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l_notes13 - Lecture XIII TwoVariable Test Constrained...

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Lecture XIII Two-Variable Test Constrained Maximum-Minimum Problems 1 The two-variable test Recall that by the Critical Point Theorem, onlyif the gradient of a function f at P is 0 (i.e. P is a critical point for f ), can P be an extreme point of f . Let f ( x, y ) = x 3 + xy + y 3 . Then f = (3 x 2 + y ) ˆ i + ( x + 3 y 2 ) ˆ j, so if P = ( x, y ) is a critical point, then 2 2 x = 3 y , y = 3 x , 1 1 hence x + 27 x 4 = 0, so the only critical points are P 1 (0 , 0) and P 2 ( 3 , 3 ). Now we have to find out if these points are extreme points. For this we can use the following test. Theorem 1 (The two-variable test) Let f be a two-variable C 2 function, and let P be an interior critical point for f . We define the functions H 1 and H 2 , called Hessian functions in the following manner: H 1 ( P ) = f xx ( P ) , H 2 ( P ) = f xx ( P ) f xy ( P ) f yx ( P ) f yy ( P ) = f xx ( P ) f yy ( P ) f 2 xy ( P ) Then: (a) If H 1 ( P ) > 0 and H 2 ( P ) > 0 then P is a local minimum point for f .
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