50)
Is this the graph of the function f(x) =x3+ 3x2– 45x + 10?
Enter your answer as just the word “yes” or “no” (lower case).
50)
51)
Determine the inflection point of f(x) = 2x3– 9x2+ 12x – 1.
Enter your answer exactly as just an ordered pair of fractions: a
b, c
d
51)
52)
Is this the graph of f(x) = 9x + 1 +1
x, x > 0?
Enter just “yes” or “no” (lower case).
52)
21
53)
Is this the graph of a function having the following properties?
(I) defined for x –1
(II) horizontal asymptote at y = 3
(III) increasing for all x –1
Enter just the word “yes” or the word “no” (lower case).
53)
54)
Is this the graph of f(x) =1
x + 2 ?
Enter just the word “yes” or “no” (lower case).
54)
55)
Is this the graph of a function having the following properties?
(I) f'(x) > 0 for all x
(II) f”(x) > 0 for all x < 0, f”(x) < 0 for x >0
(III) asymptotes at y =
2, y = –
2
Enter your answer as just “yes” or “no” (lower case).
55)
56)
Determine the values of x for which f(x) =x3– 6x is concave down.
Enter your answer as just an interval in standard interval notation.
56)
57)
Determine the maximum and minimum values of f(x) = 1 –x3+ 3x + 2 on 0 x 3 .
Enter your answer exactly as: a,b both integers where a is the minimum of f and b is the
maximum of f.
57)
58)
An airline flies 120,000 passengers per week to Florida when charging $100 per flight. It
estimates that for each $1 increase in price it will lose 400 passengers. By how much
should the fare be increased (or decreased) to maximize total revenue? Enter your answer
as just an integer (no symbols or words).
58)
59)
Determine the relative minimum point of f(x) =x2+ 4x + 5.
Enter your answer exactly as just an ordered pair of integers: (a, b)
59)
60)
Determine the minimum value of f(x) =x +1
x – 1 on x > 1.
Enter just an integer.
60)
23
61)
Points A, B, and C lie on the graph of a function f(x), as shown on the diagram. What are
the signs of f(x), f'(x), and f”(x) at the point C? Enter your answer as just “neg”, “pos”, or 0
in the order given above separated by commas.
61)
62)
Compute the maximum profit when the demand function is p(x) =x2– 3x + 2 and the total
cost function is C(x) =2x3
3–1
2x2– 2x.
Enter just a reduced fraction of form a
b.
62)
63)
Determine the minimum value of f(x) =x3– 3x2+ 2 on 1
x 3.
Enter just an integer.
63)
64)
Is this the graph of a function having the following properties?
(I) x–intercept at x = – 2
(II) absolute maximum at x = – 1
(III) relative maximum at x = 1
(IV) concave up for x
2
Enter just the word “yes” or the word “no” (lower case).
64)
65)
Is this the graph of f(x) =1
4x4–x3+ 2?
Enter just the word “yes” or “no” (lower case).
65)
66)
If a function has second derivative f ”(x) =x2(x2– 4) is (0, 0) an inflection point?
Enter just the word “yes” or “no” (lower case).
66)
67)
Is this the graph of the function having the following properties?
(I) inflection point at x = 3
(II) no relative maximum point
(III) asymptote at x = 5
Enter just the word “yes” or the word “no” (lower case).
67)
68)
Determine the maximum and minimum values of f(x) =x3– 3x2– 9x on –4 x 4 .
Enter your answer exactly as: a,b both integers where a is the minimum of f and b is the
maximum of f.
68)
69)
A homebuilder‘s advertisement promises a house with a finished recreation room of 300
square feet. Two perpendicular walls of the room are to be paneled at a cost of $5 per
running foot. A third side will be built out of windows at a cost of $10 per running foot.
What dimensions should the room have to minimize the homebuilder’s cost? Enter your
answer as: length of side using cinder, length of other side.
69)
70)
A health food store stocks bottles of multivitamins. It orders equal quantities of stock from
its wholesaler at equally spaced points throughout the year. The cost of replacing each
order is $250. Moreover, the cost of keeping a jar of vitamins in inventory is $1 per year.
The store predicts that it will sell 12,500 bottles of vitamins in the next year. How many
orders of how many bottles each will result in a minimum cost to the health food store?
Enter your answer exactly as: a, b (integers) where a represents the number of orders and
b represents the number of bottles in each order (no units or words).
70)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
71)
At which labeled point(s) is the graph concave up?
71)
A)
C, E
B)
C, D
C)
C, D, E
D)
D
Solve the problem.
72)
A certain company produces potting soil and sells it in 50 lb bags. Suppose that 100,000 bags are to
be produced each year. It costs $4 per year to store a bag of potting soil, and it costs $2000 to set up
the facility to produce a batch of bags. Find the number of bags per batch that should be produced.
72)
A)
100,000
B)
14,121
C)
10,000
D)
9574
Graph the function by first finding the relative extrema.
26
73)
f(x) =x2–6x +9
73)
A)
B)
C)
D)
27
74)
Which of the following is (are) true of f(x) = 5 + 3x2–x3?
(I) (1, 7) is a point of inflection
(II) f(2) is a relative maximum point
(III) f has a relative minimum point at x = 0
(IV) f is increasing on (2, )
74)
A)
II, III, and IV
B)
I, II, and III
C)
II and III
D)
all of these
E)
none of these
Find the relative extrema of the function, if they exist.
75)
f(x) =2
3x3+1
2x2–21x + 2
75)
A)
Relative maximum at 3, –77
2
B)
Relative maximum at –7
2, 1273
24 ; relative minimum at 7
2, –883
24
C)
Relative maximum at –7
2, 1273
24 ; relative minimum at 3, –77
2
D)
Relative maximum at –3, 103
2; relative minimum at 7
2, –883
24
Find the points of inflection.
76)
f(x) =4
3x3– 12x2+ 10x +46
76)
A)
(3, 4)
B)
(0, 4)
C)
(3, 0)
D)
(3, –26)
Solve the problem.
77)
The cost function for a manufacturer is C(x) dollars, where x is the number of units of goods
produced. The graph below shows C(x) and C(x).
60
y =C(x)
40
20
20 40 60 80 100
y =C(x)
What is the marginal cost when 90 units of good are produced?
77)
A)
$70 per unit
B)
$25 per unit
C)
$35 per unit
D)
$45 per unit
78)
Find the x coordinates of all relative extreme points of f(x) =1
4x4+2
3x3–3
2x2+ 4
78)
A)
x = – 1, 0, 3
B)
x = 0
C)
x = – 3, 0, 1
D)
x = – 3, 1
E)
x = – 1, 3
Graph the function.
79)
f(x) =x3–3x2
79)
29
A)
B)
C)
D)
Graph the function by first finding the relative extrema.
80)
f(x) =x3+2x2– x –2
80)
30
A)
B)
C)
D)
81)
Which of the following is (are) true of f(x) = 3x2/3 – 2x?
(I) f is decreasing on (1, )
(II) (1, f(1)) is a relative minimum point
(III) f is concave down everywhere
(IV) (0, 0) is an inflection point
81)
A)
I, II, and III
B)
II, III, and IV
C)
I, III, and IV
D)
all of these
E)
I and III
Solve the problem.
82)
The graph on the left shows the population (in millions) of a colony of bacteria after t hours as
given by the function f(t). The graph on the right shows f(t).
100 y = f(t)
60
20
2 4 6 8
12
y =f(t)
8
4
2 4 6 8
How fast was the population growing after 2 hours?
82)
A)
12 million per hour
B)
10 million per hour
C)
14 million per hour
D)
16 million per hour
Graph the function by first finding the relative extrema.
83)
f(x) =x3–1
83)
A)
B)
32
C)
D)
Sketch the graph and show all extrema, inflection points, and asymptotes where applicable.
84)
f(x) =6x2+ 60x
84)
A)
Rel min: (–10, –60)
No inflection points
B)
Rel min: (5, –150)
No inflection points
33
C)
Rel min: (–5, –150)
No inflection points
D)
Rel min: (10, –60)
No inflection points
Solve the problem.
85)
Find the maximum profit P if C(x) =80 + 8x and p =36 – 2x.
85)
A)
$902
B)
$98
C)
$982
D)
$18
86)
Suppose a ball is thrown into the air and after t seconds has a height of h(t) = – 16t2+ 80t feet.
When will it reach its maximum height?
86)
A)
300 sec
B)
5 sec
C)
2.5 sec
D)
0.5 sec
E)
none of these
34
87)
A health club offers memberships at the rate of $300, provided that at least 50 people join. For each
member in excess of 50, the membership fee will be reduced by $2 for all members. Due to space
limitations, at most 125 memberships will be sold. How many memberships should the club sell in
order to maximize its revenue?
87)
A)
75
B)
50
C)
100
D)
125
E)
none of these
Solve the problem.
88)
The cost of a computer system increases with increased processor speeds. The cost C of a system as
a function of processor speed is estimated as C =13S2– 6S + 1800, where S is the processor speed in
MHz. Find the processor speed for which cost is at a minimum. Round to the nearest tenth if
necessary.
88)
A)
0.3 MHz
B)
0.2 MHz
C)
4.6 MHz
D)
1.8 MHz
89)
Suppose c(x) =x3–22x2+10,000x is the cost of manufacturing x items. Find a production level that
will minimize the average cost of making x items.
89)
A)
12 items
B)
13 items
C)
11 items
D)
10 items
90)
Find the x coordinates of all relative extreme points of f(x) =1
2x4–2
3x3– 6x2– 100
90)
A)
x = – 3, 2
B)
x = – 2, 3
C)
x = – 2, 0, 3
D)
x = 0
E)
x = – 3, 0, 2
91)
Which labeled point has the most positive slope?
91)
A)
B
B)
A
C)
F
D)
E
Solve the problem.
92)
A local office supply store has an annual demand for 40,000 cases of photocopier paper per year. It
costs $3 per year to store a case of photocopier paper, and it costs $90 to place an order. Find the
optimum number of cases of photocopier paper per order.
92)
A)
2,400,000
B)
490
C)
1095
D)
1549
Find the relative extrema of the function, if they exist.
93)
s(x) = – x2–6x – 5
93)
A)
Relative maximum at (–6, –5)
B)
Relative maximum at (3, 4)
C)
Relative minimum at (6, –5)
D)
Relative maximum at (–3, 4)
94)
Find the relative minimum point(s) of f(x) =x4
4–x3– 5x2–10.
94)
A)
(0, f(0))
B)
(–2, f(–2)) and (0, f(0))
C)
(2, f(2)) and (–5, f(–5))
D)
(–2, f(–2)) and (5, f(5))
E)
none of these
E)
Solve the problem.
95)
Find the dimensions that produce the maximum floor area for a one–story house that is rectangular
in shape and has a perimeter of 127 ft. Round to the nearest hundredth, if necessary.
95)
A)
31.75 ft x 31.75 ft
B)
63.5 ft x 63.5 ft
C)
10.58 ft x 31.75 ft
D)
31.75 ft x 127 ft
A
B)
96)
A farmer decides to make three identical pens with 96 feet of fence. The pens will be next to each
other sharing a fence and will be up against a barn. The barn side needs no fence.
What dimensions for the total enclosure (rectangle including all pens) will make the area as large as
possible?
96)
A)
12 ft by 48 ft
B)
24 ft by 24 ft
C)
12 ft by 12 ft
D)
16 ft by 80 ft
A
B)
D
B)
97)
Which of the following is (are) true of f(x) =x4–x3?
(I) (0, 0) is an inflection point
(II) 1
2, –1
16 is an inflection point
(III) (0, f(0)) is a relative minimum point
(IV) f is increasing on (0, 2)
97)
A)
I, II, and III
B)
I, III, and IV
C)
I and II
D)
I, II, and IV
E)
all of these
98)
Suppose that 20,000 fans will go to a ball game when the price of a ticket is $5.00, and that 500
fewer fans will go for each $1.00 increase in ticket price. By how much should ticket prices be
increased (or decreased) in order to maximize revenue?
98)
A)
increase price by $45.00
B)
decrease price by $1.00
C)
increase price by $17.50
D)
increase price by $22.50
Solve the problem.
99)
Suppose c(x) =x3–20x2+10,000x is the cost of manufacturing x items. Find a production level that
will minimize the average cost of making x items.
99)
A)
12 items
B)
10 items
C)
9 items
D)
11 items
Find the relative extrema of the function, if they exist.
100)
f(x) = x2– 14x + 51
100)
A)
Relative minimum at (7, 2)
B)
Relative maximum at (7, 2)
C)
Relative minimum at (2, 7)
D)
Relative maximum at (2, 7)
38
101)
Let F(x) be the function graphed below. For what value is F'(x) > 0?
101)
A)
x < 0 and x > p
B)
p < x < q
C)
x > p
D)
0 < x < q
E)
none of these
102)
Find the interval(s) where f is concave down for f(x) = – 4x4+ 3x3– 5x + 10.
102)
A)
( , 0)
B)
0, 3
8
C)
, 3
8
D)
( , 0)
(3, )
E)
( , 0) 3
8,
E
Find the x–intercepts of the function.
103)
f(x) = x2+ 8x – 7
103)
A)
(–4±23, 0)
B)
(–1±23, 0)
C)
(4 +23, 0)
D)
(–4±223, 0)
A
39
C
Solve the problem.
104)
Of all numbers whose sum is 440, find the two that have the maximum product. That is, maximize
Q = xy, where x + y =440.
104)
A)
10 and 430
B)
220 and 220
C)
1 and 439
D)
219 and 221
105)
A company wishes to manufacture a box with a volume of 28 cubic feet that is open on top and is
twice as long as it is wide. Find the width of the box that can be produced using the minimum
amount of material. Round to the nearest tenth, if necessary.
105)
A)
2.7 ft
B)
6.6 ft
C)
3.3 ft
D)
5.4 ft
Find the points of inflection.
106)
f(x) =x3+7x +2
106)
A)
(0, 7)
B)
(0, 2)
C)
(2, 0)
D)
(2, 7)
Graph the function by first finding the relative extrema.
107)
h(x) =x4– 2x2+4
107)
40