13.8 Extrema of Functions of Two Variables
828
saddle point:
saddle point:
____ 7. Examine the function for relative extrema and
saddle points.
relative minimum:
relative minimum:
relative maximum:
saddle point:
saddle point:
____ 8.
Examine the function
for relative extrema and saddle
points.
a.
; relative minimum:
b.
;
relative minimum:
c.
;
relative maximum:
d.
;
relative maximum:
e.
,
829 Chapter 13: Functions of Several Variables
____ 9. Examine the function for relative extrema and saddle points.
saddle point:
relative minimum:
relative maximum:
relative minimum:
no critical points
____ 10. Determine whether there is a relative maximum, a relative minimum, a saddle point,
or insufficient information to determine the nature of the function at the critical point
, if .
relative maximum
relative minimum
saddle point
insufficient information
____ 11.
List the critical points of the function
for which the
Second Partial Test fails.
a.
Test fails at
and
.
b.
Test fails only at
.
c.
Test fails at
.
d.
Test fails at
.
e.
Test fails only at
.
13.8 Extrema of Functions of Two Variables
830
____ 12.
Find the critical points of the function
,
and from the form of the function, determine whether a relative maximum or a relative
minimum occurs at each point.
relative minimum at
relative maximum at
relative maximum at
relative minimum at
no relative extrema
____ 13. Find the critical points of the function , and
from the form of the function, determine whether a relative maximum or a relative minimum occurs
at each point.
relative maxima at , , , where are arbitrary
real numbers
relative minima at , , , where are arbitrary
real numbers
relative minimum at
relative maximum at
no relative extrema
831 Chapter 13: Functions of Several Variables
____ 14. Find the absolute extrema of on the region bounded
by the square with vertices
absolute minimum:
absolute maximum:
absolute minimum:
absolute maximum:
absolute minimum:
absolute maximum:
absolute minimum:
absolute maximum:
absolute minimum:
absolute maximum:
____ 15.
Find the absolute extrema of the function
over the triangular
region in the xy–plane with vertices
.
a.
absolute maximum:
at
absolute minimum:
at
and along the line
b.
absolute maximum:
at
absolute minimum:
at
and along the line
c.
absolute maximum:
at
absolute minimum:
at
and along the line
absolute maximum: at
absolute minimum: at
absolute maximum: at
absolute minimum: at
13.8 Extrema of Functions of Two Variables
832
____ 16.
Find the absolute extrema of
on the region
.
absolute minimum:
absolute maximum:
absolute minimum:
absolute maximum:
absolute minimum:
absolute maximum:
absolute minimum:
absolute maximum:
absolute minimum:
absolute maximum:
833 Chapter 13: Functions of Several Variables
13.8 Extrema of Functions of Two Variables
Answer Section
13.9 Applications of Extrema
834
13.9 Applications of Extrema
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the minimum distance from the point to the plane .
a.
b.
c.
d.
e.
____ 2. Find the minimum distance from the point to the surface
.
429
415
55
607
3 7
____ 3. Find three positive numbers x, y, and z whose sum is 24 and product is a maximum.
a.
b.
c.
d.
e.
835
Chapter 13: Functions of Several Variables
____
4.
Find three positive numbers x, y, and z whose sum is 30 and the sum of the squares is
a minimum.
a.
b.
c.
d.
e.
____ 5. Suppose a home improvement contractor is painting the walls and ceiling of a rectangular
room. The volume of the room is 640 cubic feet. The cost of wall paint is $0.076 per square foot
and the cost of ceiling paint is $0.19 per square foot. Let x, y, and z be the length, width, and height
of a rectangular room respectively. Find the room dimensions that result in a minimum cost for the
paint. Round your answers to two decimal places.
a.
b.
c.
d.
e.
____ 6. Suppose a home improvement contractor is painting the walls and ceiling of a rectangular
room. The volume of the room is 360 cubic feet. The cost of wall paint is $0.054 per square foot and
the cost of ceiling paint is $0.18 per square foot. Let x, y, and z be the length, width, and height of a
rectangular room respectively. Identify the room dimensions that result in a minimum cost for the
paint and use these dimensions to find the minimum cost for the paint. Round your answer to the
nearest cent.
$54.00
$8.64
$7.00
$19.44
$21.60
____ 7. The material for constructing the base of an open box costs 1.5 times as much per unit
area as the material for constructing the sides. For a fixed amount of money $150.00, find the
dimensions of the box of largest volume that can be made.
a.
b.
c.
d.
e.
13.9 Applications of Extrema
836
____ 8.
Suppose a trough with trapezoidal cross sections is formed by turning up the edges of
a 30-inch-wide sheet of aluminum (see figure). Find the cross section of maximum area.
a.
b.
c.
d.
e.
____ 9.
A company manufactures two types of sneakers: running shoes and basketball shoes.
The total revenue from
units of running shoes and
units of basketball shoes is:
,
where and are in thousands of units. Find and so as to maximize the revenue.
a.
b.
c.
d.
e.
837
Chapter 13: Functions of Several Variables
____
10.
Suppose a corporation manufactures candles at two locations. The cost of producing
units at location 1 is
and the cost of producing
units at location 2 is
. The candles sell for $43 per unit. Find the quantity that should be
produced at each location to maximize the profit .
a.
b.
c.
d.
e.
____ 11. Find the least squares regression line for the points shown in the graph.
a.
b.
c.
d.
e.
13.9 Applications of Extrema 838
____ 12. The least squares regression line for the points shown in the graph is .
Calculate S, the sum of the squared errors.
a.
b.
c.
d.
e.
839
Chapter 13: Functions of Several Variables
____
13.
Find the least squares regression line for the points
.
a.
b.
c.
d.
e.
____ 14. Find the least squares regression line for the points
. Round numerical values in your answer to two
decimal places.
a.
b.
c.
d.
e.
____ 15. Find three positive numbers x, y, and z whose sum is 48 and is a maximum.
x = 10, y = 7, z = 31
x = 7, y = 31, z = 10
x = 31, y = 10, z = 7
x = 12, y = 24, z = 12
x = 12, y = 12, z = 24
13.9 Applications of Extrema
840
13.9 Applications of Extrema
Answer Section
841 Chapter 13: Functions of Several Variables
3.10 Lagrange Multipliers
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Use Lagrange multipliers to minimize the function subject to the
following constraint:
Assume that x and y are positive.
a.
b.
no absolute minimum
____ 2.
Use Lagrange multipliers to find the maximum value of
where
and
subject to the constraint
.
a.
b.
c.
d.
e.
____ 3. Use Lagrange multipliers to maximize the function subject
to the following constraint.
Assume that x and y are positive.
a.
13.10 Lagrange Multipliers
842
b.
no absolute maximum
____ 4. Use Lagrange multipliers to minimize the function subject
to the following constraint.
Assume that x, y, and z are positive.
a.
b.
c.
d.
e.
____ 5. Use Lagrange multipliers to find the maximum value of where
and , subject to the constraint .
a.
b.
c.
d.
e.
____ 6. Use Lagrange multipliers to find the maximum value of where
subject to the constraint .
a.
maxima: ; minima:
b.
maxima: ; minima:
c.
maxima: ; minima:
843 Chapter 13: Functions of Several Variables
d.
maxima: ; minima:
e.
maxima: ; minima:
____ 7. Use Lagrange multipliers to minimize the function
subject to the following two constraints.
Assume that x, y, and z are nonnegative.
a.
b.
c.
d.
e.
____ 8. Use Lagrange multipliers to find the minimum distance from the line to the
point
a.
b.
c.
d.
e.
____ 9. Use Lagrange multipliers to find the minimum distance from the parabola to
the point . Round your answer to two decimal places.
13.10 Lagrange Multipliers
844
a.
b.
c.
d.
e.
____ 10. Use Lagrange multipliers to find the minimum distance from the circle
to the point . Round your answer to two decimal places.
2.07
103.64
10.18
____ 11.
Use Lagrange multipliers to find the minimum distance from the plane
to the point
. Round your answer to two decimal places.
a.
3.00
b.
c.
d.
9.81
e.
96.33
____ 12. Find the highest point on the curve of intersection of the following surfaces.
Cone: , Plane:
a.
b.
c.
d.
e.
845 Chapter 13: Functions of Several Variables
____ 13. A cargo container (in the shape of a rectangular solid) must have a volume of 580 cubic
feet. The bottom will cost $6 per square foot to construct and the sides and the top will cost $4 per
square foot to construct. Use Lagrange multipliers to find the dimensions of the container of this
volume that has minimum cost.
a.
b.
c.
d.
e.
____ 14. Let represent the temperature at each point on the sphere
. Find the maximum temperature on the curve formed by the intersection of
the sphere and the plane .
a.
b.
c.
d.
e.
____ 15. Find the maximum production level if the total cost of labor (at
$72 per unit) and capital (at $40 per unit) is limited to $270,000, where x is the number of units
of labor and y is the number of units of capital. Round your answer to the nearest integer.
42,016 units produced
15,796 units produced
29,115 units produced
291,147 units produced
19,792 units produced
13.10 Lagrange Multipliers
846
____ 16.
Find the minimum cost of producing 55,000 units of a product
,
where x is the number of units of labor (at $76 per unit) and y is the number of units of capital (at
$56 per unit). Round your answer to the nearest cent.
$68,837.29
$50,201.54
$42,169.29
$57,823.32
$54,500.66
847 Chapter 13: Functions of Several Variables
3.10 Lagrange Multipliers
Answer Section