Having trouble with simple analysis problem

Hi, I need help with the following problem.

Suppose that $\displaystyle f(x)$ and $\displaystyle g(x+1)$ are real valued functions on $\displaystyle \mathbb R$ having period 1 and having continuous first derivatives.

(a) Prove that $\displaystyle f'(c)=g'(c)$ for some non-negative $\displaystyle c\in \mathbb R$.

(b) Prove that there is a smallest such value of c.

I proved part a by letting h=f-g, looking for h'=0 and using the mean value theorem, but I can't seem to figure out part b.

Any help would be appreciated.