## Having trouble with simple analysis problem

Hi, I need help with the following problem.
Suppose that $f(x)$ and $g(x+1)$ are real valued functions on $\mathbb R$ having period 1 and having continuous first derivatives.
(a) Prove that $f'(c)=g'(c)$ for some non-negative $c\in \mathbb R$.
(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.