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Example
Created on 2024-06-20Asked by Samuel Johnson (Solvelet student)
Use the method of Lagrange multipliers to find the maximum value of f(x,y)=xy f(x,y) = xy subject to the constraint x2+y2=1 x^2 + y^2 = 1 .

Solution

To find the maximum value of f(x,y)=xy f(x,y) = xy subject to the constraint x2+y2=1 x^2 + y^2 = 1 using the method of Lagrange multipliers, follow these steps: 1. Define the Lagrangian function: L(x,y,λ)=xy+λ(1−x2−y2). \mathcal{L}(x, y, \lambda) = xy + \lambda (1 - x^2 - y^2). 2. Take the partial derivatives and set them to zero: ∂L∂x=y−2λx=0, \frac{\partial \mathcal{L}}{\partial x} = y - 2\lambda x = 0, ∂L∂y=x−2λy=0, \frac{\partial \mathcal{L}}{\partial y} = x - 2\lambda y = 0, ∂L∂λ=1−x2−y2=0. \frac{\partial \mathcal{L}}{\partial \lambda} = 1 - x^2 - y^2 = 0. 3. Solve the system of equations: y=2λx  ⟹  x=2λy. y = 2\lambda x \implies x = 2\lambda y. Substitute into x2+y2=1 x^2 + y^2 = 1 : (2λy)2+y2=1  ⟹  4λ2y2+y2=1  ⟹  y2(4λ2+1)=1. (2\lambda y)^2 + y^2 = 1 \implies 4\lambda^2 y^2 + y^2 = 1 \implies y^2(4\lambda^2 + 1) = 1. y=±14λ2+1  ⟹  x=±2λ4λ2+1. y = \pm \frac{1}{\sqrt{4\lambda^2 + 1}} \implies x = \pm \frac{2\lambda}{\sqrt{4\lambda^2 + 1}}. 4. Find λ \lambda by substituting x x and y y : 2λ4λ2+1=2λ⋅±14λ2+1  ⟹  λ=±12. \frac{2\lambda}{\sqrt{4\lambda^2 + 1}} = 2\lambda \cdot \pm \frac{1}{\sqrt{4\lambda^2 + 1}} \implies \lambda = \pm \frac{1}{2}. 5. Evaluate f(x,y) f(x, y) : x=±12,y=±12  ⟹  f(x,y)=(12)(12)=12. x = \frac{\pm 1}{\sqrt{2}}, \quad y = \frac{\pm 1}{\sqrt{2}} \implies f(x, y) = \left( \frac{1}{\sqrt{2}} \right) \left( \frac{1}{\sqrt{2}} \right) = \frac{1}{2}. Therefore, the maximum value of f(x,y) f(x, y) is 12 \frac{1}{2} . Solved on Solvelet with Basic AI Model
Some of the related questions asked by Camila Clark on Solvelet
1. Apply the method of Lagrange multipliers to optimize a function subject to equality constraints.2. Use the simplex method to solve a linear programming problem with inequality constraints.
DefinitionMathematical methods that seeks to find the best result to a problem by minimizing or maximizing a function Methods include calculus-based methods (e.g. Lagrange multipliers), linear programming and heuristic methods (e.g. genetic algorithms). Case study : Use of Lagrange multipliers for finding extrema for f(x,y)=xy under the constraint x2+y2=1.
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