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Notes for 12.8,12.9 - Second Partials Test Let f have...

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Calculus III – Sections 12.8 and 12.9 – Extrema of Functions of Two Variables Extreme Value Theorem Let f be a continuous function of two variables x and y defined on a closed bounded region R in the xy -plane. 1) There is at least one point in R where f takes on a minimum value. 2) There is at least one point in R where f takes on a maximum value. Furthermore, the absolute maximum and absolute minimum values must occur at either a critical point or along the boundary of the region R . Definition of a Critical Point Let f be defined on an open region R containing ( 29 0 0 , y x . The point ( 29 0 0 , y x is a critical point of f if one of the following is true. 1) 0 ) ( 0 , 0 = y x f x and 0 ) ( 0 , 0 = y x f y 2) ) ( 0 , 0 y x f x or ) ( 0 , 0 y x f y does not exist. Relative Extrema Occur Only at Critical Points
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Unformatted text preview: Second Partials Test Let f have continuous second partial derivatives on an open region containing a point ( a,b ) for which ) , ( = b a f x and ) , ( = b a f y . To test for relative extrema of f, consider the quantity [ ] 2 ) , ( ) , ( ) , ( b a f b a f b a f d xy yy xx-= . 1) If d > 0 and ) , ( b a f xx , then f has a relative minimum at ( a,b ). 2) If d > 0 and ) , ( < b a f xx , then f has a relative maximum at ( a,b ). 3) If d < 0, then ( a, b, f (a,b) ) is a saddle point . 4) The test is inconclusive if d = 0. #10 #25 #54 Section 12.9 #6 #20...
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