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Unformatted text preview: Week 9 Optimization Local extrema Our main goal in optimization of 2variable functions will be to find their local extrema: f ( x,y ) has a local maximum at ( a,b ) if f ( x,y ) f ( a,b ) for all ( x,y ) around ( a,b ). f ( x,y ) has a local minimum at ( a,b ) if f ( x,y ) f ( a,b ) for all ( x,y ) around ( a,b ). Local means it is the largest/smallest value in a neighborhood around ( a,b ); there could be larger/smaller values further away. The largest/smallest value overall is called the abso lute maximum/minimum, but we wont cover those in this course. To find local extrema, we wont really use these definitions directly, we will use critical points. Critical Points The point ( a,b ) is called a critical point of f ( x,y ) if f x ( a,b ) = 0 and f y ( a,b ) = 0, or if f x or f y does not exist at ( a,b ). Actually, we wont see the second type (nonexisting partial derivatives) much in this course, well focus on the type where the partial derivatives are zero. Compare this definition to that for 1variable functions f ( x ): they have a critical point at x = a if f ( a ) = 0 or if f ( a ) does not exist. Just like for 1variable functions, the reason for considering critical points of 2variable functions is this: Fact: If f ( x,y ) has a local maximum or minimum at ( a,b ), then ( a,b ) is a critical point....
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 Fall '10
 MalabikaPramanik
 Calculus

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