Page 342 - A Course in Linear Algebra with Applications

326 Chapter Nine: Advanced Topics

2
Let q — ax 2 + 2bxy + cy . If q is negative definite, then
/(XQ + h, yo + k) < f(x 0, yo) when h and k are small and P is
a local maximum. On the other hand, if q is positive definite,
then P is a local minimum since f(xo + h, yo + k) > /(XQ, yo)
for sufficiently small h and k. Finally, should q be indefinite,
the expression f(xo + h, yo+k)—f(xo, yo) can be both positive
and negative, so P is neither a local maximum nor a local
minimum, but a saddle point.
Thus the crucial quadratic form which provides us with
a test for P to be a local maximum or minimum arises from
the matrix

TT / Jxx Jxy \
\ Jxy JyyJ

If the matrix H(xo, yo) is positive definite or negative definite,
then / will have a local minimum or local maximum respec-
tively at P. If, however, H(x 0, yo) is indefinite, then P will
/
be a saddle point of . Combining this result with 9.2.3, we
obtain

Theorem 9.2.5
Let f be a function of x and y and assume that f and its
partial derivatives of order < 3 are continuous in some region
containing the critical point P(xo, yo)- Let D — fxxfyy — f xy :
f
(a) If D(x Q, yo) > 0 and xx{x Q, y 0) < 0, then P is a
local maximum of f;
f
(b) If D(XQ, yo) > 0 and xx(x 0, y 0) > 0, then P is a
local minimum of f;
(c) If D < 0, then P is a saddle point of f.

The argument just given for a function of two variables
can be applied to a function / of n variables x\,..., x n. The
relevant quadratic form in this case is obtained from the