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Lagrange Multipliers

Lagrange Multipliers.

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Lagrange Multipliers

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  1. LAGRANGE MULTIPLIERS By - Rahul Joshi

  2. Lagrange Multipliers: A General Definition Mathematical tool for constrained optimization of differentiable functions. Provides a strategy for finding the maximum or minimum of a function subject to constraint.

  3. Key Terms Gradient - a normal vector to a curve (in two dimensions) or a surface (in higher dimensions). Lagrange Multiplier - a constant that is required in the Lagrange function because although both gradient vectors are parallel, the directions and magnitudes of the gradient vectors are generally not equal.

  4. How to Use the Method of Lagrange Multipliers Step 1: Form the Lagrangian V (x, y, λ) = f(x, y) + λ(g(x, y)-c) - Lagrangian f(x, y) - Optimization function g(x, y) = c - Constraint function λ - Lagrange multiplier

  5. How to Use the Method of Lagrange Multipliers (Continued) Step 2: Take the gradient of the Lagrangian and set the equations equal to zero δɅ/δx = f(x, y)/8x + λ(g(x, y) - c)/8x = 0 δɅ/δy = f(x, y)/8y + λ(g(x, y) - c)/8y = 0 δΛ/δλ = g(x, y) - c = 0 Step 3: Find all critical values by solving the system Step 4: Evaluate f(x, y) at each set of values to determine maxima/minima

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