I'm trying to work out how an existance of a fixed point is linked to the constraint on the differential of that function.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Function has a fixed point?

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