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4 Cards in this Set

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The method of Lagrange Multipliers

Let x₀∈U and g(x₀)=c, and set S be the level set for g with value c (this is the set of points x∈Rⁿ satisfying g(x)=c) assume ∀g(x₀)≠0. If f|S, (f restricted to S) has a local maximum or minimum on S at x₀, then there is a real number λ such that ∀f(x₀) =λ∀g(x₀)

Terms for point where ∀f(x₀)=λ∀g(x₀)

Critical point

Lagrange Multiplier Strategy for Finding Absolute Maxima and Minima on Regions with Boundary

1. Locate all critical points of f in U


2. Use the method of Lagrange multiplier to locate all the critical points of f|dU


3. Compute the values of f at all these critical points


4. Select the largest and the smallest

Definition of smooth for Lagrange multipliers

We say that dU is smooth if dU is the level set of a smooth function g whose gradient ∀g never vanishes (ie, ∀g≠0)