absolute maximum in [a,b)
Suppose that the function 
is continuous on )
and the limit })
exists.
(i) Prove that if there is an })
such that >L)
, then has an absolute maximum in .
How to approach this question?
(ii) If there is an })
such that =L)
, does necessarily have an absolute maximum in ?
Re: absolute maximum in [a,b)
I'm assuming L is a finite real number, if that's the case then
(i) By the definition of limit there is an 
such that if 
then -L\leq |f(x)-L|< f(x_0)-L)
so that <f(x_0))
on )
. From here it should be obvious.
(ii) If we define =L)
then is continous on ![[a,b]](http://latex.codecogs.com/png.latex?[a,b])
, by the extreme value theorem we have an absolute maximum 
, then 
. Consider the cases 
, 
.
Re: absolute maximum in [a,b)
Quote:
Originally Posted by
Jose27 
I'm assuming L is a finite real number, if that's the case then
(ii) If we define
then
is continous on
, by the extreme value theorem we have an absolute maximum
, then
. Consider the cases
,
.
This method can also be adapted to do part (i) thus saving work by using the same idea for both.
CB
Re: absolute maximum in [a,b)
