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mthsc206-fall-2010-notes-15.7

# mthsc206-fall-2010-notes-15.7 - 15.7 Maximum and Minimum...

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Unformatted text preview: 15.7 Maximum and Minimum Values Definition 15.7.1 We say that f ( x,y ) has a relative (or local) maximum at ( x ,y ) if there is some open disk D r ( x ,y ) where f ( x ,y ) ≥ f ( x,y ) for all points ( x,y ) ∈ D r ( x ,y ) . We say that f ( x,y ) has a relative (or local) minimum at ( x ,y ) if there is some open disk D r ( x ,y ) where f ( x ,y ) ≤ f ( x,y ) for all points ( x,y ) ∈ D r ( x ,y ) . Definition 15.7.2 We say that f has an absolute maximum at ( x ,y ) if f ( x ,y ) ≥ f ( x,y ) for all ( x,y ) in the domain of f . We say that f has an absolute minimum at ( x ,y ) if f ( x ,y ) ≤ f ( x,y ) for all ( x,y ) in the domain of f . Question: Is every absolute extrema a relative extrema? Support your answer. Example 15.7.1 Let f ( x,y ) = 4- x 2- y 2 . Are there any local extrema? any absolute extrema? What is true of the tangent plane when there is a relative extrema?...
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mthsc206-fall-2010-notes-15.7 - 15.7 Maximum and Minimum...

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