Concavity changes 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.
 A: Say you have a polynomial $y = p_{m-1}(x)$ and there exist some real solutions of $p_{m-1}(x) = x^m$. 
A function $f(x)$ is concave at point $x$ if $f''(x)<0$ and convex if $f''(x)>0$. The change in concavity occurs when $f''(x)=0$.
If you are asking whether that implies that $p''_{m-1}(x)$ is zero at those points, than I think it does not. Consider example of $m=4$ and $p_3(x) = x^3-x+1$. The equation $p_3(x) = x^4$ admits two real solutions $x=\pm1$. The second derivative of the polynomial is $6x$ and is not zero at those points. 
A: About the "also" part of the question:
It is clear that if $m < 3$ there are no changes in concavity.
Suppose that $m \ge  3$.  Your polynomial has degree $m-1$.  So its second derivative has degree $m-3$.  
Everywhere that there is a change in concavity, the second derivative is $0$.  But a polynomial of degree $k$ has at most $k$ zeros.  So the function can change concavity at most $m-3$ times.
Side Comment: The $0$ polynomial does not have degree $0$, or at least not many mathematicians say that it does.  One convention is that it does not have a degree.  Another convention is that it has degree $-\infty$.  I have also seen $-1$ offered as the degree of the $0$ polynomial.
