Maximizing a Continuous Function on A: Analysis Homework Help

[Janez25]

Homework Statement

Let A= [0,1). Find a continuous function g: from A to R that does not attain a maximum value.

Homework Equations

The Attempt at a Solution

I believe that g(x)=x or g(x)=e^x represent such a function, but I do not know how to use the IVT to prove that either of them work. Please help! I need as much information as possible without completely giving away the answer.

 

[tiny-tim]

Welcome to PF!

Hi Janez25! Welcome to PF!

… I believe that g(x)=x or g(x)=e^x represent such a function, but I do not know how to use the IVT to prove that either of them work.

Yes, either will do …

now, what property (of either) means that the maximum is never reached?

(i don't think you need the IVT for this)

 

[Janez25]

I am not sure what property I should use. I also am not sure how to show that either function does not attain a maximum value on the interval [0,1) formally.

 

[tiny-tim]

I am not sure what property I should use. I also am not sure how to show that either function does not attain a maximum value on the interval [0,1) formally.

Hint: asssume it does attain its maximum, or any local maximum, at x = a in [0,1) …

is that possible for x or for e^x?

what property would f(x) have if that isn't possible?

 

[Janez25]

Thanks for your help!