Use Lagrange multipliers to find the minimum value of the function f(x, y) = x² + y² subject to the constraint xy = 6. Minimum:
Use Lagrange multipliers to find the minimum value of the function f(x, y) = x² + y² subject to the constraint xy = 6. Minimum:
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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![**Exercise: Lagrange Multipliers Method**
Use Lagrange multipliers to find the minimum value of the function \( f(x, y) = x^2 + y^2 \) subject to the constraint \( xy = 6 \).
**Objective:**
- Find the minimum value.
**Solution:**
\[ \text{Minimum: } \_\_\_ \]
**Explanation:**
To solve this problem using Lagrange multipliers, you need to:
1. Define the function \( f(x, y) = x^2 + y^2 \).
2. Identify the constraint \( g(x, y) = xy - 6 = 0 \).
3. Form the Lagrangian:
\[ \mathcal{L}(x, y, \lambda) = x^2 + y^2 + \lambda(xy - 6) \]
4. Take partial derivatives and solve:
\[
\frac{\partial \mathcal{L}}{\partial x} = 2x + \lambda y = 0
\]
\[
\frac{\partial \mathcal{L}}{\partial y} = 2y + \lambda x = 0
\]
\[
\frac{\partial \mathcal{L}}{\partial \lambda} = xy - 6 = 0
\]
5. Solve these equations simultaneously to find the values of \( x \), \( y \), and \( \lambda \) that satisfy both the original function and the constraint.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F49444d66-96b7-45b8-992f-0f6c51b0e4d0%2F57bd621d-fbff-419a-b8be-e5db5b69e2a0%2Fhq42vil_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise: Lagrange Multipliers Method**
Use Lagrange multipliers to find the minimum value of the function \( f(x, y) = x^2 + y^2 \) subject to the constraint \( xy = 6 \).
**Objective:**
- Find the minimum value.
**Solution:**
\[ \text{Minimum: } \_\_\_ \]
**Explanation:**
To solve this problem using Lagrange multipliers, you need to:
1. Define the function \( f(x, y) = x^2 + y^2 \).
2. Identify the constraint \( g(x, y) = xy - 6 = 0 \).
3. Form the Lagrangian:
\[ \mathcal{L}(x, y, \lambda) = x^2 + y^2 + \lambda(xy - 6) \]
4. Take partial derivatives and solve:
\[
\frac{\partial \mathcal{L}}{\partial x} = 2x + \lambda y = 0
\]
\[
\frac{\partial \mathcal{L}}{\partial y} = 2y + \lambda x = 0
\]
\[
\frac{\partial \mathcal{L}}{\partial \lambda} = xy - 6 = 0
\]
5. Solve these equations simultaneously to find the values of \( x \), \( y \), and \( \lambda \) that satisfy both the original function and the constraint.
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