Suppose that
has continuous second partial derivatives in a neighbourhood of
and that
at
Define
at
Form the discriminant
If
thenis a saddle point.
If
then
has a local minimum at
If
and a local maximum atif
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Suppose thathas continuous second partial derivatives in a neighbourhood of and thatat
DefineatForm the discriminantIfthenis a saddle point.
Ifthenhas a local minimum at
Ifand a local maximum atif