Module 13 : Maxima, Minima and Saddle Points, Constrained maxima and minima
Lecture 39 : Absolute maxima / minima [Section 39.1]
39.1.4
Note:
To find the absolute maximum and the absolute minimum of a function on , we compare the values of at the critical points of in and the absolute maximum and the absolute minimum of the restriction of to the boundary of . The latter can often be found by reducing it to a one variable problem.
39.1.5
Examples:
(i)
Suppose
and is given by
Since is a closed bounded set and is a continuous function, it has both, absolute maximum and absolute minimum in For both the partial derivatives exist everywhere and
and the absolute minimum 
of a function 
on 
, we compare the values of 
at the critical points of 
in 
and the absolute maximum and the absolute minimum of the restriction of 
to the boundary of 
. The latter can often be found by reducing it to a one variable problem. 
is given by 
is a closed bounded set and 
is a continuous function, it has both, absolute maximum and absolute minimum in 
both the partial derivatives exist everywhere and 
is a boundary point if 
