Page 341 - A Course in Linear Algebra with Applications

9.2: Quadratic Forms 325

For example, the function f(x, y) = x 2 — y 2 has a saddle
point at the origin, as shown in the diagram.

The problem is to devise a test which can distinguish
local maxima and minima from saddle points. Such a test is
furnished by the criterion for a quadratic form to be positive
definite, negative definite or indefinite.
For simplicity we assume that / is a function of two vari-
ables x and y. Assume further that / and its partial deriva-
tives of degree at most three are continuous inside a region R
of the plane, and that (xo, yo) is a critical point of / in R.
Apply Taylor's Theorem to the function / at the point
(x 0, y 0), keeping in mind that (x 0,y 0) = 0 = {x 0, y 0). If h
f x
f y
and k are sufficiently small, then f(xo + h, yo + k) — f(xo, yo)
equals

2 2
-^{h f xx(xo, yo) + 2hkf xy(x 0, yo) + k f yy(x 0, y 0)) + S :

here S is a remainder term which is a polynomial of degree 3
or higher in h and k. Write a = f xx(x Q, y Q), b = f xy(x Q, y 0)
and c = f yy(xo, y 0); then

2
+ + k)- /(xo, yo) = -^{ah 2 + 2bhk + ck ) + S.
f(x 0 h,y 0
Here S is small compared to the other terms of the sum if h
and k are small.