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 , then has an absolute maximum in .

How to approach this question?

(ii) If there is an such that , 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 so that on . From here it should be obvious.

(ii) If we define then is continous on , 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)