The Student Room Group

Second order mean value theorem.

I normally avoid seeking help with example sheet problems but this one seems to be more a case of me not understanding the question:

Let f be cts on [-1,1] and twice diff. on (-1,1). Define ø(x) = [f(x)-f(0)]/x for x≠0 and ø(0) = f'(0).

By using a second order MVT for f, show that ø'(x) = [f''(tx)]/2 for some 0<t<1.


I've already shown that ø is cts and diff. on [-1,1] and (-1,1) respectively in an earlier part of the question but I'm struggling to see an easy way to progress on this part. Is the question just asking me to use a second order form of Taylor's with either the Cauchy or Lagrange remainder? Or does "2nd order MVT" mean something else entirely?

Any help would be much appreciated!
Original post by Farhan.Hanif93

Any help would be much appreciated!


Wouldn't attempt it myself, but this thread may be useful:

Thread
Original post by ghostwalker
Wouldn't attempt it myself, but this thread may be useful:

Thread

Thanks, I should have mentioned that I found that thread immediately when I googled "second order MVT" but I didn't continue to read it due to the lack of spoilers - the last thing I want is for the question to be ruined. :p:

At a brief glance, the discussion doesn't seem to actually come onto what a 2nd order MVT is.
Original post by Farhan.Hanif93
Thanks, I should have mentioned that I found that thread immediately when I googled "second order MVT" but I didn't continue to read it due to the lack of spoilers - the last thing I want is for the question to be ruined. :p:

At a brief glance, the discussion doesn't seem to actually come onto what a 2nd order MVT is.


Fair point.

A search for "second order mean value theroem" in quotes produces some definitions on the first page of hits. If you search "Books" in google, rather than just a general search, then the first hit, page 29 may be useful.

PS: That's just a definition, so shouldn't spoil the question.
(edited 11 years ago)

Quick Reply