14-8 - Math 200, Spring 2010 Handout #16 Section 14-8...

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Unformatted text preview: Math 200, Spring 2010 Handout #16 Section 14-8 Lagrange Multipliers: Optimizing with a Constraint Theorem: Lagrange Multipliers Assume that ( ) , f x y and ( ) , g x y are differentiable functions. If ( ) , f x y has a local minimum or maximum on the constraint curve ( ) , g x y k = at ( ) , P a b = , and if ( ) , g a b , then there is a scalar such that ( ) ( ) , , f a b g a b = [The gradient vectors are parallel.] The real number is called the Lagrange multiplier . The vector equation is called the Lagrange condition which, equating components, yields Lagrange equations: ( ) ( ) , , x x f a b g a b = and ( ) ( ) , , y y f a b g a b = The point ( ) , P a b = is called a critical point and ( ) , f a b is called a critical value for the optimization problem. The Lagrange Multiplier Theorem is valid in any number of variables. For example, if ( ) , , f x y z has a local minimum or maximum on the constraint surface ( ) , , g x y z k = at the point ( ) , , P a b c...
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14-8 - Math 200, Spring 2010 Handout #16 Section 14-8...

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