Chapter 4
1.
Estimate the inflection points of
()fx
if the following graph is the graph of
A.
()fx
B.
‘( )fx
C.
‘‘( )fx
List inflection points in increasing order of x-coordinates; separate each point with a
comma.
Chapter 4
2.
The following figure is a graph of a derivative function,
. Indicate on the graph the
x-values that are critical points and label each as a local maximum, a local minimum, or
neither.
3.
For which interval(s) is the function
3
( ) 27f x x x=−
decreasing?
A)
x
-9 or 9
x
B)
-9
x
9
C)
-9
x
0
D)
0
x
9
E)
–10 < x < 10
Chapter 4
4.
Sketch a graph of a function such that
( ) 0fx
=
at x = –1,
()fx
> 0 when x< –1,
()fx
Ans:
answers vary
> 0 when x > –1.
5.
Find all of the critical points of
32
( ) –2 3 12f x x x x= + +
. List them from smallest to
largest, separated by commas.
Ans:
derivative. difficulty: medium section: 4.1
6.
Find a value of a such that the function
2
() ax
f x x e=
has a critical point at
–1x=
.
-5
-4
-3
-2
-1
1
2
3
4
5
-5 -4 -3 -2 -1 12345
x
y
Ans:
2
derivative. difficulty: medium section: 4.1
Chapter 4
7.
Suppose f has a continuous derivative whose values are given in the following table.
x
-4
-3
-2
-1
0
1
2
3
4
‘( )fx
-2
-1
1
3
2
-1
-3
1
2
Is x=–2.5 a potential local minimum, local maximum, or neither?
A)
local minimum
B)
neither
C)
local maximum
Ans: A Learning Objectives: Identify local extrema of a function given
information about its derivative. difficulty: easy section: 4.1
8.
The following figure represents the function f, with
( ) sin( )
ax
f x e x
−
=
,
0a
and
0.x
A. Determine the first two positive x-intercepts of f. Separate them by a comma.
B. What must the value of a be so that
0
π
4
x=
?
C. If
0
π
4
x=
, calculate
1
x
.
Learning Objectives: Use the derivative to obtain useful information about a graph or
function. difficulty: medium section: 4.1
Chapter 4
9.
A brick is heated in an oven and taken out to cool off after a certain time. The
temperature T of the brick at any time t is given by
2
( 1)
103 t
Te
−−
=
for
0t
, with T in
degrees Celsius and t in minutes. What is the temperature of the brick when it is placed
in the oven (to the nearest degree)?
10.
A stone is thrown vertically upward so that its height, measured in feet, after t seconds is
given by
2
( ) 88 16s t t t=−
. What is the maximum height of the stone (in feet)?
11.
The following is a graph of
()fx
. Which of the following statements about
‘( )fx
are
true?
A)
‘( )fx
changes sign at
1
x
,
3
x
, and
5
x
B)
‘( )fx
changes sign at
2
x
and
4
x
C)
‘( )fx
has a local maximum or minimum at
1
x
,
3
x
, and
5
x
D)
‘( )fx
has a local maximum or minimum at
2
x
and
4
x
Chapter 4
12.
Given the curve
32
y ax bx cx d= + + +
, with
0a
, find the relation between the
parameters a, b, and c that will ensure that the curve has no turning points.
minima, or neither using the first and second derivative tests. difficulty: easy
section: 4.1
13.
The graph of the function
32
2 6 18 130y x x x= + − −
is:
A. increasing and concave up on what interval?
B. increasing and concave down on what interval?
C. decreasing and concave upon what interval?
D. decreasing and concave down on what interval?
Learning Objectives: Find inflection points given a formula or a graph and determine
concavity using the second derivative. difficulty: medium section: 4.2
14.
Assume that the polynomial f has exactly one local maximum, two local minima, and
two inflection points. What is the largets number of zeros f could have?
Ans:
4
Learning Objectives: Find inflection points given a formula or a graph and determine
concavity using the second derivative. difficulty: easy section: 4.2
Chapter 4
15.
Sketch the curve
8(1 )
t
ye
−
=−
.
16.
If
( ) cos
x
f x e x
−
=
for
02πx
, what is
‘‘( )fx
?
A)
2 sin
x
ex
−
B)
2 sin
x
ex
−
C)
cos sin
xx
e x e x
−−
+
D)
cos sin
xx
e x e x
−−
−−
Chapter 4
17.
If
( ) cos
x
f x e x
−
=
for
02πx
, which of the following are local and/or global
extrema of
()fx
?
A)
0
B)
π
2
C)
3π
4
D)
5π
4
E)
7π
4
F)
π
G)
3π
2
section: 4.2
18.
Which of the following is a minimum point? (select all that apply)
DB
A
C
E
-5
-4
-3
-2
-1
1
2
3
4
5
-5 -4 -3 -2 -1 1 2 3 4 5
x
y
Ans:
A
Chapter 4
19.
If
( ) cos
x
f x e x
−
=
for
02πx
, which of the following are inflection points of
()fx
?
A)
0
B)
π
4
C)
3π
4
D)
π
E)
7π
4
determine concavity using the second derivative. difficulty: easy section: 4.2
20.
()fx
is a function with local minima at x=–2 and x=7, with x=–2 also a global
minimum; a local maximum at x=2 but no global maximum; and inflection points at
x=1 and x=4. Could
()fx
be of the form
Ans:
no
Chapter 4
21.
The point x=1 on the following closed graph corresponds to
A)
a local minimum
B)
a local maximum
C)
neither a maximum nor a minimum
D)
a local and global minimum
E)
a local and global maximum
Ans: A Learning Objectives: Find global extrema of a continuous function defined
on a closed interval. difficulty: easy section: 4.3
22.
For
2
2
() x
f x x e=+
and
–0.5 1x
, find the value(s) of x for which
A
()fx
has a local minimum.
B
()fx
has a local maximum.
C.
()fx
has a global minimum.
D.
()fx
has a global maximum.
List the value(s) from smallest to largest, separated by commas.
Learning Objectives: Find global extrema of a continuous function defined on a closed
interval. difficulty: easy section: 4.3
Chapter 4
23.
The distance, s, traveled by a runner in a 20 mile race is given in the following figure,
where time, t, is in hours. At which of the following values of t is the runner’s speed is
the slowest?
A)
4
B)
3
C)
1
D)
2
on a closed interval. difficulty: medium section: 4.3
24.
For
2
( ) 4cos sinf x x x=−
and
0πx
, which of the following statements are true?
A)
()fx
has global maxima at
0x=
and
πx=
B)
()fx
has a global minima at
0x=
and
πx=
C)
()fx
has a global maximum at
π
2
x=
D)
()fx
has a global minimum at
π
2
x=
defined on a closed interval. difficulty: easy section: 4.3
Chapter 4
25.
Consider the following graph of a function. Assume the entire graph is shown. How
many local minima does the function have?
Ans:
2
interval. difficulty: easy section: 4.3
26.
The quantity of a medication in the bloodstream t hours after it is ingested is given, in
mg, by
( ) 300 t
q t te−
=
. What is the maximum quantity of the medication in the
bloodstream?
A)
110 mg
B)
300 mg
C)
815 mg
D)
150 mg
Ans: A Learning Objectives: Find global extrema of a continuous function defined
on an open interval or on the entire real line. difficulty: medium section: 4.3
27.
Daily production levels in a plant can be modeled by the function
2
( ) –2 12 15G t t t= + +
, which gives units produced at t, the number of hours since the factory opened at 8 am.
Factory productivity is at a maximum at _____ am.
Ans:
11
Learning Objectives: Find global extrema of a continuous function defined on a closed
interval. difficulty: medium section: 4.3
28.
The number of plants in a terrarium is given by the function
2
( ) –1.2 4 12P c c c= + +
,
where c is the number of mg of plant food added to the terrarium. The amount of plant
food that produces the highest number of plants is _____ mg (round to the nearest
hundredth).
Ans:
1.67
Learning Objectives: Find global extrema of a continuous function defined on an open
interval or on the entire real line. difficulty: medium section: 4.3
Chapter 4
Page 13
29.
The function
2
0.3( 2) 3 3y x x= + − +
gives the population of a town (in 1000’s of
people) at time x where x is the number of years since 1980. The population was a
minimum in the year _____.
30.
A normal distribution in statistics is modeled by the function
2
2
1
() 2
x
N x e
−
=
.
Determine where the maximum value of the function would occur.
A)
–3
B)
–2
C)
0
D)
–3
E)
e–3
Ans: C Learning Objectives: Find global extrema of a continuous function defined
on an open interval or on the entire real line. difficulty: hard section: 4.3
31.
In the function y = 3sin (x) + 5, in the interval from 0
x
, at which value(s) of x
does the function contain a global maximum?
A)
and
56
B)
0 and
3
C)
2
only
D)
4
E)
2
Ans: C Learning Objectives: Find global extrema of a continuous function defined
on a closed interval. difficulty: easy section: 4.3
Ans:
1983
Learning Objectives: Find global extrema of a continuous function defined on an open
interval or on the entire real line. difficulty: medium section: 4.3
Chapter 4
32.
The following table shows cost and revenue for a product (in dollars).
A. What is the price of the product?
B. At what value of q is profit is maximized?
q
0
1000
2000
3000
4000
5000
R(q)
0
500
1000
1500
2000
2500
C(q)
100
250
400
800
1400
2000
Part A:
A. $0.50
Part B:
B. 3000
Learning Objectives: Find maximum profit. difficulty: easy section: 4.4
33.
With x people aboard, a South African airline makes a profit of (1200-4x) rands per
person for a specific flight. What is the maximum number of passengers that can board
such that the airline still profits?
Ans:
299
Learning Objectives: Find maximum profit. difficulty: medium section: 4.4
34.
Given the following table of production quantities with their corresponding marginal
revenue and marginal cost, estimate the production level that maximizes profit.
q
0
10
20
30
40
50
MR
100
100
100
100
100
100
MC
40
75
100
120
150
190
Ans:
20
Learning Objectives: Find maximum profit. difficulty: easy section: 4.4
35.
If the total revenue and total cost (in dollars) are given by
2
( ) 5 0.003R x x x=−
( ) 300 0.9C x x=+
?
What quantity of gadgets, to the nearest whole number should be produced to maximize
profit? What is the maximum profit?
Learning Objectives: Find maximum profit. difficulty: medium section: 4.4
36.
What quantity of gadgets should be produced to maximize profit if they sell for $500
per unit and the total cost (in dollars) of producing x units is given by
2
( ) 8,500 2C x x=+
? What is that profit?
Ans:
125,$Ans2
Chapter 4
37.
Total cost and revenue are approximated by the functions
1200 3.9Cq=+
and
5Rq=
,
both in dollars.
A. What is the fixed cost?
B. What is the profit function?
Part A:
$1200.00
Learning Objectives: Find maximum profit. difficulty: medium section: 4.4
38.
Write a formula for total cost, C, as a function of quantity r when fixed costs are
$45,000 and and variable costs are $2,000 per item.
Learning Objectives: Find maximum profit. difficulty: easy section: 4.4
39.
The revenue for selling q items is
2
( ) 500 3R q q q=−
and the total cost is
( ) 100 50C q q=+
.
A. Write a function that gives total profit earned.
B. Find the quantity that maximizes profit.
A.
B. 75
Learning Objectives: Find maximum profit. difficulty: medium section: 4.4
40.
The function
2
( ) 15 50C r r=−
gives cost in dollars of producing r items. What is the
marginal cost of increasing r by 1 item from the current production level of r=3?
Ans:
$90
Learning Objectives: Find maximum revenue using the demand equation.
difficulty: medium section: 4.4
41.
A. Find the marginal cost for q=110 when the fixed costs in dollars are 2,000, the
variable costs are 100 per item, and each sells for $400.
B. Find the marginal revenue under the same conditions.
Ans:
A. $100
B. $400
Chapter 4
Page 16
42.
Let C(q) represent the cost, R(q), the revenue and π(q) the profit, in dollars of producing
q items. If
(89) 83C=
and
(89) 77R=
approximately, how much profit is earned by
the
90th
item?
43.
The total revenue, R, in dollars,when selling q items is
2
( ) ln(1 1000 )R q q=+
.
Calculate and interpret the marginal revenue if q= 10.
approximately $0.20 in additional revenue.
difficulty: medium section: 4.4
44.
The total cost, C, in dollars,when producing q items is
52
( ) 0.003( 7) 120C q q q= − + +
.
Calculate and interpret the marginal cost if q= 6.
approximately $0.01 in additional cost.
difficulty: medium section: 4.4
45.
The cost of producing q items is
( ) 1000 8C q q=+
dollars. What is the average cost of
producing the 20th item?
Ans:
$58.00
section: 4.5
46.
A factory produces a product that sells for $12. They currently produce 2200 items per
month, at an average cost of $3 per item. The marginal cost at this level is $2.
Assume that the factory can sell all the items that it produces.
A. What is the profit at this production level?
B. Would increasing production increase or decrease profit?
Part A:
A. $19,800_
Part B:
B. increase
section: 4.5
Ans:
$–6
difficulty: medium section: 4.4
Chapter 4
47.
The graph of a cost function is given in the following figure. Estimate the value of q at
which average cost is minimized.
Ans:
30
Learning Objectives: Find minimum average cost. difficulty: easy
section: 4.5
48.
400 items are produced at an average cost of $100 per item. Find the cost of producing
the 401st item if the marginal cost to produce the 401st item is $110
Ans:
$100.02
Learning Objectives: Find minimum average cost. difficulty: medium
section: 4.5
49.
You sell hot dogs at a baseball game for $2.75 each. You have 100 hot dogs to sell at
an average cost to you of $1.75 each. The marginal cost at q=100 is $1.90. Assume
you can always sell out of hot dogs. Will increasing the number of hot dogs you have
to sell increase or decrease the average cost?
A)
increase
B)
decrease
Ans: A Learning Objectives: Find minimum average cost. difficulty: hard
section: 4.5
50.
You sell hot dogs at a baseball game for $3.25 each. You have 100 hot dogs to sell at
an average cost to you of $2.00 each. The marginal cost at q=100 is $1.95. Assume
you can always sell out of hot dogs. Will increasing the number of hotdogs you have to
sell increase or decrease your profit?
A)
increase
B)
decrease
Ans: A Learning Objectives: Find minimum average cost. difficulty: hard
section: 4.5
Chapter 4
Page 18
51.
Which average cost function corresponds to the total cost function shown in the
following figure?
A)
B)
C)