Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
7 please
![20
YA
0
14.8 EXERCISES
=
8. Estimate the maximum and minimum values
1. Pictured are a contour map of f and a curve with equation
g(x, y)
of f subject to the constraint that g(x, y) = 8. Explain your
reasoning.
BRE
I hen x
z = 1 = x + y = 1 = 7/√29. The corresponding values of f are
70
g(x, y) = 8
som
60
50
40
30
20
10
It
√29
√29
Therefore the maximum value of f on the given curve is 3 + √29.
X
2
√29
2. (a) Use a graphing calculator or computer to graph the
circle x² + y² = 1. On the same screen, graph several
curves of the form x² + y = c until you find two that
just touch the circle. What is the significance of the
values of c for these two curves?
work of
2
(b) Use Lagrange multipliers to find the extreme values of
f(x, y) = x² + y subject to the constraint x² + y² = 1.
Compare your answers with those in part (a).
3-14 Each of these extreme value problems has a solution with
both a maximum value and a minimum value. Use Lagrange
multipliers to find the extreme values of the function subject to
Ju9 (d)
the given constraint.
3. f(x, y) = x² - y²; x² + y² = 1
3x
4. f(x, y) = 3x + y;
+ y; x² + y² = 10
5. f(x, y) = xy;
4x² + y² = 8
6. f(x, y) =xe; x + y = 2
7. f(x, y, z) = 2x + 2y + z; x² + y² + z² = 9
8. f(x, y, z) = exy²; 2x² + y² + z² = 24
9. f(x, y, z) = xy²z; x² + y² + z² = 4
10. f(x, y, z) = ln (x² + 1) + In(y² + 1) + In(z² + 1);
x² + y² + z² = 12
+
+ 2 ( ==
√29, y = ±5/√√29, and, from (20),
5
+3
+ 3(1 ±
7
=
= 3 ± √29
11. f(x, y, z) = x² + y² + z²;
12. f(x, y, z) = x² + y² + z²;
13. f(x, y, z, t) = x+y+z+t;
x₁ + x₂
14. f(x₁, x2,...,xn) = x₁ + x₂ +
x² + x² + ... + x² = 1
x² + y² + z = 1
x² + y² + z² = 1
x² + y² + z² + 1² = 1
+ ... + xn;
15. The method of Lagrange multipliers assumes that the
extreme values exist, but that is not always the case.
Show that the problem of finding the minimum value of
f(x, y) = x² + y² subject to the constraint
xy 1 can be
solved using Lagrange multipliers, but f does not have a
maximum value with that constraint.
=
16. Find the minimum value of f(x, y, z) = x² + 2y² + 3z²
subject to the constraint x + 2y + 3z = 10. Show that f
has no maximum value with this constraint.
tado adh asing, hum
17-20 Find the extreme values of f subject to both constraints.
17. f(x, y, z) = x+y+z; x² + z² = 2, x + y = 1
18. f(x, y, z) = z; x² + y² = z², x+y+z=24
19. f(x, y, z)= yz + xy; xy = 1, y² + z² = 1
20. f(x, y, z) = x² + y² + z²; x - y = 1, y² - z² = 1
21-23 Find the extreme values of f on the region described by
the inequality.
21. f(x, y) = x² + y² + 4x - 4y, x² + y² ≤ 9
22. f(x, y) = 2x² + 3y² - 4x - 5, x² + y² ≤ 16
23. f(x, y) = exy, x² + 4y² ≤ 1
UN
24. Consider the problem of maximizing the function
f(x, y) = = 2x + 3y subject to the constraint √√x + √y = 5.
(a) Try using Lagrange multipliers to solve the problem.
(b) Does f(25, 0) give a larger value than the one in part (a)?
(c) Solve the problem by graphing the constraint equation
and several level curves of f.
(d) Explain why the method of Lagrange multipliers fails to
solve the problem.
(e) What is the significance of f(9, 4)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe5db299-428d-40d7-abe3-e166485c95d7%2F061e6b53-bd9a-4b57-8e45-6afd187e8d43%2F319dj98_processed.jpeg&w=3840&q=75)
Transcribed Image Text:20
YA
0
14.8 EXERCISES
=
8. Estimate the maximum and minimum values
1. Pictured are a contour map of f and a curve with equation
g(x, y)
of f subject to the constraint that g(x, y) = 8. Explain your
reasoning.
BRE
I hen x
z = 1 = x + y = 1 = 7/√29. The corresponding values of f are
70
g(x, y) = 8
som
60
50
40
30
20
10
It
√29
√29
Therefore the maximum value of f on the given curve is 3 + √29.
X
2
√29
2. (a) Use a graphing calculator or computer to graph the
circle x² + y² = 1. On the same screen, graph several
curves of the form x² + y = c until you find two that
just touch the circle. What is the significance of the
values of c for these two curves?
work of
2
(b) Use Lagrange multipliers to find the extreme values of
f(x, y) = x² + y subject to the constraint x² + y² = 1.
Compare your answers with those in part (a).
3-14 Each of these extreme value problems has a solution with
both a maximum value and a minimum value. Use Lagrange
multipliers to find the extreme values of the function subject to
Ju9 (d)
the given constraint.
3. f(x, y) = x² - y²; x² + y² = 1
3x
4. f(x, y) = 3x + y;
+ y; x² + y² = 10
5. f(x, y) = xy;
4x² + y² = 8
6. f(x, y) =xe; x + y = 2
7. f(x, y, z) = 2x + 2y + z; x² + y² + z² = 9
8. f(x, y, z) = exy²; 2x² + y² + z² = 24
9. f(x, y, z) = xy²z; x² + y² + z² = 4
10. f(x, y, z) = ln (x² + 1) + In(y² + 1) + In(z² + 1);
x² + y² + z² = 12
+
+ 2 ( ==
√29, y = ±5/√√29, and, from (20),
5
+3
+ 3(1 ±
7
=
= 3 ± √29
11. f(x, y, z) = x² + y² + z²;
12. f(x, y, z) = x² + y² + z²;
13. f(x, y, z, t) = x+y+z+t;
x₁ + x₂
14. f(x₁, x2,...,xn) = x₁ + x₂ +
x² + x² + ... + x² = 1
x² + y² + z = 1
x² + y² + z² = 1
x² + y² + z² + 1² = 1
+ ... + xn;
15. The method of Lagrange multipliers assumes that the
extreme values exist, but that is not always the case.
Show that the problem of finding the minimum value of
f(x, y) = x² + y² subject to the constraint
xy 1 can be
solved using Lagrange multipliers, but f does not have a
maximum value with that constraint.
=
16. Find the minimum value of f(x, y, z) = x² + 2y² + 3z²
subject to the constraint x + 2y + 3z = 10. Show that f
has no maximum value with this constraint.
tado adh asing, hum
17-20 Find the extreme values of f subject to both constraints.
17. f(x, y, z) = x+y+z; x² + z² = 2, x + y = 1
18. f(x, y, z) = z; x² + y² = z², x+y+z=24
19. f(x, y, z)= yz + xy; xy = 1, y² + z² = 1
20. f(x, y, z) = x² + y² + z²; x - y = 1, y² - z² = 1
21-23 Find the extreme values of f on the region described by
the inequality.
21. f(x, y) = x² + y² + 4x - 4y, x² + y² ≤ 9
22. f(x, y) = 2x² + 3y² - 4x - 5, x² + y² ≤ 16
23. f(x, y) = exy, x² + 4y² ≤ 1
UN
24. Consider the problem of maximizing the function
f(x, y) = = 2x + 3y subject to the constraint √√x + √y = 5.
(a) Try using Lagrange multipliers to solve the problem.
(b) Does f(25, 0) give a larger value than the one in part (a)?
(c) Solve the problem by graphing the constraint equation
and several level curves of f.
(d) Explain why the method of Lagrange multipliers fails to
solve the problem.
(e) What is the significance of f(9, 4)?
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 2 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781285741550/9781285741550_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
![Thomas' Calculus (14th Edition)](https://www.bartleby.com/isbn_cover_images/9780134438986/9780134438986_smallCoverImage.gif)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
![Calculus: Early Transcendentals (3rd Edition)](https://www.bartleby.com/isbn_cover_images/9780134763644/9780134763644_smallCoverImage.gif)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781285741550/9781285741550_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
![Thomas' Calculus (14th Edition)](https://www.bartleby.com/isbn_cover_images/9780134438986/9780134438986_smallCoverImage.gif)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
![Calculus: Early Transcendentals (3rd Edition)](https://www.bartleby.com/isbn_cover_images/9780134763644/9780134763644_smallCoverImage.gif)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781319050740/9781319050740_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
![Precalculus](https://www.bartleby.com/isbn_cover_images/9780135189405/9780135189405_smallCoverImage.gif)
![Calculus: Early Transcendental Functions](https://www.bartleby.com/isbn_cover_images/9781337552516/9781337552516_smallCoverImage.gif)
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning