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31-38 Find the absolute maximum and minimum values o the set D. 12: 31. f(x, y) = x² + y² - 2x, D is the closed triangular re with vertices (2, 0), (0, 2), and (0, -2)

31-38 Find the absolute maximum and minimum values o the set D. 12: 31. f(x, y) = x² + y² - 2x, D is the closed triangular re with vertices (2, 0), (0, 2), and (0, -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
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968 CHAPTER 14 Partial Derivatives 4. f(x, y) = 3x - x³ - 2y² + y4 1.5. y 6. f(x, y) = xy - 2x - 2y - x² - y² - 7. f(x, y) = (x − y)(1 − xy) 8. f(x, y) = y(ex - 1) 9. f(x, y) = x² + y² + 2xy 1651 10. f(x, y) = 2x² + 2x² - y² 1.5 5-20 Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 5. f(x, y) = x² + xy + y² + y 16. f(x, y) = xye 11. f(x, y) = x³ - 3x + 3xy² 294085dw 12. f(x, y) = x³ + y³ − 3x² − 3y² – 9x 13. f(x, y) = x² - 2x² + y³ - 3y 14. f(x, y) = y cos x 15. f(x, y) = e* cos y ‚—(x² + y²)/2 16X 17. f(x, y) = xy + e¯xy 18. ƒ(x, y) = (x² + y²)e¯x 19. f(x, y) = y² - 2y cos x, -1 ≤x≤7 20. f(x, y) = sin x sin y, 100 0 <1 anoisibong -T<x<T, -π<y< π + 2 has an infinite = 21. Show that f(x, y) = x² + 4y² - 4xy number of critical points and that D 0 at each one. Then show that f has a local (and absolute) minimum at each critical point. 22. Show that f(x, y) = x²ye-²-² has maximum values at (±1, 1/√2) and minimum values at (±1, -1/√2). Show also that f has infinitely many other critical points and D = 0 at each of them. Which of them give rise to maximum values? Minimum values? Saddle points? 23-26 Use a graph or level curves or both to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. -2 23. f(x, y) = x² + y² + x-²y-² 24. f(x, y) = (x - y)e-x²-y² 25. f(x, y) = sin x + sin y + sin(x + y), 0≤x≤2T, 0≤ y ≤ 2T v) A 26. f(x, y) = sin x + sin y + cos(x + y), 0≤x≤ π/4, 0≤ y ≤ π/4 27-30 Use a graphing device as in Example 4 (or Newton's method or solve numerically using a calculator or computer) to find the critical points of f correct to three decimal places. Then classify the critical points and find the highest or lowest points on the graph, if any. 27. f(x, y) = x² + y² - 4x²y + 2y 28. f(x, y) = y6 - 2y + x² - y² + y 29. f(x, y) = x² + y³ - 3x² + y² + x - 2y + 1 30. f(x, y) = 20e-²-sin 3x cos 3y, |x|<1, |y|<1 31-38 Find the absolute maximum and minimum values of f on the set D. 31. f(x, y) = x² + y² - 2x, D is the closed triangular region with vertices (2, 0), (0, 2), and (0, -2) 32. f(x, y) = x + y - xy, D is the closed triangular region with vertices (0, 0), (0, 2), and (4,0) its massi 33. f(x, y) = x² + y² + x²y + 4, M D = {(x, y) | |x| ≤ 1, |y| ≤ 1} 34. f(x, y) = x² + xy + y² - 6y, D = {(x, y) | -3 ≤x≤ 3,0 ≤ y ≤ 5} 35. f(x, y) = x² + 2y² - 2x - 4y + 1, D = {(x, y) | 0 ≤ x ≤ 2,0 ≤ y ≤ 3} !)szoqq2 $.10 10100 36. f(x, y) = xy², D = {(x, y) | x ≥ 0, y = 0,₁ x² + y² ≤ 3} 37. f(x, y) = 2x³ + y4, D = {(x, y) | x² + y² ≤ 1} 38. f(x, y) = x³ - 3x - y3 + 12y, D is the quadrilateral whose vertices are (-2, 3), (2, 3), (2, 2), and (-2,-2) PARDEX FAR 39. For functions of one variable it is impossible for a continuous function to have two local maxima and no local minimum. But for functions of two variables such functions exist. Show that the function 5/6 f(x, y) = (x² - 1)² - (x²y - x - 1)² has only two critical points, but has local maxima at both of them. Then use a computer to produce a graph with a care- fully chosen domain and viewpoint to see how this is possible. souns oss is mudunum 10 muraiz
Transcribed Image Text:968 CHAPTER 14 Partial Derivatives 4. f(x, y) = 3x - x³ - 2y² + y4 1.5. y 6. f(x, y) = xy - 2x - 2y - x² - y² - 7. f(x, y) = (x − y)(1 − xy) 8. f(x, y) = y(ex - 1) 9. f(x, y) = x² + y² + 2xy 1651 10. f(x, y) = 2x² + 2x² - y² 1.5 5-20 Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 5. f(x, y) = x² + xy + y² + y 16. f(x, y) = xye 11. f(x, y) = x³ - 3x + 3xy² 294085dw 12. f(x, y) = x³ + y³ − 3x² − 3y² – 9x 13. f(x, y) = x² - 2x² + y³ - 3y 14. f(x, y) = y cos x 15. f(x, y) = e* cos y ‚—(x² + y²)/2 16X 17. f(x, y) = xy + e¯xy 18. ƒ(x, y) = (x² + y²)e¯x 19. f(x, y) = y² - 2y cos x, -1 ≤x≤7 20. f(x, y) = sin x sin y, 100 0 <1 anoisibong -T<x<T, -π<y< π + 2 has an infinite = 21. Show that f(x, y) = x² + 4y² - 4xy number of critical points and that D 0 at each one. Then show that f has a local (and absolute) minimum at each critical point. 22. Show that f(x, y) = x²ye-²-² has maximum values at (±1, 1/√2) and minimum values at (±1, -1/√2). Show also that f has infinitely many other critical points and D = 0 at each of them. Which of them give rise to maximum values? Minimum values? Saddle points? 23-26 Use a graph or level curves or both to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. -2 23. f(x, y) = x² + y² + x-²y-² 24. f(x, y) = (x - y)e-x²-y² 25. f(x, y) = sin x + sin y + sin(x + y), 0≤x≤2T, 0≤ y ≤ 2T v) A 26. f(x, y) = sin x + sin y + cos(x + y), 0≤x≤ π/4, 0≤ y ≤ π/4 27-30 Use a graphing device as in Example 4 (or Newton's method or solve numerically using a calculator or computer) to find the critical points of f correct to three decimal places. Then classify the critical points and find the highest or lowest points on the graph, if any. 27. f(x, y) = x² + y² - 4x²y + 2y 28. f(x, y) = y6 - 2y + x² - y² + y 29. f(x, y) = x² + y³ - 3x² + y² + x - 2y + 1 30. f(x, y) = 20e-²-sin 3x cos 3y, |x|<1, |y|<1 31-38 Find the absolute maximum and minimum values of f on the set D. 31. f(x, y) = x² + y² - 2x, D is the closed triangular region with vertices (2, 0), (0, 2), and (0, -2) 32. f(x, y) = x + y - xy, D is the closed triangular region with vertices (0, 0), (0, 2), and (4,0) its massi 33. f(x, y) = x² + y² + x²y + 4, M D = {(x, y) | |x| ≤ 1, |y| ≤ 1} 34. f(x, y) = x² + xy + y² - 6y, D = {(x, y) | -3 ≤x≤ 3,0 ≤ y ≤ 5} 35. f(x, y) = x² + 2y² - 2x - 4y + 1, D = {(x, y) | 0 ≤ x ≤ 2,0 ≤ y ≤ 3} !)szoqq2 $.10 10100 36. f(x, y) = xy², D = {(x, y) | x ≥ 0, y = 0,₁ x² + y² ≤ 3} 37. f(x, y) = 2x³ + y4, D = {(x, y) | x² + y² ≤ 1} 38. f(x, y) = x³ - 3x - y3 + 12y, D is the quadrilateral whose vertices are (-2, 3), (2, 3), (2, 2), and (-2,-2) PARDEX FAR 39. For functions of one variable it is impossible for a continuous function to have two local maxima and no local minimum. But for functions of two variables such functions exist. Show that the function 5/6 f(x, y) = (x² - 1)² - (x²y - x - 1)² has only two critical points, but has local maxima at both of them. Then use a computer to produce a graph with a care- fully chosen domain and viewpoint to see how this is possible. souns oss is mudunum 10 muraiz
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