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Unformatted text preview: Week 9 Optimization Local extrema Our main goal in optimization of 2-variable 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 (non-existing partial derivatives) much in this course, well focus on the type where the partial derivatives are zero. Compare this definition to that for 1-variable 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 1-variable functions, the reason for considering critical points of 2-variable 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