Explanation Given: The function is f ( x , y ) = x y 3 . The cost of a unit of x is $2, the cost of a unit of y is $4 and $80 is available. Definition used: Lagrange Multipliers 1. The form of the constraint is g ( x , y ) = 0 . 2. The form of the Lagrange function F ( x , y , λ ) = f ( x , y ) − λ ⋅ g ( x , y ) . 3 Form the system of equations F x ( x , y , λ ) = 0 , F y ( x , y , λ ) = 0 and F λ ( x , y , λ ) = 0 , compute the solution of these system and also the relative extrema among these solution. Calculation: The available cost is $80. Here, the constraint is 2 x + 4 y = 80 . The constraint in the form g ( x , y ) = 0 . Thus, the given constrain can be written as g ( x , y ) = 2 x + 4 y − 80 The form of the Lagrange function is, F ( x , y , λ ) = f ( x , y ) − λ ⋅ g ( x , y ) . The multivariable function f ( x , y ) = x y 3 . Substitute f ( x , y ) = x y 3 and g ( x , y ) = 2 x + 4 y − 80 in F ( x , y , λ ) = f ( x , y ) − λ ⋅ g ( x , y ) , F ( x , y , λ ) = x y 3 − λ ( 2 x + 4 y − 80 ) = x y 3 − 2 λ x − 4 λ y + 80 λ Thus, the Lagrange function is F ( x , y , λ ) = x y 3 − 2 λ x − 4 λ y + 80 λ . Find the partial derivative of the function F ( x , y , λ ) with respect to x , take y and λ are constants. F x ( x , y , λ ) = ∂ ∂ x ( x y 3 − 2 λ x − 4 λ y + 80 λ ) = ∂ ∂ x ( x y 3 ) − ∂ ∂ x ( 2 λ x ) − ∂ ∂ x ( 4 λ y ) + ∂ ∂ x ( 80 λ ) = ( y 3 ) ∂ ∂ x ( x ) − ( 2 λ ) ∂ ∂ x ( x ) − ∂ ∂ x ( 4 λ y ) + ∂ ∂ x ( 80 λ ) Simplify the terms and obtain the value as follows, F x ( x , y , λ ) = ( y 3 ) ( 1 ) − ( 2 λ ) ( 1 ) − 0 + 0 = y 3 − 2 λ Therefore, the partial derivative with respect to x is F x ( x , y , λ ) = y 3 − 2 λ . Find the partial derivative of the function F ( x , y , λ ) with respect to y , take x and λ are constants. F y ( x , y , λ ) = ∂ ∂ y ( x y 3 − 2 λ x − 4 λ y + 80 λ ) = ∂ ∂ y ( x y 3 ) − ∂ ∂ y ( 2 λ x ) − ∂ ∂ y ( 4 λ y ) + ∂ ∂ y ( 80 λ ) = ( x ) ∂ ∂ y ( y 3 ) − ∂ ∂ y ( 2 λ x ) − ( 4 λ ) ∂ ∂ y ( y ) + ∂ ∂ y ( 80 λ ) Simplify the terms and obtain the value as follows, F y ( x , y , λ ) = ( x ) ( 3 y 2 ) − 0 − ( 4 λ ) ( 1 ) + 0 = 3 x y 2 − 4 λ Therefore, the partial derivative with respect to y is F y ( x , y , λ ) = 3 x y 2 − 4 λ

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Chapter 17, Problem 91RE

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Consider the following system of equations, Ax=b : x+2y+3z - w = 2 2x4z2w = 3 -x+6y+17z7w = 0 -9x-2y+13z7w = -14 a. Find the solution to the system. Write it as a parametric equation. You can use a computer to do the row reduction. b. What is a geometric description of the solution? Explain how you know. c. Write the solution in vector form? d. What is the solution to the homogeneous system, Ax=0?

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Chapter 17 Solutions

EBK FINITE MATHEMATICS AND CALCULUS WIT

Ch. 17.1 - For the function in Example 1, find f(2, 3)....Ch. 17.1 - Prob. 2YT Ch. 17.1 - Prob. 3YT Ch. 17.1 - Prob. 4YT Ch. 17.1 - Prob. 1WE Ch. 17.1 - Prob. 2WE Ch. 17.1 - Prob. 3WE Ch. 17.1 - Prob. 4WE Ch. 17.1 - Prob. 5WE Ch. 17.1 - Prob. 6WE

Ch. 17.1 - Let f(x, y) = 2x 3y + 5. Find the following....Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E Ch. 17.1 - Prob. 7E Ch. 17.1 - Prob. 8E Ch. 17.1 - Graph the first-octant portion of each plane. 9.2x...Ch. 17.1 - Prob. 10E Ch. 17.1 - Prob. 11E Ch. 17.1 - Prob. 12E Ch. 17.1 - Prob. 13E Ch. 17.1 - Prob. 14E Ch. 17.1 - Prob. 15E Ch. 17.1 - Prob. 16E Ch. 17.1 - Prob. 17E Ch. 17.1 - Prob. 18E Ch. 17.1 - Prob. 19E Ch. 17.1 - Prob. 20E Ch. 17.1 - Prob. 21E Ch. 17.1 - Prob. 22E Ch. 17.1 - Match each equation in Exercises 2328 with its...Ch. 17.1 - Prob. 24E Ch. 17.1 - Prob. 25E Ch. 17.1 - Match each equation in Exercises 2328 with its...Ch. 17.1 - Prob. 27E Ch. 17.1 - Prob. 28E Ch. 17.1 - Prob. 29E Ch. 17.1 - Prob. 30E Ch. 17.1 - Prob. 31E Ch. 17.1 - Prob. 32E Ch. 17.1 - Prob. 33E Ch. 17.1 - Prob. 34E Ch. 17.1 - Production Find the level curve at a production of...Ch. 17.1 - Prob. 36E Ch. 17.1 - Prob. 37E Ch. 17.1 - Individual Retirement Accounts The multiplier...Ch. 17.1 - Individual Retirement Accounts The multiplier...Ch. 17.1 - Heat Loss The rate of heal loss (in watts) in...Ch. 17.1 - Prob. 41E Ch. 17.1 - Prob. 42E Ch. 17.1 - Pollution Intolerance According to research at the...Ch. 17.1 - Dengue Fever In tropical regions, dengue fever is...Ch. 17.1 - Prob. 45E Ch. 17.1 - Prob. 46E Ch. 17.1 - Prob. 47E Ch. 17.1 - Prob. 48E Ch. 17.1 - Prob. 49E Ch. 17.1 - Prob. 50E Ch. 17.1 - Prob. 51E Ch. 17.2 - Prob. 1YT Ch. 17.2 - Prob. 2YT Ch. 17.2 - Prob. 3YT Ch. 17.2 - Prob. 4YT Ch. 17.2 - Prob. 1WE Ch. 17.2 - Prob. 2WE Ch. 17.2 - Prob. 3WE Ch. 17.2 - Prob. 4WE Ch. 17.2 - Prob. 5WE Ch. 17.2 - Prob. 6WE Ch. 17.2 - Prob. 7WE Ch. 17.2 - Prob. 8WE Ch. 17.2 - Prob. 1E Ch. 17.2 - Prob. 2E Ch. 17.2 - Prob. 3E Ch. 17.2 - Prob. 4E Ch. 17.2 - Prob. 5E Ch. 17.2 - Prob. 6E Ch. 17.2 - Prob. 7E Ch. 17.2 - Prob. 8E Ch. 17.2 - Prob. 9E Ch. 17.2 - Prob. 10E Ch. 17.2 - Prob. 11E Ch. 17.2 - Prob. 12E Ch. 17.2 - Prob. 13E Ch. 17.2 - Prob. 14E Ch. 17.2 - Prob. 15E Ch. 17.2 - Prob. 16E Ch. 17.2 - Prob. 17E Ch. 17.2 - Prob. 18E Ch. 17.2 - Prob. 19E Ch. 17.2 - Prob. 20E Ch. 17.2 - Prob. 21E Ch. 17.2 - Prob. 22E Ch. 17.2 - Prob. 23E Ch. 17.2 - Prob. 24E Ch. 17.2 - Prob. 25E Ch. 17.2 - Prob. 26E Ch. 17.2 - Prob. 27E Ch. 17.2 - Prob. 28E Ch. 17.2 - Prob. 29E Ch. 17.2 - Prob. 30E Ch. 17.2 - Prob. 31E Ch. 17.2 - Prob. 32E Ch. 17.2 - Prob. 33E Ch. 17.2 - Prob. 34E Ch. 17.2 - Prob. 35E Ch. 17.2 - Prob. 36E Ch. 17.2 - Prob. 37E Ch. 17.2 - Prob. 38E Ch. 17.2 - Prob. 39E Ch. 17.2 - Prob. 40E Ch. 17.2 - Find fx(x,y,z),fy(x,y,z),fz(x,y,z), and fyz(x,y,z)...Ch. 17.2 - Prob. 42E Ch. 17.2 - Prob. 43E Ch. 17.2 - Prob. 44E Ch. 17.2 - Business and Economics 45.Manufacturing Cost...Ch. 17.2 - Prob. 46E Ch. 17.2 - Prob. 47E Ch. 17.2 - Prob. 48E Ch. 17.2 - Prob. 49E Ch. 17.2 - Prob. 50E Ch. 17.2 - Prob. 51E Ch. 17.2 - Prob. 52E Ch. 17.2 - Prob. 53E Ch. 17.2 - Prob. 54E Ch. 17.2 - Prob. 55E Ch. 17.2 - Prob. 56E Ch. 17.2 - Prob. 57E Ch. 17.2 - Prob. 58E Ch. 17.2 - Prob. 59E Ch. 17.2 - Prob. 60E Ch. 17.2 - Prob. 61E Ch. 17.2 - Prob. 62E Ch. 17.2 - Drug Reaction The reaction to x units of a drug t...Ch. 17.2 - Prob. 64E Ch. 17.2 - Prob. 66E Ch. 17.2 - Prob. 67E Ch. 17.2 - Prob. 68E Ch. 17.2 - Gravitational Attraction The gravitational...Ch. 17.2 - Velocity In 1931, Albert Einstein developed the...Ch. 17.2 - Prob. 71E Ch. 17.3 - Prob. 1YT Ch. 17.3 - Prob. 2YT Ch. 17.3 - Prob. 1WE Ch. 17.3 - Prob. 2WE Ch. 17.3 - Prob. 1E Ch. 17.3 - Prob. 2E Ch. 17.3 - Prob. 3E Ch. 17.3 - Prob. 4E Ch. 17.3 - Prob. 5E Ch. 17.3 - Prob. 6E Ch. 17.3 - Prob. 7E Ch. 17.3 - Prob. 8E Ch. 17.3 - Prob. 9E Ch. 17.3 - Prob. 10E Ch. 17.3 - Prob. 11E Ch. 17.3 - Prob. 12E Ch. 17.3 - Prob. 13E Ch. 17.3 - Prob. 14E Ch. 17.3 - Prob. 15E Ch. 17.3 - Prob. 16E Ch. 17.3 - Prob. 17E Ch. 17.3 - Prob. 18E Ch. 17.3 - Prob. 19E Ch. 17.3 - Prob. 20E Ch. 17.3 - Prob. 21E Ch. 17.3 - Prob. 22E Ch. 17.3 - Prob. 23E Ch. 17.3 - Figures (a)(f) show the graphs of the functions...Ch. 17.3 - Prob. 25E Ch. 17.3 - Prob. 26E Ch. 17.3 - Prob. 27E Ch. 17.3 - Prob. 28E Ch. 17.3 - Prob. 29E Ch. 17.3 - Prob. 30E Ch. 17.3 - Prob. 31E Ch. 17.3 - Prob. 32E Ch. 17.3 - Prob. 33E Ch. 17.3 - Prob. 34E Ch. 17.3 - Prob. 35E Ch. 17.3 - Prob. 36E Ch. 17.3 - Prob. 37E Ch. 17.3 - Prob. 38E Ch. 17.3 - Prob. 39E Ch. 17.3 - Prob. 40E Ch. 17.3 - Prob. 42E Ch. 17.4 - Prob. 1YT Ch. 17.4 - Prob. 1WE Ch. 17.4 - Prob. 2WE Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Find the relative maxima or minima in Exercises...Ch. 17.4 - Prob. 11E Ch. 17.4 - Find positive numbers x and y such that x + y = 48...Ch. 17.4 - Prob. 13E Ch. 17.4 - Prob. 14E Ch. 17.4 - Find the maximum and minimum values of f(x, y) =x3...Ch. 17.4 - Prob. 16E Ch. 17.4 - Prob. 17E Ch. 17.4 - Prob. 18E Ch. 17.4 - Prob. 19E Ch. 17.4 - Prob. 20E Ch. 17.4 - Prob. 21E Ch. 17.4 - Discuss the advantages and disadvantages of the...Ch. 17.4 - Prob. 23E Ch. 17.4 - Prob. 24E Ch. 17.4 - Prob. 25E Ch. 17.4 - Prob. 26E Ch. 17.4 - Prob. 27E Ch. 17.4 - Prob. 28E Ch. 17.4 - Prob. 29E Ch. 17.4 - Prob. 30E Ch. 17.4 - Prob. 31E Ch. 17.4 - Prob. 32E Ch. 17.4 - Prob. 33E Ch. 17.4 - Prob. 34E Ch. 17.4 - Prob. 35E Ch. 17.4 - Prob. 36E Ch. 17.4 - Prob. 37E Ch. 17.4 - Prob. 38E Ch. 17.4 - Prob. 39E Ch. 17.4 - Prob. 40E Ch. 17.4 - Prob. 41E Ch. 17.4 - Prob. 42E Ch. 17.5 - Prob. 1YT Ch. 17.5 - Prob. 2YT Ch. 17.5 - Prob. 3YT Ch. 17.5 - Prob. 1WE Ch. 17.5 - Prob. 2WE Ch. 17.5 - Prob. 3WE Ch. 17.5 - Prob. 4WE Ch. 17.5 - Prob. 1E Ch. 17.5 - Prob. 2E Ch. 17.5 - Prob. 3E Ch. 17.5 - Prob. 4E Ch. 17.5 - Prob. 5E Ch. 17.5 - Prob. 6E Ch. 17.5 - Prob. 7E Ch. 17.5 - Prob. 8E Ch. 17.5 - Prob. 9E Ch. 17.5 - Prob. 10E Ch. 17.5 - Prob. 11E Ch. 17.5 - Prob. 12E Ch. 17.5 - Prob. 13E Ch. 17.5 - Prob. 14E Ch. 17.5 - Prob. 15E Ch. 17.5 - Prob. 16E Ch. 17.5 - Volume of a Coating An industrial coaling 0.1 in....Ch. 17.5 - Prob. 18E Ch. 17.5 - Prob. 19E Ch. 17.5 - Prob. 20E Ch. 17.5 - Prob. 21E Ch. 17.5 - Prob. 22E Ch. 17.5 - Prob. 23E Ch. 17.5 - Prob. 24E Ch. 17.5 - Life Span As we saw in Exercise 62 of Section...Ch. 17.5 - Prob. 26E Ch. 17.5 - Prob. 27E Ch. 17.5 - Prob. 28E Ch. 17.5 - Prob. 29E Ch. 17.5 - Prob. 30E Ch. 17.5 - Prob. 31E Ch. 17.5 - Prob. 32E Ch. 17.5 - Prob. 33E Ch. 17.5 - Prob. 34E Ch. 17.5 - Prob. 35E Ch. 17.6 - Prob. 1YT Ch. 17.6 - Evaluate 02[13(6x2y2+4xy+8x3+10y4+3)dy]dx and then...Ch. 17.6 - Prob. 3YT Ch. 17.6 - Prob. 4YT Ch. 17.6 - Prob. 5YT Ch. 17.6 - Prob. 1WE Ch. 17.6 - Evaluate each definite integral. W2.23x+6dx Ch. 17.6 - Prob. 3WE Ch. 17.6 - Prob. 4WE Ch. 17.6 - Prob. 5WE Ch. 17.6 - Prob. 6WE Ch. 17.6 - Prob. 1E Ch. 17.6 - Prob. 2E Ch. 17.6 - Prob. 3E Ch. 17.6 - Prob. 4E Ch. 17.6 - Prob. 5E Ch. 17.6 - Prob. 6E Ch. 17.6 - Prob. 7E Ch. 17.6 - Prob. 8E Ch. 17.6 - Prob. 9E Ch. 17.6 - Prob. 10E Ch. 17.6 - Prob. 11E Ch. 17.6 - Prob. 12E Ch. 17.6 - Prob. 13E Ch. 17.6 - Prob. 14E Ch. 17.6 - Prob. 15E Ch. 17.6 - Prob. 16E Ch. 17.6 - Prob. 17E Ch. 17.6 - Prob. 18E Ch. 17.6 - Prob. 19E Ch. 17.6 - Prob. 20E Ch. 17.6 - Prob. 21E Ch. 17.6 - Prob. 22E Ch. 17.6 - Prob. 23E Ch. 17.6 - Prob. 24E Ch. 17.6 - Prob. 25E Ch. 17.6 - Prob. 26E Ch. 17.6 - Prob. 27E Ch. 17.6 - Prob. 28E Ch. 17.6 - Prob. 29E Ch. 17.6 - Prob. 30E Ch. 17.6 - Prob. 31E Ch. 17.6 - Prob. 32E Ch. 17.6 - Prob. 33E Ch. 17.6 - Prob. 34E Ch. 17.6 - Prob. 35E Ch. 17.6 - Prob. 36E Ch. 17.6 - Prob. 37E Ch. 17.6 - Prob. 38E Ch. 17.6 - Prob. 39E Ch. 17.6 - Prob. 40E Ch. 17.6 - Prob. 41E Ch. 17.6 - Prob. 42E Ch. 17.6 - Prob. 43E Ch. 17.6 - Use the region R with the indicated boundaries to...Ch. 17.6 - Prob. 45E Ch. 17.6 - Prob. 46E Ch. 17.6 - Prob. 47E Ch. 17.6 - Prob. 48E Ch. 17.6 - Prob. 49E Ch. 17.6 - Prob. 50E Ch. 17.6 - Prob. 51E Ch. 17.6 - Prob. 52E Ch. 17.6 - Prob. 53E Ch. 17.6 - Prob. 54E Ch. 17.6 - Prob. 55E Ch. 17.6 - Prob. 56E Ch. 17.6 - Prob. 57E Ch. 17.6 - Prob. 58E Ch. 17.6 - Prob. 59E Ch. 17.6 - Prob. 60E Ch. 17.6 - Prob. 61E Ch. 17.6 - Prob. 62E Ch. 17.6 - Prob. 63E Ch. 17.6 - Prob. 64E Ch. 17.6 - Prob. 65E Ch. 17.6 - Prob. 66E Ch. 17.6 - Prob. 67E Ch. 17.6 - Prob. 68E Ch. 17.6 - Prob. 69E Ch. 17.6 - Time In Exercise 39 of Section 17.3, we saw that...Ch. 17.6 - Profit In Exercise 38 of Section 17.3, we saw that...Ch. 17 - Determine whether each of the following statements...Ch. 17 - Prob. 2RE Ch. 17 - Prob. 3RE Ch. 17 - Prob. 4RE Ch. 17 - Prob. 5RE Ch. 17 - Prob. 6RE Ch. 17 - Prob. 7RE Ch. 17 - Prob. 8RE Ch. 17 - Prob. 9RE Ch. 17 - Prob. 10RE Ch. 17 - Prob. 11RE Ch. 17 - Prob. 12RE Ch. 17 - Prob. 13RE Ch. 17 - Prob. 14RE Ch. 17 - Prob. 15RE Ch. 17 - Prob. 16RE Ch. 17 - Prob. 17RE Ch. 17 - Prob. 18RE Ch. 17 - Prob. 19RE Ch. 17 - Prob. 20RE Ch. 17 - Prob. 21RE Ch. 17 - Prob. 22RE Ch. 17 - Prob. 23RE Ch. 17 - Prob. 24RE Ch. 17 - Prob. 25RE Ch. 17 - Prob. 26RE Ch. 17 - Prob. 27RE Ch. 17 - Prob. 28RE Ch. 17 - Prob. 29RE Ch. 17 - Prob. 30RE Ch. 17 - Prob. 31RE Ch. 17 - Prob. 32RE Ch. 17 - Prob. 33RE Ch. 17 - Prob. 34RE Ch. 17 - Prob. 35RE Ch. 17 - Prob. 36RE Ch. 17 - Prob. 37RE Ch. 17 - Prob. 38RE Ch. 17 - Prob. 39RE Ch. 17 - Prob. 40RE Ch. 17 - Prob. 41RE Ch. 17 - Prob. 42RE Ch. 17 - Prob. 43RE Ch. 17 - Prob. 44RE Ch. 17 - Prob. 45RE Ch. 17 - Prob. 46RE Ch. 17 - Prob. 47RE Ch. 17 - Prob. 48RE Ch. 17 - Prob. 49RE Ch. 17 - Prob. 50RE Ch. 17 - Prob. 51RE Ch. 17 - Prob. 52RE Ch. 17 - Prob. 53RE Ch. 17 - Prob. 54RE Ch. 17 - Prob. 55RE Ch. 17 - Prob. 56RE Ch. 17 - Prob. 57RE Ch. 17 - Prob. 58RE Ch. 17 - Evaluate dz using the given information. 59.z =...Ch. 17 - Prob. 60RE Ch. 17 - Prob. 61RE Ch. 17 - Prob. 62RE Ch. 17 - Prob. 63RE Ch. 17 - Prob. 64RE Ch. 17 - Prob. 65RE Ch. 17 - Prob. 66RE Ch. 17 - Prob. 67RE Ch. 17 - Prob. 68RE Ch. 17 - Prob. 69RE Ch. 17 - Prob. 70RE Ch. 17 - Prob. 71RE Ch. 17 - Prob. 72RE Ch. 17 - Prob. 73RE Ch. 17 - Prob. 74RE Ch. 17 - Prob. 75RE Ch. 17 - Prob. 76RE Ch. 17 - Prob. 77RE Ch. 17 - Prob. 78RE Ch. 17 - Prob. 79RE Ch. 17 - Prob. 80RE Ch. 17 - Prob. 81RE Ch. 17 - Prob. 82RE Ch. 17 - Prob. 83RE Ch. 17 - Evaluate each double integral. If the function...Ch. 17 - Prob. 85RE Ch. 17 - Prob. 86RE Ch. 17 - Prob. 87RE Ch. 17 - Prob. 88RE Ch. 17 - Prob. 89RE Ch. 17 - Prob. 90RE Ch. 17 - Prob. 91RE Ch. 17 - Prob. 92RE Ch. 17 - Prob. 93RE Ch. 17 - Prob. 94RE Ch. 17 - Prob. 95RE Ch. 17 - Prob. 96RE Ch. 17 - Profit The total profit from 1 acre of a certain...Ch. 17 - Prob. 98RE Ch. 17 - Prob. 99RE Ch. 17 - Prob. 100RE Ch. 17 - Prob. 101RE Ch. 17 - Prob. 102RE Ch. 17 - Prob. 103RE Ch. 17 - Prob. 104RE Ch. 17 - Prob. 105RE Ch. 17 - Prob. 106RE

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Chapter 17 Solutions