14209an8.7-9 - a b is a critical point f x a b)= 0 and f y...

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c Kendra Kilmer April 23, 2009 Section 8.3 - Maxima and Minima Example 1: Graph f ( x , y ) = x 2 + 4 y 2 a) Draw the plane y = 2. b) Draw the plane x = 2 Definition: Let z = f ( x , y ) be a function of two variables. The value f ( a , b ) is a) a local maximum if there exists a circular region in the domain of f with ( a , b ) as the center such that f ( a , b ) f ( x , y ) for all ( x , y ) in the region. b) a local minimum if there exists a circular region in the domain of f with ( a , b ) as the center such that f ( a , b ) f ( x , y ) for all ( x , y ) in the region. Definition: A critical point of z = f ( x , y ) is where both first partial derivatives are zero: 7
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c Kendra Kilmer April 23, 2009 Example 2: Determine the critical points of f ( x , y ) = 2 x 2 + 3 y 2 + 2 xy + 4 x - 8 y + 3 Example 3: Graph the following curves: a) f ( x , y ) = 10 - x 2 - 4 y 2 b) f ( x , y ) = 3 + 4 x 2 + 2 y 2 c) f ( x , y ) = x 2 - 6 y 2 Second Derivative Test for Local Extrema Let z = f ( x , y ) be a function of two variables such that f xx ( x , y ) , f yy ( x , y ) , and f xy ( x , y ) exist for every point inside a circle centered at ( a
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Unformatted text preview: ( a , b ) is a critical point ( f x ( a , b )= 0 and f y ( a , b )= 0) we define a number D to be D = f xx ( a , b ) · f yy ( a , b )-[ f xy ( a , b )] 2 Then, 1. If D ( a , b ) > 0 and f xx ( a , b ) < 0 then f has a at ( a , b ) . 2. If D ( a , b ) > 0 and f xx ( a , b ) > 0 then f has a at ( a , b ) . 3. If D ( a , b ) < 0 then f has a at ( a , b ) . 4. If D ( a , b ) = 0 then no conclusion can be made about f ( a , b ) . 8 c ± Kendra Kilmer April 23, 2009 Example 4: Find all critical points and determine whether each is a saddle point, local max, or local min. a) f ( x , y ) =-x 2-y 2 + 6 x + 8 y-21 b) f ( x , y ) = x 3 + y 3-6 xy Section 8.3 Homework Problems: 3,7,9,13,19,21,29 9...
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