Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
S 19. f(x, y, z)= yz + xy; xy = 1, y² + z² = 1 20 f(x y z) = 2 V 2

S 19. f(x, y, z)= yz + xy; xy = 1, y² + z² = 1 20 f(x y z) = 2 V 2

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

19 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)?
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
steps

Step by step

Solved in 3 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS