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!