Name: __________________
Class:
Date: _____________
(copy A)
1
For each of the numbers a, b, c, d, r, s, and t, state whether the function whose graph is shown has an absolute maximum or
minimum, a local maximum or minimum, or neither a maximum nor a minimum.
x
min
a
b
c
d
r
s
t
PAGE 1
max local absolute
Name: __________________
2
Class:
Date: _____________
(a) Sketch the graph of a function that has a local maximum at 2 and is differentiable at 2.
a.
b.
c.
PAGE 2
(copy A)
Name: __________________
Class:
Date: _____________
d.
e.
(b) Sketch the graph of a function that has a local maximum at 2 and is continuous but not differentiable at 2.
a.
PAGE 3
(copy A)
Name: __________________
b.
c.
d.
e.
PAGE 4
Class:
Date: _____________
(copy A)
Name: __________________
Class:
Date: _____________
(c) Sketch the graph of a function that has a local maximum at 2 and is not continuous at 2.
a.
b.
c.
PAGE 5
(copy A)
Name: __________________
d.
e.
PAGE 6
Class:
Date: _____________
(copy A)
Name: __________________
3
Class:
Date: _____________
(a) Sketch the graph of a function on [0, 3] that has an absolute maximum but no absolute minimum.
a.
b.
c.
PAGE 7
(copy A)
Name: __________________
Class:
Date: _____________
(copy A)
d.
e.
(b) Sketch the graph of a function on [0, 3] that is discontinuous but has both an absolute maximum and an absolute minimum.
a.
PAGE 8
Name: __________________
b.
c.
d.
e.
PAGE 9
Class:
Date: _____________
(copy A)
Name: __________________
Class:
Date: _____________
4 Find the critical numbers of the function.
f (x ) = 5x
2
+ x
________
5 Find the critical numbers of the function.
3z + 9
f (z ) =
5z
2
+ 3z + 9
________
6 Find the critical numbers of the function.
5
G (x ) =
x
2
5x
________
7 Find the critical numbers of the function.
f ( ) = 2sin
2
+ cos 8 Find the absolute maximum and absolute minimum values of f on the given interval.
f (t ) = 3t
64 2
t , [
5, 8]
9 Find the absolute maximum and absolute minimum values of f on the given interval.
f (x ) = sinx PAGE 10
cosx ,
, 5
2
6
(copy A)
Name: __________________
Class:
Date: _____________
(copy A)
10 Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that
satisfy the conclusion of Rolle's Theorem.
f (x ) = x
3
3x
2
+ 2x + 5, [0, 2]
11 Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that
satisfy the conclusion of Rolle's Theorem.
f (x ) = x x + 12 , [ 12, 0]
12 Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that
satisfy the conclusion of the Mean Value Theorem.
f (x ) = 2x
c =
2
+ 8x + 6,
[
1, 5]
________
13 Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that
satisfy the conclusion of the Mean Value Theorem.
f (x ) =
3
x , [0, 1]
14 Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that
satisfy the conclusion of the Mean Value Theorem.
f (x ) =
x , [2, 4]
x + 2
Round the answer to the nearest hundredth.
15 How many real roots does the following equation have?
8 + 9x + 7x
3
+ 8x
5
= 0
a.
exactly two real roots
b.
at most one real root
c.
exactly one real root
d.
no real roots
e.
at most two real roots
PAGE 11
Name: __________________
16
Class:
How many real roots can the equation x
a.
at most two real roots
b.
at most one real root
c.
exactly two real roots
d.
exactly three real roots
e.
at most three real roots
3
Date: _____________
36x + c = 0 have in the interval [ 1, 1] ?
17 Does there exist a function f such that f (4) = 7, f (6) = 16, and f ' (x ) 2 for all x ?
________
18 For which of the following pairs of functions does f ' (x ) = g ' (x ) for all x in their domains?
a.
b.
c.
d.
e.
PAGE 12
f (x ) = 5x and g(x ) =
f (x ) =
f (x ) =
1
x 5
1
x 5
{ 5x 5x
and g(x ) =
and g(x ) =
{
{
if x > 0
if x < 0
1
x if x > 5
5
1
x if x < 5
5
1
x if x > 5
5
1 +
1
x 5
f (x ) = 1 and g(x ) =
x
{
1
x
1 + x
if x > 0
f (x ) = 5x and g(x ) =
{ 5x1 + 5x
if x > 0
if x < 0
if x < 0
if x < 5
(copy A)
ANSWER KEY
Name: __________________
Class:
x min max local absolute
1.
a
t
f
f
t
b
f
t
t
f
c
t
f
t
f
d
f
t
t
f
r
t
f
t
f
s
f
f
f
f
t
f
t
f
t
d
2.
d
d
PAGE 1
Date: _____________
= +n 2
3 + n
2
+ n
2
+2 n
2
+2 n
2
3 + n
2
3 +2 n
2
3 +2 n
2
+ n
2
7.
+n 2
= 3 + n
2
= + n
2
= +2 n
2
+2 n
=
2
3
=
+ n
2
= 3 +2 n
2
3
+2 n
=
2
= + n
2
(copy A)
c=
13.
f ( 5 ) = 15 39,f ( 4 2 ) =96
8. f ( 5 ) = 15 39,f ( 32 ) =96
f ( 5 ) = 15 39,f ( 32 ) =96
14.
3
9
3
9
1
27
c=2.90
2.90
ANSWER KEY
Name: __________________
Class:
Date: _____________
f
f
2
2
a
3. d
=1,f
3
4
3
4
= 2
= 2
9.
f
f
2
2
1
1
10
10.
5. z= 6,z=0
6. x=0,x=
PAGE 2
5 ,x=5
2
=1,f
=1,f
3
4
3
4
15. c
0.5
=2
3
3
3
c=1+
,c=1 3
3
3
1+
,1 3
3
c= 8
11.
8
12. 2
=2
1
2
3
3
c=1
4. x=
=1,f
(copy A)
3
3
16. b
17. no
18. c,e
Name: __________________
Class:
Date: _____________
(copy B)
1
For each of the numbers a, b, c, d, r, s, and t, state whether the function whose graph is shown has an absolute maximum or
minimum, a local maximum or minimum, or neither a maximum nor a minimum.
x
min
a
b
c
d
r
s
t
PAGE 1
max local absolute
Name: __________________
2
Class:
Date: _____________
(a) Sketch the graph of a function that has a local maximum at 1 and is differentiable at 1.
a.
b.
c.
PAGE 2
(copy B)
Name: __________________
Class:
Date: _____________
d.
e.
(b) Sketch the graph of a function that has a local maximum at 1 and is continuous but not differentiable at 1.
a.
PAGE 3
(copy B)
Name: __________________
b.
c.
d.
e.
PAGE 4
Class:
Date: _____________
(copy B)
Name: __________________
Class:
Date: _____________
(c) Sketch the graph of a function that has a local maximum at 1 and is not continuous at 1.
a.
b.
c.
PAGE 5
(copy B)
Name: __________________
d.
e.
PAGE 6
Class:
Date: _____________
(copy B)
Name: __________________
3
Class:
Date: _____________
(a) Sketch the graph of a function on [ 1, 2] that has an absolute maximum but no absolute minimum.
a.
b.
c.
PAGE 7
(copy B)
Name: __________________
Class:
Date: _____________
(copy B)
d.
e.
(b) Sketch the graph of a function on [ 1, 2] that is discontinuous but has both an absolute maximum and an absolute minimum.
a.
PAGE 8
Name: __________________
b.
c.
d.
e.
PAGE 9
Class:
Date: _____________
(copy B)
Name: __________________
Class:
Date: _____________
4 Find the critical numbers of the function.
f (x ) = 5x
2
+ 2x
________
5 Find the critical numbers of the function.
3z + 9
f (z ) =
2z
2
+ 3z + 9
________
6 Find the critical numbers of the function.
5
G (x ) =
x
2
3x
________
7 Find the critical numbers of the function.
f ( ) = 2sin
2
+ cos 8 Find the absolute maximum and absolute minimum values of f on the given interval.
f (t ) = 3t
64 2
t , [
5, 8]
9 Find the absolute maximum and absolute minimum values of f on the given interval.
f (x ) = sinx PAGE 10
cosx ,
, 5
2
6
(copy B)
Name: __________________
Class:
Date: _____________
(copy B)
10 Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that
satisfy the conclusion of Rolle's Theorem.
f (x ) = x
3
6x
2
+ 5x + 7, [0, 5]
11 Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that
satisfy the conclusion of Rolle's Theorem.
f (x ) = x x + 15 , [ 15, 0]
12 Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that
satisfy the conclusion of the Mean Value Theorem.
f (x ) = 2x
c =
2
+ 9x + 6,
[
1, 1]
________
13 Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that
satisfy the conclusion of the Mean Value Theorem.
f (x ) =
3
x , [0, 8]
14 Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that
satisfy the conclusion of the Mean Value Theorem.
f (x ) =
x , [2, 3]
x + 2
Round the answer to the nearest hundredth.
15 How many real roots does the following equation have?
3 + 9x + 8x
3
+ 4x
5
= 0
a.
exactly one real root
b.
at most one real root
c.
exactly two real roots
d.
no real roots
e.
at most two real roots
PAGE 11
Name: __________________
16
Class:
How many real roots can the equation x
a.
exactly three real roots
b.
at most three real roots
c.
exactly two real roots
d.
at most two real roots
e.
at most one real root
3
Date: _____________
48x + c = 0 have in the interval [ 2, 2] ?
17 Does there exist a function f such that f (4) = 11, f (8) = 26, and f ' (x ) 2 for all x ?
________
18 For which of the following pairs of functions does f ' (x ) = g ' (x ) for all x in their domains?
a.
b.
c.
d.
e.
PAGE 12
f (x ) =
1
x 7
and g(x ) =
f (x ) = 5x and g(x ) =
f (x ) =
1
x 7
{
1
x x { 5x1 + 5x
and g(x ) =
{
f (x ) = 1 and g(x ) =
x
{
f (x ) = 5x and g(x ) =
{ 5x 5x
if x > 7
7
1
if x < 7
7
if x > 0
if x < 0
1
x 1 +
1
x
1 + x
if x > 7
7
1
x 7
if x > 0
if x < 0
if x > 0
if x < 0
if x < 7
(copy B)
ANSWER KEY
Name: __________________
Class:
x min max local absolute
1.
a
f
f
f
f
b
t
f
t
f
c
t
f
t
f
d
f
t
t
t
r
t
f
t
f
s
f
t
t
f
t
t
f
f
t
b
2.
c
e
PAGE 1
Date: _____________
= +n 2
3 + n
2
+ n
2
+2 n
2
+2 n
2
3 + n
2
3 +2 n
2
3 +2 n
2
+ n
2
7.
+n 2
= 3 + n
2
= + n
2
= +2 n
2
+2 n
=
2
3
=
+ n
2
= 3 +2 n
2
3
+2 n
=
2
= + n
2
(copy B)
8 3
9
8 3
13.
9
8
27
c=
f ( 5 ) = 15 39,f ( 4 2 ) =96
8. f ( 5 ) = 15 39,f ( 32 ) =96
f ( 5 ) = 15 39,f ( 32 ) =96
14.
c=2.47
2.47
ANSWER KEY
Name: __________________
Class:
f
f
Date: _____________
2
2
a
3. d
=1,f
=1,f
3
4
3
4
= 2
= 2
9.
f
f
2
2
=1,f
=1,f
3
4
3
4
21
3
21
2
3
10.
21
c=2+
,c=2 3
21
21
2+
,2 3
3
c= 10
11.
10
(copy B)
=2
1
2
15. a
0.5
=2
c=2
4. x=
1
5
5. z= 6,z=0
6. x=0,x=
PAGE 2
3 ,x=3
2
12. 0
21
3
16. e
17. no
18. b,c