Math 22 Extrema of Functions of Two Variables
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Relative Extrema of a Function of Two Variables
Let be a function defined on a region containing . The function has a relative maximum at when there is a circular region centered at such that for all in .
The function has a relative minimum at when there is a circular region centered at such that for all in .
First-Partials Test for Relative Extrema
If has a relative extremum at on an open region in the xy-plane, and the first partial derivatives of exist in , then and
Example: Find relative extrema of:
|Consider: , so|
|and: , so|
|Therefore, there is a relative extrema at|
The Second-Partials Test for Relative Extrema
Let have continuous second partial derivatives on an open region containing for which and Then, consider
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