Do you know the theorem that says: If is continuous then on any closed interval f attains a maximum and a minimum in that interval?
If so, then what about the interval ?
If not, then you do need to prove that theorem.
Hi. I'd really appreciate any help you can give me on this.
If is a continuous, periodic function, then prove has a maximum and minimum.
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A periodic function is where if you have any K>0, then for all , . And we need to show that a set has a maximum and minimum.
I'm thinking that we need to show it is bounded first of all. But I have no idea how I could do that. I'm just jotting down some thoughts, so I don't know whether they are relevant.
for all
This question in the textbook is under the section of the "The extreme value theorem" so I think that must be relavent.
Please help.
Regards,
Joel.
Thank you, Plato.
I think I mentioned that theorem in my post. If I'm not mistaken that is the extreme value theorem (at least that's what is written in a textbook - not sure if that is the universal name for it).
Over the interval [0, K], I'm not sure. Over that interval, does ? and ? Don't know where to go with this. *embarrassed*
Yes, I realised and edited my mistake just before your post. My concern is that these max and minimums don't hold. (I know they do, but I don't understand why)
A function is periodic if there exists a K>0 such that for all real x,
We need to show that for all real x,
But we only considered the period [0,K]. There has to be a number greater than K in the set of real numbers. What if k=98. Then consider the interval [-7,K+99]
which could be greater than . :S
Apologies if I'm going off track. I'm just trying to understand this.