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wn words, describe how to apply Langranges

wn words, describe how to apply Langranges

Advanced Engineering Mathematics
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 17: In your own words, describe how to apply Lagrange's Method.** --- Lagrange's Method, commonly used for finding the maxima and minima of functions subject to constraints, involves several steps: 1. **Identify the Objective Function:** - Determine the function \( f(x, y, \ldots) \) that you want to optimize (maximize or minimize). 2. **Identify the Constraint:** - Determine the constraint equation \( g(x, y, \ldots) = 0 \) that must be satisfied. 3. **Formulate the Lagrangian:** - Construct the Lagrangian function \( \mathcal{L} = f(x, y, \ldots) + \lambda(g(x, y, \ldots)) \), where \( \lambda \) is the Lagrange multiplier. 4. **Take Partial Derivatives:** - Calculate the partial derivatives of the Lagrangian with respect to each variable and the Lagrange multiplier \( \lambda \). 5. **Set the Equations to Zero:** - Solve the system of equations formed by setting each partial derivative equal to zero. 6. **Solve the System:** - Use the resulting equations to find the values of the variables and \( \lambda \) that satisfy both the objective and constraint equations. 7. **Interpret the Solution:** - Verify that the solution meets the original constraints and provides the desired optimization (maximum or minimum).
Transcribed Image Text:**Question 17: In your own words, describe how to apply Lagrange's Method.** --- Lagrange's Method, commonly used for finding the maxima and minima of functions subject to constraints, involves several steps: 1. **Identify the Objective Function:** - Determine the function \( f(x, y, \ldots) \) that you want to optimize (maximize or minimize). 2. **Identify the Constraint:** - Determine the constraint equation \( g(x, y, \ldots) = 0 \) that must be satisfied. 3. **Formulate the Lagrangian:** - Construct the Lagrangian function \( \mathcal{L} = f(x, y, \ldots) + \lambda(g(x, y, \ldots)) \), where \( \lambda \) is the Lagrange multiplier. 4. **Take Partial Derivatives:** - Calculate the partial derivatives of the Lagrangian with respect to each variable and the Lagrange multiplier \( \lambda \). 5. **Set the Equations to Zero:** - Solve the system of equations formed by setting each partial derivative equal to zero. 6. **Solve the System:** - Use the resulting equations to find the values of the variables and \( \lambda \) that satisfy both the objective and constraint equations. 7. **Interpret the Solution:** - Verify that the solution meets the original constraints and provides the desired optimization (maximum or minimum).
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