Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive. Maximize f(x, y) = 7x + 7xy +y Constraint: 7x + y = 700
Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive. Maximize f(x, y) = 7x + 7xy +y Constraint: 7x + y = 700
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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Question
13.10 #1
![**Title: Using Lagrange Multipliers to Find Extrema**
**Objective:**
Use Lagrange multipliers to find the indicated extrema, assuming that \(x\) and \(y\) are positive.
**Problem Statement:**
- **Maximize:**
\[
f(x, y) = 7x + 7xy + y
\]
- **Subject to the Constraint:**
\[
7x + y = 700
\]
**Solution:**
1. Define the function to be maximized:
\[
f(x, y) = 7x + 7xy + y
\]
2. Define the constraint:
\[
g(x, y) = 7x + y - 700 = 0
\]
3. Set up the Lagrangian:
\[
\mathcal{L}(x, y, \lambda) = 7x + 7xy + y + \lambda (700 - 7x - y)
\]
4. Calculate the partial derivatives and solve the system of equations:
\[
\frac{\partial \mathcal{L}}{\partial x} = 0, \quad \frac{\partial \mathcal{L}}{\partial y} = 0, \quad \frac{\partial \mathcal{L}}{\partial \lambda} = 0
\]
5. Solve for \(x\), \(y\), and \(\lambda\).
6. Substitute \(x\) and \(y\) back into \(f(x, y)\) to find the maximum value:
\[
f( \text{____} , \text{____} ) = \text{Maximum Value}
\]
**Additional Resources:**
- **Need Help?**
- Click "Read It" for detailed guidance.
- Click "Talk to a Tutor" for personalized assistance.
**Note:** When working through this problem, ensure all steps are checked for algebraic accuracy, and verify solutions satisfy both the original function and the constraint.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadf70025-15e9-4d93-968e-9d86049b61bd%2F46cf4f2e-dc09-482f-917e-873d9fc530a1%2F57th8ym_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Using Lagrange Multipliers to Find Extrema**
**Objective:**
Use Lagrange multipliers to find the indicated extrema, assuming that \(x\) and \(y\) are positive.
**Problem Statement:**
- **Maximize:**
\[
f(x, y) = 7x + 7xy + y
\]
- **Subject to the Constraint:**
\[
7x + y = 700
\]
**Solution:**
1. Define the function to be maximized:
\[
f(x, y) = 7x + 7xy + y
\]
2. Define the constraint:
\[
g(x, y) = 7x + y - 700 = 0
\]
3. Set up the Lagrangian:
\[
\mathcal{L}(x, y, \lambda) = 7x + 7xy + y + \lambda (700 - 7x - y)
\]
4. Calculate the partial derivatives and solve the system of equations:
\[
\frac{\partial \mathcal{L}}{\partial x} = 0, \quad \frac{\partial \mathcal{L}}{\partial y} = 0, \quad \frac{\partial \mathcal{L}}{\partial \lambda} = 0
\]
5. Solve for \(x\), \(y\), and \(\lambda\).
6. Substitute \(x\) and \(y\) back into \(f(x, y)\) to find the maximum value:
\[
f( \text{____} , \text{____} ) = \text{Maximum Value}
\]
**Additional Resources:**
- **Need Help?**
- Click "Read It" for detailed guidance.
- Click "Talk to a Tutor" for personalized assistance.
**Note:** When working through this problem, ensure all steps are checked for algebraic accuracy, and verify solutions satisfy both the original function and the constraint.
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