= Solution
The difference $d=p-q$ has $d(x_i)=d'(x_i)=0$ at every node. <Rolle's theorem> supplies an additional zero of $d'$ in each of the $n$ open intervals between successive nodes. Together with the $n+1$ nodal derivative zeros, these are $2n+1$ distinct zeros of $d'$, including the two endpoints. But $d'$ is a polynomial of degree at most $2n$, so it is identically zero. Since $d(x_0)=0$, $d$ is also identically zero.
Thus \b[the interpolation polynomial is unique]. Once $d'=0$ identically, it indeed has arbitrarily many zeros in the open interval as requested. The initial Rolle count naturally includes the endpoints, rather than initially supplying $2n+1$ open-interval zeros. Equivalently, every node is a double root of $d$, so a nonzero difference would be divisible by the degree-$2n+2$ polynomial $\prod_i(x-x_i)^2$, exceeding its allowed degree.
Back to article page
