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# 14-7 - Math 200 Spring 2010 Handout#15 Section 14-7...

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Math 200, Spring 2010 Handout #15 Section 14-7 Optimization in Several Variables Definition: Local Extreme Values A function ( ) , f x y has a local extremum at ( ) , P a b = if there exists an open disk ( ) , D P r such that: * Local maximum : ( ) ( ) , , f x y f a b for all ( ) ( ) , , x y D P r * Local minimum : ( ) ( ) , , f x y f a b for all ( ) ( ) , , x y D P r Definition: Critical Point (or Stationary Point) A point ( ) , P a b = in the domain of ( ) , f x y is called a critical point if: * ( ) , 0 x f a b = and ( ) ( ) , 0 , 0 y f a b f a b = = , or * At least one of the partial derivatives ( ) , x f a b , ( ) , y f a b does not exist. 1. Find all critical points of 2 2 ( , ) 4 4 2 f x y x y x y = + + Fermat’s Theorem in Several Variables If ( ) , f x y has a local minimum or maximum at ( , ) P a b = , then ( , ) P a b = is a critical point of ( ) , f x y Note that the converse of this theorem in not true. Not all critical points give rise to minima or maxima. 2. Find the extreme values of 2 2 ( , ) f x y x y = .

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Theorem: Second Derivatives Test Suppose the 2 nd partial derivatives of f are continuous on an open disk with center ( , ) a b , and suppose that ( , )
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14-7 - Math 200 Spring 2010 Handout#15 Section 14-7...

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