For functions f(x,y,z) = 5xyz and g(x,y,z) = x2 + 3y? + 4z2 – 7. write the Lagrange multiplier conditions that must be satisfied by a point that maximizes or minimizes f subject to the constraint g(x,y,z) = 0- O A. 5yz = (2x), 5xz=A(6y), 5xy=A(8z), 5xyz = 0. O B. 5xyz = (2x), 5xyz=(6y), 5xyz= A(8z), x² + 3y² + 4z2 – 7=0. O C. 5yz = (x2), 5xz= (3y?), 5xy= X(4z²), x² + 3y² +4z² – 7=0. O D. 5yz= (2x), 5xz=(6y), 5xy= A(82), x² + 3y² + 4z² – 7= 0.
For functions f(x,y,z) = 5xyz and g(x,y,z) = x2 + 3y? + 4z2 – 7. write the Lagrange multiplier conditions that must be satisfied by a point that maximizes or minimizes f subject to the constraint g(x,y,z) = 0- O A. 5yz = (2x), 5xz=A(6y), 5xy=A(8z), 5xyz = 0. O B. 5xyz = (2x), 5xyz=(6y), 5xyz= A(8z), x² + 3y² + 4z2 – 7=0. O C. 5yz = (x2), 5xz= (3y?), 5xy= X(4z²), x² + 3y² +4z² – 7=0. O D. 5yz= (2x), 5xz=(6y), 5xy= A(82), x² + 3y² + 4z² – 7= 0.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Optimizing Functions Using Lagrange Multipliers**
When working with constrained optimization problems, the method of Lagrange multipliers is an invaluable tool. Suppose you are given the functions \(f(x,y,z)\) and \(g(x,y,z)\), and you are asked to find the conditions under which \(f\) has an extremum subject to the constraint \(g(x,y,z) = 0\).
Consider the following example:
Let \( f(x,y,z) = 5xyz \) and \( g(x,y,z) = x^2 + 3y^2 + 4z^2 - 7 \). To find the Lagrange multiplier conditions that must be satisfied by a point that maximizes or minimizes \(f\) subject to the constraint \(g(x,y,z) = 0\), examine the options provided below:
**A.**
\[ 5yz = \lambda(2x),\ 5xz = \lambda(6y),\ 5xy = \lambda(8z),\ 5xyz = 0. \]
**B.**
\[ 5xyz = \lambda(2x),\ 5xyz = \lambda(6y),\ 5xyz = \lambda(8z),\ x^2 + 3y^2 + 4z^2 - 7 = 0. \]
**C.**
\[ 5yz = \lambda(x^2),\ 5xz = \lambda(3y^2),\ 5xy = \lambda(4z^2),\ x^2 + 3y^2 + 4z^2 - 7 = 0. \]
**D.**
\[ 5yz = \lambda(2x),\ 5xz = \lambda(6y),\ 5xy = \lambda(8z),\ x^2 + 3y^2 + 4z^2 - 7 = 0. \]
### Explanation
To find the correct Lagrange conditions, recall that the gradients of \(f\) and \(g\) must be proportional at the extrema, i.e., \(\nabla f = \lambda \nabla g\), where \(\lambda\) is the Lagrange multiplier. This results in the system of equations given by the gradients equated with the inclusion of the constraint:
\[
\n](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faba5746d-e0e8-46ca-a0bb-ec14000055d1%2Fd4998223-1598-41fe-9799-54ae88f05f1f%2Fhc81qsdw_processed.png&w=3840&q=75)
Transcribed Image Text:**Optimizing Functions Using Lagrange Multipliers**
When working with constrained optimization problems, the method of Lagrange multipliers is an invaluable tool. Suppose you are given the functions \(f(x,y,z)\) and \(g(x,y,z)\), and you are asked to find the conditions under which \(f\) has an extremum subject to the constraint \(g(x,y,z) = 0\).
Consider the following example:
Let \( f(x,y,z) = 5xyz \) and \( g(x,y,z) = x^2 + 3y^2 + 4z^2 - 7 \). To find the Lagrange multiplier conditions that must be satisfied by a point that maximizes or minimizes \(f\) subject to the constraint \(g(x,y,z) = 0\), examine the options provided below:
**A.**
\[ 5yz = \lambda(2x),\ 5xz = \lambda(6y),\ 5xy = \lambda(8z),\ 5xyz = 0. \]
**B.**
\[ 5xyz = \lambda(2x),\ 5xyz = \lambda(6y),\ 5xyz = \lambda(8z),\ x^2 + 3y^2 + 4z^2 - 7 = 0. \]
**C.**
\[ 5yz = \lambda(x^2),\ 5xz = \lambda(3y^2),\ 5xy = \lambda(4z^2),\ x^2 + 3y^2 + 4z^2 - 7 = 0. \]
**D.**
\[ 5yz = \lambda(2x),\ 5xz = \lambda(6y),\ 5xy = \lambda(8z),\ x^2 + 3y^2 + 4z^2 - 7 = 0. \]
### Explanation
To find the correct Lagrange conditions, recall that the gradients of \(f\) and \(g\) must be proportional at the extrema, i.e., \(\nabla f = \lambda \nabla g\), where \(\lambda\) is the Lagrange multiplier. This results in the system of equations given by the gradients equated with the inclusion of the constraint:
\[
\n
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