Module 4 : Local / Global Maximum / Minimum and Curve Sketching
Lecture 11 : Absolute Maximum / Minimum [Section 11.1]
Since is continuous, it is bounded and hence, attains its absolute maximum and absolute minimum. Note that if and only if and or and . Thus,
Since is not differentiable at , and , for , the critical points of , are and 2. Further,
.
So, has absolute maximum at and absolute minimum at and . Since
, has local maximum at . The absolute maximum for is 6 and absolute minimum is 0.
11.1.6
Remarks:
(i)
Even if a function is continuous at and has a global (local) maximum at , it is not necessary that will be increasing on or decreasing on for some . For example, consider the function
The function is continuous at and has global maximum at But takes all values between at points arbitrarily close to Hence it is neither increasing nor decreasing in any neighborhood of the point
is continuous, it is bounded and hence, attains its absolute maximum and absolute minimum. Note that 
if and only if 
and 
or 
and 
. Thus, 
is not differentiable at 
, and 
, for 
, the critical points of 
, are 
and 2. Further, 
. 
has absolute maximum at 
and absolute minimum at 
and 
. Since 
, 
has local maximum at 
. The absolute maximum for 
is continuous at 
and has a global (local) maximum at 
, it is not necessary that 
will be increasing on 
or decreasing on 
for some 
. For example, consider the function 
and has global maximum at 
But 
takes all values between 
at points arbitrarily close to 
Hence it is neither increasing nor decreasing in any neighborhood of the point 
