This one has been stumping me for quite some time know...
Given continuous on and differentiable on such that there exists an so that for all , show that there exists a unique point such that .
I can show that if there is a point, then it is unique, but showing the point exists is challenging.
Hint: Pick and let .
I have used repeated applications of the mean value theorem to achieve the following inequality
Unfortunately, this is not Cauchy necessarily, so I am very stuck. Any help would be greatly appreciated!