80)
Find all the critical values of y=x3–x2–x+ 2.
80)
81)
The revenue equation for a company is given by R(x) = 1296x– 0.12x3. Determine when
relative extrema occur on the interval (0, ).
81)
82)
Suppose that the total number of units produced by a worker in t hours of an 8–hour shift
can be modeled by the production function P(t) = 19t+21
2t2–1
2t3. Find where the
function is concave up and where the function is concave down.
82)
83)
If y= 3x4– 6x2, use the second–derivative test to find all values of x for which (a) relative
maxima occur (b) relative minima occur.
83)
84)
If f(x) =2x – 3
x– 4 , determine the intervals on which f is increasing and the intervals on which
f is decreasing.
84)
85)
Find the intervals where the graph of y=f(x)is concave up and where it is concave down.
Do not sketch its graph. Also determine its points of inflection. f(x) =x4
2–x3+ 5.
85)
86)
Let y=x4– 4x3.
(a) Determine y’ and y”.
(b) Determine intervals on which the function is increasing; determine intervals on which
the function is decreasing.
(c) Determine the coordinates of all relative maximum and relative minimum points.
(d) Determine intervals on which the function is concave up; determine intervals on
which the function is concave down;
(e) Determine the coordinates of all inflection points.
(f) With the aid of the information obtained in parts (a)–(e), give a reasonable sketch of
the curve.
86)
87)
Use your graphing calculator to estimate the relative extrema of the function
y=x5+x4–x3+x2+x(estimate to 3 decimal places).
87)
88)
Determine the equations of the vertical asymptotes and non–vertical asymptotes for the
graph of y=x2– 4
4x2– 25 .
88)
89)
Sensitivity to a drug depends on the dosage size x according to the equation s= 50x–x2.
Find the dosage that maximizes sensitivity.
89)
90)
Using fencing, a rectangular area is to be enclosed next to the wall of a building. The
building will form one of the sides and no fencing will be placed there. The cost of the
fencing for the side opposite the building is $5 per foot. The cost of the fencing for the other
two sides is $3 per foot. The area to be enclosed is 3000 ft2. In order to keep expenses of the
three sides to a minimum, how many feet of each type of fencing must be bought?
90)
91)
The profit equation for a taco stand is given by P(x) = – 0.4x2+ 100x–100, where x is the
number of tacos sold, and P(x) is the profit in dollars. Determine where the graph of the
equation is concave up and where it is concave down and find any inflection points.
91)
23
92)
Let y=x2
x– 1
(a) Determine all x– and y– intercepts.
(b) Determine equations of all vertical and non–vertical asymptotes.
(c) Determine y’ and y’’.
(d) Determine intervals on which the function is increasing; determine intervals on which
the function is decreasing.
(e) Determine the coordinates of all relative maximum and relative minimum points.
(f) Determine intervals on which the function is concave up; determine intervals on which
the function is concave down.
(g) Determine the coordinates of all inflection points.
(h) With the aid of the information obtained in parts (a) — (g), give a reasonable sketch of
the curve.
92)
93)
Determine the equations of the vertical asymptotes and non–vertical asymptotes for the
graph of y=x2– 4
2.
93)
94)
Let f(x) =x
x– 1 . Find the x values of (a) all relative maxima, (b) all relative minima, and (c)
all inflection points.
94)
95)
If f(x) =x3–6x2– 15x– 7, determine the intervals on which f is increasing and the
intervals on which f is decreasing.
95)
96)
Given y=f(x) = 4x2– 4x+ 5 on the interval 0, 4 , find the values of x at which absolute
maxima and absolute minima occur.
96)
97)
The cost equation for a cookie store is given by C(x) =x3– 6x2+ 250, where x is the
number of cookies made (in dozens), and C(x) is the cost in dollars. Determine where the
graph of the equation is concave up and where it is concave down and find any inflection
points.
97)
98)
If C= – 3x– 2x2+x3 is a cost function, sketch the graph of this function with the aid of
intercepts, symmetry, and the first–derivative test.
98)
99)
It is estimated that under a short–term energy–assistance plan, n million households will
receive direct aid in meeting home–fuel costs after t years, where
n= 2t3– 15t2+ 36t, 0 t
4. For what value of t will n be a maximum?
99)
100)
Find two numbers whose sum is 30 and whose product is a maximum.
100)
101)
A rectangular plot of ground is to be fenced in so that it has an area of 15,000 square feet.
The plot is to be divided in half with a fence parallel to one pair of sides. Find the
dimensions of the plot if the least amount of fencing is to be used.
101)
102)
Use your graphing calculator to graph f(x) =1
20 x5+1
6x4–1
4x. Looking at the graph,
estimate the intervals where the function is concave up and concave down. Now find these
intervals using the second derivative.
102)
103)
Let f(x) =5
x– 2 . Determine the intervals of which f is (a) concave up and (b) concave down.
(c) Find the x–values of all inflection points.
103)
104)
Find the vertical asymptotes of the following function. Do not sketch its graph.
y=x2+ 3x– 1
(2x+ 1)ex
104)
105)
Find the absolute extrema for y=x2
ex on the interval 0, 3 and where they occur.
105)
106)
The cost equation for a company is C(x) = 2x3– 39x2+ 180x+ 21,200. Use the
second–derivative test, if applicable, to find the relative maxima and the relative minima.
106)
107)
If f(x) =x2
x– 3 , determine the intervals on which f is increasing and the intervals on which f
is decreasing.
107)
108)
Use your graphing calculator to graph y=x3– 6x2+ 9x– 2. Inspect the graph to estimate
where the relative extrema occur. Then use the first–derivative test to find the relative
extrema.
108)
109)
Find the intervals where the graph of y=f(x) is concave up and where it is concave down.
Do not sketch its graph. Also determine its points of inflection. f(x) =x2
ex
109)
110)
A cable TV company has 200 customers paying $10 each month. For each $1 reduction in
price, the company attracts 50 more customers. Find the price that yields maximum
revenue.
110)
111)
Let f(x) = 3x5– 10x4+ 7x. Determine the intervals on which f is (a) concave up and (b)
concave down.
111)
112)
Find the vertical and non–vertical asymptotes of the following function. Do not sketch its
graph.
y=x2+ 5x– 4
x– 5
112)
113)
Find the vertical and non–vertical asymptotes of the following function. Do not sketch its
graph.
y=3x3
x2– 5
113)
114)
Determine the equations of the vertical asymptotes and non–vertical asymptotes for the
graph of y=x– 2
x2+ 5x.
114)
116)
Let f(x) =x
x– 2 .
(a) Determine the intervals on which f is increasing.
(b) Determine the intervals on which f is decreasing.
(c) Based on your answers to parts (a) and (b), find the values of x for which f has relative
maxima.
(d) Based on your answers to parts (a) and (b), find the values of x for which f has relative
minima.
116)
117)
The cost equation for a company is C(x) = 3x3– 27x2+ 45x+ 100. Use the
117)
118)
If y=x3+ 4x2– 3x+ 4, use the second–derivative test to find all values of x for which (a)
relative maxima occur (b) relative minima occur.
118)
119)
Find all the critical values of f(x) =2x– 1.
119)
120)
If f(x) = (x+ 5)3, determine the intervals on which f is increasing and the intervals on which
f is decreasing.
120)
121)
Determine the equations of the vertical asymptotes and non–vertical asymptotes for the
graph of y=4x5– 2x+ 3
x2– 2x – 8 .
121)
115)
115)
122)
Use your graphing calculator to estimate the relative extrema of the function y=x7–x5–
x3 (estimate to 3 decimal places).
122)
123)
Determine the intervals on which the function is increasing and on which it is decreasing.
Also determine the points of relative maxima and relative minima.
y=x3+x2– 3x+ 7
123)
124)
The demand equation for a monopolist’s product is p= 2700 –q2, where p is the price per
unit (in dollars) when q units are demanded.
(a) Find the value of q for which revenue is maximum.
(b) What is the maximum revenue?
124)
125)
A manufacturer has to produce annually 360 units of a product that are sold at a uniform
rate during the year. The production cost for each unit is $200, and carrying costs
(insurance, interest, storage, and so on) are estimated to be 10% of the value of average
inventory. Set–up costs per production run are $100. Find the economic lot size.
125)
126)
A released film has a revenue given by R(t) =20t
t2+ 1 , where R(t) is in millions of dollars
and t is the number of weeks after its release. Sketch the graph of this function.
126)
127)
Let f(x) =x2
x– 4 .
(a) Determine the intervals on which f is increasing.
(b) Determine the intervals on which f is decreasing.
(c) Based on your answers to parts (a) and (b), find the values of x for which f has relative
maxima.
(d) Based on your answers to parts (a) and (b), find the values of x for which f has relative
minima.
127)
128)
Suppose that the total number of units produced by a worker in t hours of an 8–hour shift
can be modeled by the production function P(t) = 22t+ 15t2–t3. Find where the function is
concave up and where the function is concave down.
128)
129)
Let f(x) =x(x–3)5.
(a) Determine the intervals on which f is increasing.
(b) Determine the intervals on which f is decreasing.
(c) Based on your answers to parts (a) and (b), find the values of x for which f has relative
maxima.
(d) Based on your answers to parts (a) and (b), find the values of x for which f has relative
minima.
129)
130)
A manufacturer has to produce annually 1000 units of a product that are sold at a uniform
rate during the year. The production cost for each unit is $400, and carrying costs
(insurance, interest, storage, and so on) are estimated to be 5% of the value of average
inventory. Set–up costs per production run are $64. Find the economic lot size.
130)
131)
A company owns an apartment building containing 80 units. If the company charges $300
per month rent, then all units can be rented out, and for every increase of $10 per month in
rent, the company will lose one customer. What rent should be charged to maximize
revenue?
131)
132)
Find the intervals where the graph of y=f(x) is concave up and where it is concave down.
Do not sketch its graph. Also determine its points of inflection. f(x) =x3+x2– 3x + 7.
132)
133)
A rectangular closed box with a square base is to have a volume of 1000 in3. Find its
dimensions if the box is to require the least possible material.
133)
134)
A company owns an apartment building containing 100 units. If the company charges $400
per month rent, then all units can be rented out, and for every increase of $20 per month in
rent, the company will lose one customer. What rent should be charged to maximize
revenue?
134)
135)
Sketch the graph of y =x3– 4x with the aid of intercepts, symmetry, and the
first–derivative test. Use your Graphing Calculator to verify your solution.
135)
136)
Use your graphing calculator to find the absolute extrema of f(x) =x3+x2– 2x– 2 +x+ 4
on –3, 1 (round answers to 3 decimal places).
136)
137)
Use the second derivative test to find the points of relative maxima and relative minima for
the function y=x4
2– 2x3+ 5.
137)
138)
Suppose that the total number of units produced by a worker in t hours of an 8–hour shift
can be modeled by the production function P(t) = 20t+ 15t2–5
6t3. Find where the function
is concave up and where the function is concave down.
138)
139)
Use your graphing calculator to graph y= ln(x) +1
x. Inspect the graph to estimate where
the relative extrema occur. Then use the first–derivative test to find the relative extrema.
139)
140)
Use the second derivative test to find the relative extremas of f(x) = – 4x5 and where they
occur. If the relative extremas can not be determined by the second derivative test, state so.
140)
141)
Determine the intervals on which the function is increasing and on which it is decreasing.
Also determine the points of relative maxima and relative minima.
f(x) = (x2– 3x +2)2
141)
142)
Determine the equations of the vertical asymptotes and non–vertical asymptotes for the
graph of y=6
x2+ 4 .
142)
143)
Find all the critical values of f(x) =x ln x.
143)
144)
Use your graphing calculator to graph f(x) =1
3x3–x2+ 2. Looking at the graph, estimate
the intervals where the function is concave up and concave down. Now find these intervals
using the second derivative.
144)
145)
Sketch the graph of y=ex– 7x with the aid of intercepts, symmetry, and the
first–derivative test. Use your graphing calculator to verify your solution.
145)
146)
The revenue equation for a company is given by R(x) = 84.375x– 0.5x3. Determine when
relative extrema occur on the interval (0, ).
146)
147)
The demand equation for a monopolist’s product is p=10,000
q2+ 25 , where p is the price per
unit (in dollars) when q units are demanded.
(a) Determine the value of q for which revenue is maximum.
(b) What is the maximum revenue?
147)
148)
Find two numbers whose sum is 50 and such that the product of one of them and five more
than the other is a maximum.
148)
149)
Use your graphing calculator to find the absolute extrema of f(x) =x3– 4x2+ 3x+ 2 on
0, 4 (round answers to 3 decimal places).
149)
150)
A manufacturer found that the total cost c of producing q units of a product is given by
c= 0.02q2+ 2q+ 800. At what level of production will average cost be a minimum?
150)
151)
The cost equation for a cookie store is given by C(x) =x3– 6x2+ 250, where x is the
number of cookies made (in dozens) and C(x) is the cost in dollars. Use the first–derivative
test to find when relative extrema occur.
151)
152)
A cable TV company has 500 customers paying $20 each month. For each $1 reduction in
price, the company attracts 50 more customers. Find the price that yields maximum
revenue.
152)
153)
Determine the intervals on which the function is increasing and on which it is decreasing.
Also determine the points of relative maxima and relative minima.
f(x) =x2
ex
153)
154)
If f(x) =x4
4–2x3
3, determine the intervals on which f is increasing and the intervals on
which f is decreasing.
154)
155)
If y=x(x–1)3, find the x–values of all inflection points.
155)
156)
Given y=f(x) = 2x3+12x2– 7 on the interval 0, 4 , find the values of x at which absolute
maxima and absolute minima occur.
156)
34