Let f and g be two continuous functions that are defined on [a,b] and differentiable on (a,b).
Claim: If f'=g' on (a,b), then there exists a constant k such that f(x)=g(x)+k for all x in [a,b]
The Attempt at a Solution
Let h:[a,b]->R be defined by h(x)=g(x)+k-f(x)
h is continuous and differentiable on [a,b] by the assumptions of f and g already made.
Since f'=g' => g'-f'=0
So h'(x) = g'(x) - f'(x) = 0
By the mean value theorem, there exists a c in (a,b) so that
0=(h(b)-h(a))/(b-a) => h(a)=h(b)
I think I showed rolle's theorem. I am trying to use my assumption that f'=g' to show the claim. Any hints on this?