Questions about Real analysis (Rolle's theorem and the mean value theorem etc.)

In general, when a question says "f is twice differentiable at a" does that implicitly say that in fact f'(x) exists in a neighborhood/ball around a, not just at a?

http://workspace.imperial.ac.uk/mathematics/public/students/ug/exampapers/2007/M2P1-2007.PDF

How would 2(ii) be done? Is it just some smart rearrangement of the difference quotient? I started with the inverse function theorem and tried to form the difference quotient. But I couldn't rearrange it in a nice way. I thought of adding 0 in a smart way but it didn't work out.

For 3(ii), do we use Rolle's theorem or the mean value theorem? I thought of Rolle's theorem but the proof becomes wordy and I'm unsure about it. Would the MVT work here? It seems that MVT would give existence of one of the x1 and x2 straight away. But then we have to do some work to get the other one.

For 3(iii), would square root of x be a valid example?

Thanks for any help.

Reply 1

DFranklin

Original post by gangsta316

In general, when a question says "f is twice differentiable at a" does that implicitly say that in fact f'(x) exists in a neighborhood/ball around a, not just at a?Yes

url]http://workspace.imperial.ac.uk/mathematics/public/students/ug/exampapers/2007/M2P1-2007.PDF

How would 2(ii) be done? Is it just some smart rearrangement of the difference quotient? I started with the inverse function theorem and tried to form the difference quotient. But I couldn't rearrange it in a nice way. I thought of adding 0 in a smart way but it didn't work out.I don't know what you mean by the difference quotient. You need to do two things, first show that the inverse has a 2nd derivative (use the inverse function theorem). Second you need to show the 2nd derivative of the inverse has the formula given. For which, I would use the first part with a particular choice of g.

For 3(ii), do we use Rolle's theorem or the mean value theorem? I thought of Rolle's theorem but the proof becomes wordy and I'm unsure about it. Would the MVT work here? It seems that MVT would give existence of one of the x1 and x2 straight away. But then we have to do some work to get the other one.

I would say if the first part asks you to prove Rolle, you would need to prove the MVT if you want to use it. (Since MVT relies on Rolle and you had to prove Rolle).

You can do it directly from Rolle, although it will be a bit like proving the MVT.

For 3(iii), would square root of x be a valid example?

Yes.

Reply 2

gangsta316

Original post by DFranklin

Yes

I don't know what you mean by the difference quotient. You need to do two things, first show that the inverse has a 2nd derivative (use the inverse function theorem). Second you need to show the 2nd derivative of the inverse has the formula given. For which, I would use the first part with a particular choice of g.

I would say if the first part asks you to prove Rolle, you would need to prove the MVT if you want to use it. (Since MVT relies on Rolle and you had to prove Rolle).

You can do it directly from Rolle, although it will be a bit like proving the MVT.

Yes.

Thank you. So for 2(ii), can we use inverse function theorem and chain rule etc. and do the question without computing limits (i.e. without differentiation from first principles)?

Reply 3

sidewalkwhenshewalks

I read the questions. I have a lot of respect for mathematicians ... you are like modern wizzards. :colonhash:

Reply 4

DFranklin

Original post by gangsta316

Thank you. So for 2(ii), can we use inverse function theorem and chain rule etc. and do the question without computing limits (i.e. without differentiation from first principles)?

Questions about Real analysis (Rolle's theorem and the mean value theorem etc.)

Quick Reply

Related discussions

Latest

Trending

Trending

Articles for you