# absolute maximum in [a,b)

• Dec 13th 2011, 05:43 PM
alphabeta89
absolute maximum in [a,b)
Suppose that the function $f$ is continuous on $[a,b)$ and the limit $L=\lim_{x\rightarrow{b^-}}{f(x)}$ exists.
(i) Prove that if there is an $x_0\in{[a,b)}$ such that $f(x_0)>L$, then $f$ has an absolute maximum in $[a,b)$.
How to approach this question?

(ii) If there is an $x_1\in{[a,b)}$ such that $f(x_1)=L$, does $f$ necessarily have an absolute maximum in $[a,b)$?
• Dec 13th 2011, 09:19 PM
Jose27
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 $s>0$ such that if $0<|x-b| then $f(x)-L\leq |f(x)-L|< f(x_0)-L$ so that $f(x) on $(b-s,b)$. From here it should be obvious.

(ii) If we define $f(b)=L$ then $f$ is continous on $[a,b]$, by the extreme value theorem we have an absolute maximum $M$, then $M\geq L$. Consider the cases $M=L$, $M>L$.
• Dec 13th 2011, 10:50 PM
CaptainBlack
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 $f(b)=L$ then $f$ is continous on $[a,b]$, by the extreme value theorem we have an absolute maximum $M$, then $M\geq L$. Consider the cases $M=L$, $M>L$.

This method can also be adapted to do part (i) thus saving work by using the same idea for both.

CB
• Dec 14th 2011, 12:03 AM
alphabeta89
Re: absolute maximum in [a,b)
(Rofl)