You are correct about using the mean value theorem.
From that we know that for .
So what is
This is what I did. Please correct any mistakes:
Given defined by .
We want to find a so that implicates that for all and in
The mean value theorem says that since f is countinous for , we have:
where
By graphing we see that this (absolute) value is never greater than 4 (or 5, or 6...). We now have that:
Let's say that , where we have been told that . Since , then .
So finally:
Therefore is a value for that makes