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!