I'm done being confused by Galois theory and am back to being confused by elementary calculus.

I have a polynomial of degree $m-1$ that is bounded by the curve $y = x^m$ and intersects it at a nonzero number of points. Does each of these intersection points represent a change in concavity (or two, given that it has to go down and up again)? Also, am I right in saying there can be at most $m-3$ changes in concavity? Thanks, I kind of forgot this stuff.