You can show it is possible to have a root using the intermediate value theorem.

To use Rolle's theorem to argue there is at most one, you can proceed thusly:

Assume, to the contrary, there are two roots (or more, but at least 2 is fine), say for and , both in the interval you are considering. and you may also assume that . then that means

and so according to Rolle's theorem, there must be a point , such that and .

Where can you get with that?