Module 13 : Maxima, Minima and Saddle Points, Constrained maxima and minima
Lecture 37 : Maxima and Minima [Section 37.1]
minimum at , then
As in the case of one variable we make the following definition.
37.1.4
Definition:
An interior point is called a critical point of if
(i)
Either, both and exist with
(ii)
Or, one or both of do not exist.
As a consequence of theorem 37.1.3, we have the following:
37.1.5
Corollary:
For a function local maxima/minima can occur only at critical point or boundary points of .
37.1.6
Example:
Let
Then, is the only critical point of , as both and do not exist at and at every other point both exist are non-zero. The point is obviously a point of local minimum, with local minimum being . Thus, the condition in theorem is only necessary, not sufficient.
, then 
is called a critical point of 
if 
and 
exist with 
do not exist.
. 
is the only critical point of 
, as both 
and 
do not exist at 
and at every other point both exist are non-zero. The point 
is obviously a point of local minimum, with local minimum being 
. Thus, the condition in theorem 
is only necessary, not sufficient. 
