 Solution

The difference $d=p-q$ has $d(x_i)=d^{\prime }(x_i)=0$ at every node. Rolle's theorem supplies an additional zero of $d^{\prime }$ 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 $d^{\prime }$, including the two endpoints. But $d^{\prime }$ is a polynomial of degree at most $2n$, so it is identically zero. Since $d(x_0)=0$, $d$ is also identically zero.

Thus the interpolation polynomial is unique. Once $d^{\prime }=0$ 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.