I'm trying to work out how an existance of a fixed point is linked to the constraint on the differential of that function. For example, i need to prove f has a fixed point if f'(x)=>2. I understand that what I have is a monotone increasing function so it is 1-1. All the fixed points are on the line f(x) = x. So conceptually it must be true that these two lines should intersect somewhere, but I can't prove this rigorously. I have a feeling I should be using the Mean Value Theorem ( f(b) - f(a)) = f'(c)(b-a) but can't get much further than that.