20e-lecture10

20e-lecture10 - Lecture 10 - Wednesday April 22nd 10.1...

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Lecture 10 - Wednesday April 22nd [email protected] 10.1 Constrained Optimization A set U R n is called closed if the complement R n \ U is open. For example, the set { ( x,y ) R 2 : x 2 + y 2 1 } is a closed set since { ( x,y ) R 2 : x 2 + y 2 > 1 } – its complement – is open. Let f : R n R be a function and let U be a closed subset of the domain of f so that f is continuous on U . In this section we use the second derivative methods we have so far developed to find global maxima and minima of f on U . Definition. Let f : R n R be a function and let U be a subset of the domain of f . A point x U is a global or absolute maximum of f on U if f ( x ) f ( y ) for all y U , and a global or absolute minimum of f on U if f ( x ) f ( y ). The following theorem guarantees the existence of absolute minima and maxima on closed sets. Theorem. Let f : R n R be a function and let U be a closed subset of the domain of f so that f is continuous on U . Then f has a global maximum and a global minimum on U . The global extremes guaranteed by this theorem need not be unique – f may have many global maxima and minima on U . With the assumption f C 2 ( U ), we can use the second derivative test on U to find these global extremes.
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This note was uploaded on 02/12/2010 for the course MATH 20E taught by Professor Enright during the Spring '07 term at UCSD.

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20e-lecture10 - Lecture 10 - Wednesday April 22nd 10.1...

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