You would probably use the Mean Value Theorem. You know that there is a such that .
I wasn't sure how to start this but I thought it had something to do with a Taylor series.Let be a continuous function.
If also f is differentiable on (0,1) prove that such that .
So I did this:
and I know that this is only valid for .
However, this isn't a proof!
Can someone help me finish this?
Ah, i'm so sorry about this but I haven't told you the entire question!
I thought that they had redefined the function f when they haven't.
The question also states that and .
This makes the questions significantly easier!
Let . It is apparent that g is continuous over and differentiable over .
Therefore by Cauchy's mean value theorem
We know that and .
This gives:
Rearranging gives: