Use Lagrange multipliers to find the absolute maximum and minimum values of the function z = f(x, y) =1-xy³ subject to the constraint equation x² +12y² = 16.
Use Lagrange multipliers to find the absolute maximum and minimum values of the function z = f(x, y) =1-xy³ subject to the constraint equation x² +12y² = 16.
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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![**Finding Absolute Maximum and Minimum Values Using Lagrange Multipliers**
---
**Objective:**
Use Lagrange multipliers to determine the absolute maximum and minimum values of the function
\[ z = f(x, y) = 1 - xy^3 \]
subject to the constraint equation
\[ x^2 + 12y^2 = 16. \]
---
**Instructions:**
1. **Understand the Problem:**
- You are given a function \( z = 1 - xy^3 \) to optimize (i.e., find its maximum and minimum values).
- There is a constraint on the variables \( x \) and \( y \) defined by the equation \( x^2 + 12y^2 = 16 \).
2. **Formulate Using Lagrange Multipliers:**
- Introduce a Lagrange multiplier \( \lambda \) (lambda).
- Set up the Lagrangian function:
\[ \mathcal{L}(x, y, \lambda) = 1 - xy^3 + \lambda (x^2 + 12y^2 - 16) \]
3. **Solve the System of Equations:**
- Compute the partial derivatives of \( \mathcal{L} \) with respect to \( x \), \( y \), and \( \lambda \):
\[ \frac{\partial \mathcal{L}}{\partial x} = -y^3 + 2\lambda x = 0 \]
\[ \frac{\partial \mathcal{L}}{\partial y} = -3xy^2 + 24\lambda y = 0 \]
\[ \frac{\partial \mathcal{L}}{\partial \lambda} = x^2 + 12y^2 - 16 = 0 \]
- Solve this system of equations to find the critical points.
4. **Determine Maximum and Minimum Values:**
- Substitute the critical points back into the original function \( z = 1 - xy^3 \) to find the corresponding values.
- Identify which points give the absolute maximum and minimum values.
---
**Example Solution:**
1. From \( \frac{\partial \mathcal{L}}{\partial x} = -y^3 + 2\lambda x = 0 \):
\[ \lambda = \frac](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F41f9fc9a-253f-4b12-9d0d-9da0ab6d6da8%2Fab1bb3da-7d3b-4f9d-8947-634f1fad07e8%2Fdx7w9y_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding Absolute Maximum and Minimum Values Using Lagrange Multipliers**
---
**Objective:**
Use Lagrange multipliers to determine the absolute maximum and minimum values of the function
\[ z = f(x, y) = 1 - xy^3 \]
subject to the constraint equation
\[ x^2 + 12y^2 = 16. \]
---
**Instructions:**
1. **Understand the Problem:**
- You are given a function \( z = 1 - xy^3 \) to optimize (i.e., find its maximum and minimum values).
- There is a constraint on the variables \( x \) and \( y \) defined by the equation \( x^2 + 12y^2 = 16 \).
2. **Formulate Using Lagrange Multipliers:**
- Introduce a Lagrange multiplier \( \lambda \) (lambda).
- Set up the Lagrangian function:
\[ \mathcal{L}(x, y, \lambda) = 1 - xy^3 + \lambda (x^2 + 12y^2 - 16) \]
3. **Solve the System of Equations:**
- Compute the partial derivatives of \( \mathcal{L} \) with respect to \( x \), \( y \), and \( \lambda \):
\[ \frac{\partial \mathcal{L}}{\partial x} = -y^3 + 2\lambda x = 0 \]
\[ \frac{\partial \mathcal{L}}{\partial y} = -3xy^2 + 24\lambda y = 0 \]
\[ \frac{\partial \mathcal{L}}{\partial \lambda} = x^2 + 12y^2 - 16 = 0 \]
- Solve this system of equations to find the critical points.
4. **Determine Maximum and Minimum Values:**
- Substitute the critical points back into the original function \( z = 1 - xy^3 \) to find the corresponding values.
- Identify which points give the absolute maximum and minimum values.
---
**Example Solution:**
1. From \( \frac{\partial \mathcal{L}}{\partial x} = -y^3 + 2\lambda x = 0 \):
\[ \lambda = \frac
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