# Proof involving derivatives

1. Dec 16, 2009

### tracedinair

1. The problem statement, all variables and given/known data

Let the symmetric derivative of f at x be

lim h->0 f(x+h) + f(x-h) - 2f(x) / h, provided the limit exists.

Prove there exists a point, x, in (0,1) where the ordinary derivative exists.

Note: f is cont. on [0,1], and the symm. deriv. exists everywhere on (0,1). Prove there is a point, x, in (0,1) where the ordinary derivative exists. Assume f(0)=f(1)=0 and it is unknown if f is differentiable.

2. Relevant equations

3. The attempt at a solution

I don't know where to start on this. I tried MVT and Rolle's Thm, but unfortunately they don't seem to apply here. Also tried setting h(x) = f(x) - f(0) - x[f(1)-f(0)] and letting g(1)=g(0)=0, but no progress there either. Thx for any help.