice-differentiable for all real
numbers.
(a) Where is the graph of f concave up? Explain your reasoning.
(b) Determine any inflection points for the graph of f. Explain why your choice(s) is an inflection
point.
(c) Suppose that f '(1)  0. Do any local extrema occur at x = 1? If so, name the extrema and
explain your reasoning.
27
10) Sketch graphs meeting the criteria below.
1)
f '( x)  0 for  , 2
2)
f '(2)  0.
3)
f "( x)  0 for  2,   .
4)
f "( x)  0 for  , 2  .
 2,  .
==========================================================================
1) f '( x )  0 for  , 4  4,   .
2)
3)
4)
f '( x)  0 for (4, 4).
f '(4)  0.
f '(4) is undefined.
5)
f "( x)  0 for  , 4
 4,  .
28
11) The graph below is a graph of f '( x ). Use this graph to answer the questions below.
the domain of f '( x)  [ 4,5].
Assume that
a) Determine where f(x) is increasing. Justify your answer.
b) Determine where f(x) is decreasing. Justify your answer.
c) Determine the x-coordinates of any local extreme values, and name whether they or
maximum or minimum points.
d) Determine where f(x) is concave up. Justify your answer.
e) Determine where f(x) is concave down. Justify your answer.
f) Suppose f(1) = 7. Write the equation of the line tangent to f at this point.
29
12) The graph below is of f ''( x). Use this graph to answer the questions below. Assume f is twicedifferentiable for all real numbers.
a. Where is f(x) concave up? Justify your answer.
b. Where is f(x) concave down? Justify your answer.
c. List the x-coordinates of any inflection points. Justify your answer.
d. Suppose f '(3)  0. Determine whether x = 3 is a local maximum or a local minimum.
Justify your answer.
30
13) Suppose g(x) is differentiable and g(x) < 0 for all x. Suppose f '( x)  ( x 2  x  12) g ( x).
a. Completely factor f '( x ).
b. Make a sign chart for f '( x ). Remember that g(x) is negative for all x!
c. Determine any critical values of x and whether a relative maximum or a relative
minimum at that point.
14) The function f has a negative first derivative and a positive second derivative. Which of the
following data tables could represent f?
a.
c.
e.
x
2
0
2 4
b.
f ( x) 13 11 9 7
x
2
0
2
4
d.
f ( x) 13 12 10 7
x
f ( x)
2 0
7
2
4
8 10 13
31
x
2
0
2 4
f ( x) 13 10 8 7
x
2
f ( x)
7
0
2
4
10 12 13
15) The function f has a positive first derivative and a positive second derivative. Which of the
following data tables could represent f?
a.
c.
e.
x
2
0
2 4
b.
f ( x) 13 11 9 7
x
2
0
2
4
d.
f ( x) 13 12 10 7
x
f ( x)
2 0
7
2
x
2
0
2 4
f ( x) 13 10 8 7
x
2
f ( x)
7
0
2
4
10 12 13
4
8 10 13
16) The function f has a negative first derivative and a negative second derivative. Which of the
following data tables could represent f?
a.
c.
e.
x
2
0
2 4
b.
f ( x) 13 11 9 7
x
2
0
2
4
d.
f ( x) 13 12 10 7
x
f ( x)
2 0
7
2
4
8 10 13
32
x
2
0
2 4
f ( x) 13 10 8 7
x
2
f ( x)
7
0
2
4
10 12 13
17) Suppose f ( x ) is differentiable over the interval 1  x  4. Additionally, suppose the domain of
f ( x ) is 1  x  4. Suppose f (1)  7 and f (4)  8.
a) What does the intermediate value theorem tell us about f ( x ) ?
b) What does the extreme value theorem tell us about f ( x ) ?
c) What does the mean value theorem tell us about f ( x ) ?
d) Suppose that this question tells you that f is continuous instead of differentiable. How does
the affect the answers to questions (a), (b) and (c)?
33
Chapter 4 Key Concepts
1) The ______________________________________ theorem simply tells us that a continuous
function over a closed interval will have a maximum and minimum value.
2) To determine when a function has an absolute maximum or minimum, we must do the following:
a. Consider _____________.
b. Determine when the derivative is ___________________ and _________________.
c. Setup a ______________________, and fill it out by substituting back into the
____________________________.
d. Sub critical values back into the ________________ equation, to determine which values
are the biggest and smallest.
3) If we aren’t given a _______________ interval, we aren’t going to worry about absolute extrema,
but instead we’ll just worry about ___________________ extrema.
4) The ____________________________ theorem applies whenever we are given a
_______________________ interval, our function is ________________________ over that
entire interval and ____________________________ on the interior of that interval.
5) When our first derivative is positive, our original function is __________________________.
6) When our first derivative is _______________, our original function is decreasing.
7) When we write increasing intervals, AP will expect us to _________________ endpoints, unless
they are explicitly excluded from the function’s domain.
8) To verify where a function is increasing or decreasing, we’ll use our number line. This is also
known as the _______________________________________ test.
9) In addition to using a number line to test to see which points are maximum/minimum points, we
can also use the __________________________________ test.
a. To use the aforementioned test, we find when the derivative equals zero and substitute
these critical points into the _________________________________.
b. If the result from (a) is positive, we have a local ________________________.
c. If the result from (a) is negative, we have a local ________________________.
10) Inflection points occur when the second derivative ___________________________________.
11) A function is ______________________ when the second derivative is positive.
12) When writing where a function is concave up or concave down, we should
___________________ endpoints.
13) When our first derivative is increasing, our original function is ________________________.
14) When our first derivative changes from increasing to decreasing or decreasing to increasing, our
original function has a(n) ____________________________________.
15) When our first derivative changes from positive to negative, our original function has a(n)
______________________. When it changes from negative to positive, our original function has
a(n) ________________________.
16) When our second derivative changes from positive to negative or vice versa, our original
function has a(n) ________________________________.
17) Suppose we are given a graph of the position function.
a. The velocity is positive when the graph is _________________________.
b. The acceleration is positive when the graph is _________________________.
34
AP Calculus
4.1-4.3 Multiple Choice Practice (No Calculator)
Name: ___________________________________________________________________Hr: _________
______ 1. What is the x-coordinate of the point of inflection on the graph of y 
(A) -4
(B) -3
(C) -1
(D) -3/10
1 5 1 4 3
x  x  ?
10
2
10
(E) 0
______ 2. The graph of f(x), a function that is continuous and twice-differentiable, is shown below.
Which of the following is true?
(A) f (2)  f '(2)  f "(2)
(B) f (2)  f "(2)  f '(2)
(C) f '(2)  f (2)  f "(2)
(D) f "(2)  f (2)  f '(2)
(E) f "(2)  f '(2)  f (2)
35
______ 3. If f "( x)  ( x  1)( x  2)3 ( x  4) 2 , f has inflection points at x = ___.
(A) -2 only
(B) 1 only
(D) -2 and 1 only
(E) -2, 1, and 4 only
(C) 1 and 4 only
______ 4. The function f is given by f ( x)   x6  x3  2. When is f decreasing?

1


2 

 1 
(E)  3 ,  
 2 





(B)  ,  3
(A) (, 0)
(D) (0, )
(C)  0, 3
1

2 
_______ 5. The minimum velocity attained on the interval [0, 4] by the particle whose velocity is
given by v(t )  t 3  4t 2  3t  2 is _______.
(A) -16
(B) -10
(C) -8
36
(D) -3
(E) -25/3
________ 6) The graph of f ' , the derivative of f is shown above. Which of the following
statements is true about f ?
a)
f is decreasing for 1  x  3.
b)
f is decreasing for 3  x  0.
c)
f is increasing for 1  x  1.
d)
f has a local maximum at x = 0.
e)
f is not differentiable at x  1 and x  1.
_______ 7) The function f ( x ) has the property that f ( x ) and f "( x ) are positive for all real values of
x, and f '( x) is negative for all real values of x. Which of the following could be the graph
of f ?
A)
B)
D)
E)
37
C)
______ 8) Let f be the function whose derivative is given by f ' 
x
5
. On which of the
x
following intervals is f decreasing?
A) (0, 5]
B) [5, )
D) [25, )
E) (0, 5 
C) (0, 25]

_______ 9) Let f be the function given by f ( x)  3xe x . The graph of f is concave down when:
A) x > 1
B) x < 1
D) x < 2
E) x > 3
38
C) x > 2
______ 10) The second derivative function f is given by f "( x)  x( x  a)2 ( x  b). The graph of f "
is shown above. For what values of x does the graph of f have a point of inflection?
A) 0 and b only
B) j and k only
D) a only
E) j only
C) 0, a and b only
______ 11) Let g be a twice-differentiable function with g '( x )  0 and g "( x)  0 for all real numbers x,
such that g (4)  12 and g (5)  16. Of the following, which is a possible value of g (6) ?
A) 19
B) 20
C) 21
D) 22
E) 23
______ 12) Let f be a function that is differentiable on the open interval (-3, 7). If f ( 1)  4,
f (2)  5 and f (6)  8, which of the following must be true?
I.
II.
III.
f has at least two zeros
f has a relative minimum at x = 2
For some c, 2 < c < 6, f(c) = 4.
(A) I only
(B) II only
(D) I and III only
(E) I, II, and III
39
(C) I and II only
______ 13) The function f is continuous for 2  x  1 and differentiable for 2  x  1. If
f (2)  5 and f (1)  4, which of the following statements could be false?
(A)
There exists c, where 2  c  1, such that f (c)  0.
(B)
There exists c, where 2  c  1, such that f '(c)  0.
(C)
There exists c, where 2  c  1, such that f (c)  3.
(D)
There exists c, where 2  c  1, such that f '(c)  3.
(E)
There exists c, where 2  c  1, such that f (c)  f ( x)
for all x on the closed interval 2  x  1.
Use the graph below for question #14
_______ 14) The graph of the derivative of a function f is shown in the figure above. The graph has
horizontal tangents at x  1, x  1, and x  3. At which of the following values of x
does f have a relative maximum?
(A) -2
(B) -1
(D) 3
(E) 4
40
(C) 1
_______ 15) For all x in the closed interval [2, 5], the function f has a positive first derivative and a
negative second derivative. Which of the following could be a table of values for f ?
A) below
X
2
3
4
5
B) below
x
2
3
4
5
f(x)
7
9
12
16
D) below
X
2
3
4
5
f(x)
16
14
11
7
f(x)
7
11
14
16
C) below
x
2
3
4
5
E) below
x
2
3
4
5
f(x)
16
13
10
7
Use the graph below for question #16.
_______ 16) The graph of f ( x ) is shown above. Which of the following statements is true?
A) f '(0)  f '(4.5)  f '(4)
B) f '(4.5)  f '(0)  f '(4)
C) f '(4)  f '(0)  f '(4.5)
D) f '(4.5)  f '(4)  f '(0)
E) f '(0)  f '(4)  f '(4.5)
41
f(x)
16
12
9
7
Use the graph below for question #17.
_______ 17) The graph of f ( x ) is shown above. Which of the following statements is true?
A) f ''(0)  f ''(4.5)  f ''( 4)
B) f ''(4.5)  f ''(0)  f ''( 4)
C) f ''(4)  f ''(0)  f ''(4.5)
D) f ''(4.5)  f ''(4)  f ''(0)
E) f ''(0)  f ''(4)  f ''(4.5)
Use the graph below for question #18.
_______ 18) The graph of f '( x) is shown above. At which value of x, in -5 < x < 5, does f ( x ) have
a relative maximum?
A) -4
B) -2
C) 0
42
D) 2
E) 4
Use the graph below for question #19.
_______ 19) The graph of f '( x) is shown above. At which value of x does f ( x ) have an inflection
point?
A) -4
B) -2
C) 0
D) 2
E) 3
Use the graph below for question #20.
_______ 20) The graph of f ''( x) is shown above. Suppose f '(3)  0. Which statement is true
regarding the graph of f ( x ) at x = 3?
A)
B)
C)
D)
E)
f ( x)
f ( x)
f ( x)
f ( x)
f ( x)
has an inflection point at x = 3
has a relative maximum at x = 3
has a relative minimum at x = 3
is not differentiable at x = 3
is not continuous at x = 3
43