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.