Let $B^n$ be the set of 1d-Bezier curves $\gamma(t)$ with control points $|C|=n$, $C_i \in \{0, 1\}$ and where they start at 0 and end at 1, i.e. $C_1=0, C_n=1$, or equivalently, $\gamma(0)=0, \gamma(1)=1$.
Now, I want to take a look at the extension of $\gamma$ from $[0,1]$ to $[0, \infty)$. You may do this by taking the polynomial which is equal to gamma on the interval [0,1] and simply using it instead in $\mathbb R$.
Define, for each $\gamma$, the polynomial $M = 1+\gamma'(t)^2$. My question is, why is the maximum root of $M$ bigger when $n=8$ than when $n=9$? You may find it through an exhaustive search, as I've found it is around 53 versus 57. What explains this phenomenon exactly, as clearly this number generally increases as you increase the dimension as the degrees of the polynomials are getting larger. Why does the best 8 beat any 9?