I'm not sure how to do this using the 'mean value theorem', I'm pretty sure the mean value theorem goes something like: (roughly) when considering an arc between two endpoints of a function, at least one point exists where tangent to the arc is parallel to the line connecting the endpoints.
However, this is how I would approach your problem:
With the expression in this form it's easier to argue the following:
As well as:
It must be true that: