Module 13 : Maxima, Minima and Saddle Points, Constrained maxima and minima
Lecture 39 : Absolute maxima / minima [Section 39.1]
39.1.3
Theorem:
Let
(i)
Let assume its absolute maximum at a point Then, either at is a
boundary point of or is a critical
point of in .
(ii)
Let assume its absolute minimum at a point Then, either is a boundary point of or is a critical
point of in .
Proof
Suppose is not a boundary point of . Then, must assume its maximum at some interior point of Thus, is an interior point of If both and exist, then by theorem 37.1.3. In any case, is a critical point of . Similar arguments hold for .
assume its absolute maximum at a point 
Then, either at 
is a
boundary point of 
or is a critical 
in 
. 
its absolute minimum at a point 
Then, either 
is a boundary point of 
or is a critical 
in 
. 
. Then, 
must assume its maximum at some interior point of 
Thus, 
If both 
and 
exist, then 
by theorem 37.1.3. In any case, 
. Similar arguments hold for 
