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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 the given constraint.

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 the given constraint.

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
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ERI YA 0 14.8 EXERCISES g(x, y) = 8. Estimate the maximum and minimum values 1. Pictured are a contour map of f and a curve with equation of f subject to the constraint that g(x, y): reasoning. = 8. Explain your 70 g(x, y) = 8 and so μ² = 29, μ = ± √29/2. Then x = +2/√29, y = ±5/√29, and, from (20), z = 1 = x + y = 1 ± 7/√29. The corresponding values of f are 60 x² + y² = 1 x² + y² = 10 5 √29 Therefore the maximum value of f on the given curve is 3 + D 50 40 Indi 30 20 10 F X 2 √29 2 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? 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 the given constraint. 307 (07 3. f(x, y) = x² - y²; 4. f(x, y) = 3x + y; 5. f(x, y) = xy; 4x² + y² = 8 = 6. f(x,y) = xe"; x +y2 =2 Z 7. f(x, y, z) = 2x + 2y + z; x² + y² +2²=9 8. f(x, y, z)=e; 2x² + y² + z² = 24 9. f(x, y, z) = xy²z; x² + y² + z² = 4 10. f(x, y, z) = ln(x² + 1) + ln(y² + 1) + In (z² + 1); x² + y² + ² = 12 SECTION 14.8 Lagrange Multipliers 2(= + 3 ( 1 = 7/20 29 977 = 3 ± √29 ... 11. f(x, y, z) = x² + y² + z²; x² + y² + z = 1 12. f(x, y, z) = x² + y² + z²; x² + y² + z² = 1 13. f(x, y, z, t) = x+y+z+t; x² + y² + z² + 1² = 1 14. f(x₁,x₂,...,xn) = x₁ + x₂ + ... + xni x² + x² + + x² = 1 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. portul 10-208 1809 sdi 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, 2 x² + y² ≤ 9 22. f(x, y) = 2x² + 3y² - 4x - 5, x² + y² ≤ 16 23. f(x, y) = e, x² + 4y² ≤ 1 SP 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:ERI YA 0 14.8 EXERCISES g(x, y) = 8. Estimate the maximum and minimum values 1. Pictured are a contour map of f and a curve with equation of f subject to the constraint that g(x, y): reasoning. = 8. Explain your 70 g(x, y) = 8 and so μ² = 29, μ = ± √29/2. Then x = +2/√29, y = ±5/√29, and, from (20), z = 1 = x + y = 1 ± 7/√29. The corresponding values of f are 60 x² + y² = 1 x² + y² = 10 5 √29 Therefore the maximum value of f on the given curve is 3 + D 50 40 Indi 30 20 10 F X 2 √29 2 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? 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 the given constraint. 307 (07 3. f(x, y) = x² - y²; 4. f(x, y) = 3x + y; 5. f(x, y) = xy; 4x² + y² = 8 = 6. f(x,y) = xe"; x +y2 =2 Z 7. f(x, y, z) = 2x + 2y + z; x² + y² +2²=9 8. f(x, y, z)=e; 2x² + y² + z² = 24 9. f(x, y, z) = xy²z; x² + y² + z² = 4 10. f(x, y, z) = ln(x² + 1) + ln(y² + 1) + In (z² + 1); x² + y² + ² = 12 SECTION 14.8 Lagrange Multipliers 2(= + 3 ( 1 = 7/20 29 977 = 3 ± √29 ... 11. f(x, y, z) = x² + y² + z²; x² + y² + z = 1 12. f(x, y, z) = x² + y² + z²; x² + y² + z² = 1 13. f(x, y, z, t) = x+y+z+t; x² + y² + z² + 1² = 1 14. f(x₁,x₂,...,xn) = x₁ + x₂ + ... + xni x² + x² + + x² = 1 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. portul 10-208 1809 sdi 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, 2 x² + y² ≤ 9 22. f(x, y) = 2x² + 3y² - 4x - 5, x² + y² ≤ 16 23. f(x, y) = e, x² + 4y² ≤ 1 SP 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)?
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