Let f be continuous on [a,b], differentiable and not zero on (a,b). If f(a)=f(b)=0, prove that the function g defined on (a,b) where g(x)=f'(x)/f(x) takes every value a in R. I apply the Mean Value Theorem to log f(x). So then there exists a c in [a,b] such that f'(c)/f(c) = (log f(b) - log f(a)) / (b-a). How does this imply that f'(x)/f(x) takes every value a in the real numbers?