FM5002-HW1-1.25.12

Sec 1 2 2 sec 1 2 tan 1 x 0 y 0 tan 1 4 sec 1 2 tan

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Unformatted text preview: .nb 72 x2 2 y2 . Note : the graph of z ethe graph of z a . Compute f x, y is the upper half of the ellipsoid x2 f x, y . The north pole appears on the graph of z 2 y2 z2 f x, y o ver x, y 2 f 0, 0 . 1 x2 b. Compute f ’ 0, 0 . c. Compute H f 0, 0 2 x2 d . Compute f ’’ 0, 0 2 x2 . . e. Compute Qf ’’ 0,0 f. Show Qf ’’ x, y is negative definite. 0,0 f a. x, y . x x x2 49 2y f , , so x 2 y2 x2 49 f 0, 0 0, 0 . 2 y2 b. Again , this is 0 0 . c. We compute : 2 2 x 2 49 x , 2y x2 x2 49 2 2x y 49 2 y2 32 , y2 49 x2 xy 32 0 1 7 d . Again , this is 0 Qf ’’ 0,0 2 2 y2 g y2 7 x 0, y 0 . 0 2 7 . 0 2 7 . e. We compute the bilinear form by Qf ’’ f. x2 2 . 0. x2 49 Therefore, H f 0, 0 7 x 0, y 0 , 1 7 1 x2 x 0, y 0 2 y2 2 2 y2 g g 4 y2 g 2 1 2 32 2 g xy 2 x2 g 0,0 x, y 1 7 xy 0 0 2 7 x y 2 y2 is a sum of a square and twice a square, and is therefore non 1 x2 2 y2 is non p ositive, or negative definite. x, y 7 1 7 x2 72. The lower half would b 2 y2 . n egative. Therefore, 0, 0 at 0, 0, 7 . FM5002− HW1− 1.25.12− 2.nb 3 x2 0037 5. Let Q x, y 3 y2. Let f x, y 4 xy 1 Q x, y . Let M H f 0, 0 2 x2 a. Show that (0, 0) is a critical p...
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