- #1
michaelxavier
- 15
- 0
Homework Statement
Assume that f is twice differentiable on the entire real line. Show that
f(-1) + f(1) - 2f(0) = f"(c) for some c in [-1,1]
Homework Equations
I'm thinking the mean value theorem will be helpful here -- the MVT states that, given a function f differentiable on [a,b], there is some point c in (a,b) s.t.
( f(b) - f(a) ) / (b-a) = f'(c).
The Attempt at a Solution
By applying to MVT to ( f(-1) - f(0) ) and ( f(1) - f(0) ) and then adding the results, I've managed to show that f(-1) + f(1) - 2f(0) = f'(d) - f'(e) for some d,e in (-1, 1). But then I am stuck. How to prove that f'(d) - f'(e) = f"(c) for some c in [-1,1]? Or am I just completely on the wrong track?
Thanks for your help!