Hey Dark Sun.

In the [0,1] range x^n =< x^m if m < n and only equal if m=n or x = 0 or 1. This rules out II.

For III., the condition is definitely satisfied as the RHS will overtake LHS and result in a root (ie LHS = RHS).

For I. I don't know whether you can say for sure and you would have to look at the specifics.

If you want to look at proofs consider the result above with the m's and n's.