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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online