Module 4 : Local / Global Maximum / Minimum and Curve Sketching
Lecture 11 : Absolute Maximum / Minimum [Section 11.1]
(iii)
A function can have a local (global) maximum / minimum at a point without being differentiable or
even continuous at For example, let
Then has global minimum at , butis not continuous at .
11.1.7
Examples:
(i) Let us analyze absolute maximum/ minimum of the function
Though the function is continuous, its domain is not a closed and bounded interval. Thus, we are not sure that the function has maximum or minimum. However, is differentiable and
gives in the domain of . Clearly,
Thus, is strictly increasing in every interval of the form , and is strictly decreasing in every interval of the form for every . Since is continuous at this implies that
can have a local (global) maximum / minimum at a point 
without being differentiable or
For example, let 
has global minimum at 
, but
is continuous, its domain is not a closed and bounded interval. Thus, we are not sure that the function has maximum or minimum. However, 
is differentiable and 
in the domain of 
. Clearly, 
is strictly increasing in every interval of the form 
, and is strictly decreasing in every interval of the form 
for every 
. Since 
is continuous at 
this implies that 
has global minimum at 
