Explanation Result used: Let the partial derivatives f x x , f y y and f x y of the function z = f ( x , y ) exists in the circular region contained in the xy -plane with center ( a , b ) and f x ( a , b ) = 0 , f y ( a , b ) = 0 and the discriminant D = f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 . (a) The function has relative maximum at ( a , b ) if D > 0 and f x x ( a , b ) < 0 . (b) The function has relative minimum at ( a , b ) if D > 0 and f x x ( a , b ) > 0 . (c) If D < 0 , then f ( a , b ) is a saddle point (neither a maximum nor a minimum) (d) If D = 0 , then the test fail. Calculation: The given function is, z = 2 x 2 − 3 y 2 + 12 y . Calculate the partial derivative of the above function with respect to x . z x = ∂ ∂ x ( 2 x 2 − 3 y 2 + 12 y ) = ∂ ∂ x ( 2 x 2 ) − ∂ ∂ x ( 3 y 2 ) + ∂ ∂ x ( 12 y ) = 4 x − 0 + 0 = 4 x Again, take the partial derivative of the function z x with respect to x . z x x = ∂ ∂ x ( z x ) = ∂ ∂ x ( 4 x ) = 4 That is, z x x = 4 (1) Calculate the partial derivative of the function z with respect to y . z y = ∂ ∂ y ( 2 x 2 − 3 y 2 + 12 y ) = ∂ ∂ y ( 2 x 2 ) − ∂ ∂ y ( 3 y 2 ) + ∂ ∂ y ( 12 y ) = 0 − 6 y + 12 = − 6 y + 12 Again, take the partial derivative of the function z y with respect to y

Calculus with Applications (11th Edition)

11th Edition

ISBN: 9780321979421

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 9, Problem 45RE

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To find: The points such that the function z=2x2−3y2+12y attains relative extrema and saddle points.

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Chapter 9, Problem 44RE

Chapter 9, Problem 46RE

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Chapter 9 Solutions

Calculus with Applications (11th Edition)

Ch. 9.1 - YOUR TURN 1 For the function in Example 1, find...Ch. 9.1 - YOUR TURN 2 For the function in Example 3, find...Ch. 9.1 - YOUR TURN 3 Graph x + 2y + 3z = 12 in the first...Ch. 9.1 - YOUR TURN 4 Find the level curve at a production...Ch. 9.1 - Prob. 1WE Ch. 9.1 - Prob. 2WE Ch. 9.1 - Prob. 3WE Ch. 9.1 - Prob. 4WE Ch. 9.1 - Prob. 5WE Ch. 9.1 - Prob. 6WE

Ch. 9.1 - 1. Let f(x, y) = 2x – 3y + 5. Find the...Ch. 9.1 - 2. Let g(x, y) = x2 – 2xy + y3. Find the...Ch. 9.1 - Prob. 3E Ch. 9.1 - 4. Let . Find the following. (a) f(10,...Ch. 9.1 - Prob. 5E Ch. 9.1 - 6. Let f(x, y) = xex+y. Find the...Ch. 9.1 - Prob. 7E Ch. 9.1 - Prob. 8E Ch. 9.1 - Graph the first-octant portion of each...Ch. 9.1 - Graph the first-octant portion of each...Ch. 9.1 - Prob. 11E Ch. 9.1 - Prob. 12E Ch. 9.1 - Prob. 13E Ch. 9.1 - Prob. 14E Ch. 9.1 - Prob. 15E Ch. 9.1 - Graph the level curves in the first quadrant of...Ch. 9.1 - Graph the level curves in the first quadrant of...Ch. 9.1 - Graph the level curves in the first quadrant of...Ch. 9.1 - Prob. 19E Ch. 9.1 - Prob. 20E Ch. 9.1 - Prob. 21E Ch. 9.1 - Prob. 22E Ch. 9.1 - Match each equation in Exercises 23–28 with its...Ch. 9.1 - Prob. 24E Ch. 9.1 - Match each equation in Exercises 23–28 with its...Ch. 9.1 - Prob. 26E Ch. 9.1 - Match each equation in Exercises 23–28 with its...Ch. 9.1 - Match each equation in Exercises 23–28 with its...Ch. 9.1 - 29. Let f(x, y) = 4x2 – 2y2, and find the...Ch. 9.1 - Prob. 30E Ch. 9.1 - Prob. 31E Ch. 9.1 - Prob. 32E Ch. 9.1 - 33. Production Production of a digital camera is...Ch. 9.1 - Production Find the level curve at a production of...Ch. 9.1 - Production Find the level curve at a production of...Ch. 9.1 - Prob. 36E Ch. 9.1 - 37. Cost If labor (x) costs $250 per unit,...Ch. 9.1 - Prob. 38E Ch. 9.1 - Prob. 39E Ch. 9.1 - Prob. 40E Ch. 9.1 - Prob. 41E Ch. 9.1 - 42. Dinosaur Running An article titled “How...Ch. 9.1 - Prob. 43E Ch. 9.1 - 44. Dengue Fever In tropical regions, dengue fever...Ch. 9.1 - Prob. 45E Ch. 9.1 - Prob. 46E Ch. 9.1 - Prob. 47E Ch. 9.1 - Prob. 48E Ch. 9.1 - Prob. 49E Ch. 9.1 - Prob. 50E Ch. 9.1 - 51. Elliptical Templates The holes cut in a roof...Ch. 9.2 - YOUR TURN 1 Let Find and . Ch. 9.2 - Prob. 2YT Ch. 9.2 - Prob. 3YT Ch. 9.2 - Prob. 4YT Ch. 9.2 - Prob. 1WE Ch. 9.2 - Prob. 2WE Ch. 9.2 - Prob. 3WE Ch. 9.2 - Prob. 4WE Ch. 9.2 - Prob. 5WE Ch. 9.2 - Prob. 6WE Ch. 9.2 - Prob. 7WE Ch. 9.2 - Prob. 8WE Ch. 9.2 - Prob. 1E Ch. 9.2 - Prob. 2E Ch. 9.2 - Prob. 3E Ch. 9.2 - In Exercises 3–20, find fx(x, y) and fy(x,y). Then...Ch. 9.2 - In Exercises 3–20, find fx(x, y) and fy(x, y)....Ch. 9.2 - Prob. 6E Ch. 9.2 - Prob. 7E Ch. 9.2 - Prob. 8E Ch. 9.2 - Prob. 9E Ch. 9.2 - Prob. 10E Ch. 9.2 - Prob. 11E Ch. 9.2 - Prob. 12E Ch. 9.2 - In Exercises 3–20, find fx(x, y) and fy(x, y)....Ch. 9.2 - Prob. 14E Ch. 9.2 - Prob. 15E Ch. 9.2 - Prob. 16E Ch. 9.2 - Prob. 17E Ch. 9.2 - Prob. 18E Ch. 9.2 - In Exercises 3–20, find fx(x, y) and fy(x, y)....Ch. 9.2 - Prob. 20E Ch. 9.2 - Prob. 21E Ch. 9.2 - Prob. 22E Ch. 9.2 - Prob. 23E Ch. 9.2 - Prob. 24E Ch. 9.2 - Prob. 25E Ch. 9.2 - Prob. 26E Ch. 9.2 - Prob. 27E Ch. 9.2 - Prob. 28E Ch. 9.2 - Prob. 29E Ch. 9.2 - Prob. 30E Ch. 9.2 - Prob. 31E Ch. 9.2 - Prob. 32E Ch. 9.2 - Prob. 33E Ch. 9.2 - Prob. 34E Ch. 9.2 - For the functions defined as follows, find all...Ch. 9.2 - For the functions defined as follows, find all...Ch. 9.2 - Prob. 37E Ch. 9.2 - Find and for the following. 38. Ch. 9.2 - Prob. 39E Ch. 9.2 - Prob. 40E Ch. 9.2 - Find and for the following. 41. Ch. 9.2 - Find and for the following. 42. Ch. 9.2 - Prob. 43E Ch. 9.2 - Prob. 44E Ch. 9.2 - Business and Economics 45. Manufacturing Cost...Ch. 9.2 - 46. Revenue The revenue from the sale of x units...Ch. 9.2 - Prob. 47E Ch. 9.2 - Prob. 48E Ch. 9.2 - 49. Marginal Productivity A manufacturer estimates...Ch. 9.2 - 50. Marginal Productivity The production function...Ch. 9.2 - Prob. 51E Ch. 9.2 - 52. Marginal Productivity A manufacturer of...Ch. 9.2 - Prob. 53E Ch. 9.2 - Prob. 54E Ch. 9.2 - Prob. 55E Ch. 9.2 - Prob. 56E Ch. 9.2 - Prob. 57E Ch. 9.2 - Prob. 58E Ch. 9.2 - Prob. 59E Ch. 9.2 - 60. Health The body mass index (BMI) is a number...Ch. 9.2 - Prob. 61E Ch. 9.2 - Prob. 62E Ch. 9.2 - 63. Drug Reaction The reaction to x units of a...Ch. 9.2 - Prob. 64E Ch. 9.2 - Prob. 66E Ch. 9.2 - Prob. 67E Ch. 9.2 - Prob. 68E Ch. 9.2 - 69. Gravitational Attraction The gravitational...Ch. 9.2 - Prob. 70E Ch. 9.2 - Prob. 71E Ch. 9.3 - YOUR TURN 1 Find all critical points for Ch. 9.3 - Prob. 2YT Ch. 9.3 - Prob. 1WE Ch. 9.3 - Prob. 2WE Ch. 9.3 - Find all points where the functions have any...Ch. 9.3 - Find all points where the functions have any...Ch. 9.3 - Find all points where the functions have any...Ch. 9.3 - Find all points where the functions have any...Ch. 9.3 - Prob. 5E Ch. 9.3 - Prob. 6E Ch. 9.3 - Prob. 7E Ch. 9.3 - Find all points where the functions have any...Ch. 9.3 - Prob. 9E Ch. 9.3 - Prob. 10E Ch. 9.3 - Prob. 11E Ch. 9.3 - Find all points where the functions have any...Ch. 9.3 - Find all points where the functions have any...Ch. 9.3 - Find all points where the functions have any...Ch. 9.3 - Prob. 15E Ch. 9.3 - Find all points where the functions have any...Ch. 9.3 - Find all points where the functions have any...Ch. 9.3 - Find all points where the functions have any...Ch. 9.3 - Prob. 19E Ch. 9.3 - Prob. 20E Ch. 9.3 - Figures (a)–(f) show the graphs of the functions...Ch. 9.3 - Prob. 22E Ch. 9.3 - Figures (a)–(f) show the graphs of the functions...Ch. 9.3 - Prob. 24E Ch. 9.3 - Prob. 25E Ch. 9.3 - Prob. 26E Ch. 9.3 - 27. Show that f(x, y) = 1 – x4 – y4 has a relative...Ch. 9.3 - 28. Show that D = 0 for f(x, y) = x3 + (x – y)2...Ch. 9.3 - Prob. 29E Ch. 9.3 - Prob. 30E Ch. 9.3 - Prob. 31E Ch. 9.3 - 32. In Exercise 9 of Section 1.3, we found the...Ch. 9.3 - Prob. 33E Ch. 9.3 - 34. Profit Suppose that the profit (in hundreds of...Ch. 9.3 - 35. Labor Costs Suppose the labor cost (in...Ch. 9.3 - 36. Cost The total cost (in dollars) to produce x...Ch. 9.3 - 37. Revenue The total revenue (in hundreds of...Ch. 9.3 - 38. Profit The profit (in thousands of dollars)...Ch. 9.3 - Prob. 39E Ch. 9.3 - 40. Political Science The probability that a...Ch. 9.3 - Prob. 42E Ch. 9.4 - YOUR TURN 1 Find the minimum value of f(x, y) = x2...Ch. 9.4 - Prob. 1WE Ch. 9.4 - Prob. 2WE Ch. 9.4 - Find the relative maxima or minima in Exercises...Ch. 9.4 - Find the relative maxima or minima in Exercises...Ch. 9.4 - Find the relative maxima or minima in Exercises...Ch. 9.4 - Prob. 4E Ch. 9.4 - Find the relative maxima or minima in Exercises...Ch. 9.4 - Prob. 6E Ch. 9.4 - Find the relative maxima or minima in Exercises...Ch. 9.4 - Prob. 8E Ch. 9.4 - Find the relative maxima or minima in Exercises...Ch. 9.4 - Find the relative maxima or minima in Exercises...Ch. 9.4 - 11. Find positive numbers x and y such that x + y...Ch. 9.4 - 12. Find positive numbers x and y such that x + y...Ch. 9.4 - Prob. 13E Ch. 9.4 - 14. Find three positive numbers whose sum is 240...Ch. 9.4 - Prob. 15E Ch. 9.4 - Prob. 16E Ch. 9.4 - Prob. 17E Ch. 9.4 - 18. Show that the function f(x, y) = xy2 in...Ch. 9.4 - 19. Show that the function f(x, y) = 8x2y in...Ch. 9.4 - Prob. 20E Ch. 9.4 - 21. Consider the problem of minimizing f(x, y) =...Ch. 9.4 - Prob. 22E Ch. 9.4 - Business and Economics Utility Maximize each of...Ch. 9.4 - Business and Economics Utility Maximize each of...Ch. 9.4 - Business and Economics Utility Maximize each of...Ch. 9.4 - Prob. 26E Ch. 9.4 - 27. Maximum Area for Fixed Expenditure Because of...Ch. 9.4 - 28. Maximum Area for Fixed Expenditure To enclose...Ch. 9.4 - Prob. 29E Ch. 9.4 - 30. Profit The profit from the sale of x units of...Ch. 9.4 - 31. Production A manufacturing firm estimates that...Ch. 9.4 - Prob. 32E Ch. 9.4 - Prob. 33E Ch. 9.4 - Prob. 34E Ch. 9.4 - Prob. 35E Ch. 9.4 - Prob. 36E Ch. 9.4 - Prob. 37E Ch. 9.4 - Prob. 38E Ch. 9.4 - Prob. 39E Ch. 9.4 - Prob. 40E Ch. 9.4 - 41. Container Construction RD Enterprises is...Ch. 9.4 - Social Sciences 42. Political Science The...Ch. 9.5 - YOUR TURN 1 For the function , find (a) dz, and...Ch. 9.5 - Prob. 2YT Ch. 9.5 - Prob. 3YT Ch. 9.5 - Prob. 1WE Ch. 9.5 - Prob. 2WE Ch. 9.5 - Prob. 3WE Ch. 9.5 - Prob. 4WE Ch. 9.5 - Evaluate dz using the given information. 1. Ch. 9.5 - Evaluate dz using the given information. 2. Ch. 9.5 - Evaluate dz using the given information. 3. Ch. 9.5 - Prob. 4E Ch. 9.5 - Evaluate dz using the given information. 5. Ch. 9.5 - Evaluate dz using the given information. 6. Ch. 9.5 - Prob. 7E Ch. 9.5 - Prob. 8E Ch. 9.5 - Use the total differential to approximate each...Ch. 9.5 - Prob. 10E Ch. 9.5 - Prob. 11E Ch. 9.5 - Prob. 12E Ch. 9.5 - Use the total differential to approximate each...Ch. 9.5 - Use the total differential to approximate each...Ch. 9.5 - Prob. 15E Ch. 9.5 - Prob. 16E Ch. 9.5 - 17. Volume of a Coating An industrial coaling 0.1...Ch. 9.5 - Prob. 18E Ch. 9.5 - Prob. 19E Ch. 9.5 - Prob. 20E Ch. 9.5 - Prob. 21E Ch. 9.5 - Prob. 22E Ch. 9.5 - 23. Blood Volume In Exercise 58 of Section 2 in...Ch. 9.5 - Prob. 24E Ch. 9.5 - Life Span As we saw in Exercise 62 of Section 9.2,...Ch. 9.5 - Prob. 26E Ch. 9.5 - Prob. 27E Ch. 9.5 - Prob. 28E Ch. 9.5 - Prob. 29E Ch. 9.5 - Prob. 30E Ch. 9.5 - Prob. 31E Ch. 9.5 - Prob. 32E Ch. 9.5 - Prob. 33E Ch. 9.5 - Prob. 34E Ch. 9.5 - Prob. 35E Ch. 9.6 - YOUR TURN 1 Evaluate Ch. 9.6 - YOUR TURN 2 Evaluate and then integrate with the...Ch. 9.6 - Prob. 3YT Ch. 9.6 - Prob. 4YT Ch. 9.6 - Prob. 5YT Ch. 9.6 - Prob. 1WE Ch. 9.6 - Prob. 2WE Ch. 9.6 - Prob. 3WE Ch. 9.6 - Prob. 4WE Ch. 9.6 - Prob. 5WE Ch. 9.6 - Prob. 6WE Ch. 9.6 - Prob. 1E Ch. 9.6 - Evaluate each integral. 2. Ch. 9.6 - Prob. 3E Ch. 9.6 - Prob. 4E Ch. 9.6 - Prob. 5E Ch. 9.6 - Evaluate each integral. 6. Ch. 9.6 - Prob. 7E Ch. 9.6 - Prob. 8E Ch. 9.6 - Prob. 9E Ch. 9.6 - Prob. 10E Ch. 9.6 - Evaluate each iterated integral. (Many of these...Ch. 9.6 - Evaluate each iterated integral. (Many of these...Ch. 9.6 - Evaluate each iterated integral. (Many of these...Ch. 9.6 - Prob. 14E Ch. 9.6 - Evaluate each iterated integral. (Many of these...Ch. 9.6 - Prob. 16E Ch. 9.6 - Evaluate each iterated integral. (Many of these...Ch. 9.6 - Prob. 18E Ch. 9.6 - Evaluate each iterated integral. (Many of these...Ch. 9.6 - Evaluate each iterated integral. (Many of these...Ch. 9.6 - Prob. 21E Ch. 9.6 - Find each double integral over the rectangular...Ch. 9.6 - Find each double integral over the rectangular...Ch. 9.6 - Prob. 24E Ch. 9.6 - Find each double integral over the rectangular...Ch. 9.6 - Find each double integral over the rectangular...Ch. 9.6 - Find each double integral over the rectangular...Ch. 9.6 - Prob. 28E Ch. 9.6 - Find the volume under the given surface and above...Ch. 9.6 - Find the volume under the given surface and above...Ch. 9.6 - Prob. 31E Ch. 9.6 - Find the volume under the given surface and above...Ch. 9.6 - Find the volume under the given surface and above...Ch. 9.6 - Prob. 34E Ch. 9.6 - Find the volume under the given surface and above...Ch. 9.6 - Find the volume under the given surface and above...Ch. 9.6 - Prob. 37E Ch. 9.6 - Although it is often true that a double integral...Ch. 9.6 - Prob. 39E Ch. 9.6 - Evaluate each double integral. 40. Ch. 9.6 - Prob. 41E Ch. 9.6 - Prob. 42E Ch. 9.6 - Prob. 43E Ch. 9.6 - Use the region R with the indicated boundaries to...Ch. 9.6 - Prob. 45E Ch. 9.6 - Prob. 46E Ch. 9.6 - Prob. 47E Ch. 9.6 - Use the region R with the indicated boundaries to...Ch. 9.6 - Prob. 49E Ch. 9.6 - Prob. 50E Ch. 9.6 - Prob. 51E Ch. 9.6 - Use the region R with the indicated boundaries to...Ch. 9.6 - Use the region R with the indicated boundaries to...Ch. 9.6 - Use the region R with the indicated boundaries to...Ch. 9.6 - Prob. 55E Ch. 9.6 - Prob. 56E Ch. 9.6 - Evaluate each double integral. If the function...Ch. 9.6 - Prob. 58E Ch. 9.6 - Prob. 59E Ch. 9.6 - Prob. 60E Ch. 9.6 - The idea of the average value of a function,...Ch. 9.6 - Prob. 62E Ch. 9.6 - Prob. 63E Ch. 9.6 - Prob. 64E Ch. 9.6 - 65. Packaging The manufacturer of a fruit juice...Ch. 9.6 - Prob. 66E Ch. 9.6 - 67. Average Production A production function is...Ch. 9.6 - 68. Average Profit The profit (in dollars) from...Ch. 9.6 - 69. Average Revenue A company sells two products....Ch. 9.6 - Prob. 70E Ch. 9.6 - Prob. 71E Ch. 9 - Prob. 1RE Ch. 9 - Prob. 2RE Ch. 9 - Prob. 3RE Ch. 9 - Prob. 4RE Ch. 9 - Prob. 5RE Ch. 9 - Prob. 6RE Ch. 9 - Prob. 7RE Ch. 9 - Prob. 8RE Ch. 9 - Prob. 9RE Ch. 9 - Prob. 10RE Ch. 9 - Prob. 11RE Ch. 9 - Prob. 12RE Ch. 9 - Prob. 13RE Ch. 9 - Prob. 14RE Ch. 9 - Prob. 15RE Ch. 9 - Prob. 16RE Ch. 9 - Prob. 17RE Ch. 9 - Find (–1, 2) and f(6, –3) for the...Ch. 9 - Prob. 19RE Ch. 9 - Prob. 20RE Ch. 9 - Prob. 21RE Ch. 9 - Prob. 22RE Ch. 9 - Prob. 23RE Ch. 9 - Prob. 24RE Ch. 9 - Prob. 25RE Ch. 9 - Prob. 26RE Ch. 9 - Prob. 27RE Ch. 9 - Prob. 28RE Ch. 9 - Prob. 29RE Ch. 9 - Prob. 30RE Ch. 9 - Prob. 31RE Ch. 9 - Prob. 32RE Ch. 9 - Prob. 33RE Ch. 9 - Prob. 34RE Ch. 9 - Prob. 35RE Ch. 9 - Prob. 36RE Ch. 9 - Prob. 37RE Ch. 9 - Prob. 38RE Ch. 9 - Prob. 39RE Ch. 9 - Prob. 40RE Ch. 9 - Prob. 41RE Ch. 9 - Prob. 42RE Ch. 9 - Prob. 43RE Ch. 9 - Prob. 44RE Ch. 9 - Prob. 45RE Ch. 9 - Prob. 46RE Ch. 9 - Prob. 47RE Ch. 9 - Prob. 48RE Ch. 9 - Prob. 49RE Ch. 9 - Prob. 50RE Ch. 9 - Prob. 51RE Ch. 9 - Prob. 52RE Ch. 9 - Prob. 53RE Ch. 9 - Prob. 54RE Ch. 9 - Prob. 55RE Ch. 9 - Prob. 56RE Ch. 9 - Prob. 57RE Ch. 9 - Prob. 58RE Ch. 9 - Prob. 59RE Ch. 9 - Prob. 60RE Ch. 9 - Prob. 61RE Ch. 9 - Prob. 62RE Ch. 9 - Prob. 63RE Ch. 9 - Prob. 64RE Ch. 9 - Prob. 65RE Ch. 9 - Prob. 66RE Ch. 9 - Prob. 67RE Ch. 9 - Prob. 68RE Ch. 9 - Prob. 69RE Ch. 9 - Prob. 70RE Ch. 9 - Prob. 71RE Ch. 9 - Prob. 72RE Ch. 9 - Prob. 73RE Ch. 9 - Prob. 74RE Ch. 9 - Prob. 75RE Ch. 9 - Prob. 76RE Ch. 9 - Prob. 77RE Ch. 9 - Prob. 78RE Ch. 9 - Prob. 79RE Ch. 9 - Prob. 80RE Ch. 9 - Prob. 81RE Ch. 9 - Prob. 82RE Ch. 9 - Prob. 83RE Ch. 9 - Prob. 84RE Ch. 9 - Prob. 85RE Ch. 9 - Prob. 86RE Ch. 9 - Prob. 87RE Ch. 9 - Prob. 88RE Ch. 9 - Prob. 89RE Ch. 9 - Prob. 90RE Ch. 9 - Prob. 91RE Ch. 9 - Prob. 92RE Ch. 9 - Prob. 93RE Ch. 9 - Prob. 94RE Ch. 9 - Prob. 95RE Ch. 9 - Prob. 96RE Ch. 9 - 97. Profit The total profit from 1 acre of a...Ch. 9 - Prob. 98RE Ch. 9 - Prob. 99RE Ch. 9 - Prob. 100RE Ch. 9 - Prob. 101RE Ch. 9 - Prob. 102RE Ch. 9 - Prob. 103RE Ch. 9 - Prob. 104RE Ch. 9 - Prob. 105RE Ch. 9 - Prob. 106RE

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The points such that the function z = 2 x 2 − 3 y 2 + 12 y attains relative extrema and saddle points.

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Chapter 9 Solutions