# Thread:

1. ## 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 $[a,b)$?

2. ## 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 $0<|x-b|<s$ then $f(x)-L\leq |f(x)-L|< f(x_0)-L$ so that $f(x)<f(x_0)$ 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 $M=L$, $M>L$.

3. ## Re: absolute maximum in [a,b)

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
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