Show that there is a unique polynomial of degree at most $2k+1$ such that $p^{[j]}(x_1)=a_j \text{ and } p^{[j]}(x_2)=b_j \text{ for } j=0,\dots, k.$

Question

Let $x_1,x_2 \in \mathbb R$ and let $(a_0, \dots, a_k), (b_0, \dots, b_k)$ be $(k+1)$-tuples of real numbers. Show that there is a unique polynomial of degree at most $2k+1$ such that $$p^{[j]}(x_1)=a_j \text{ and } p^{[j]}(x_2)=b_j \text{ for } j=0,\dots, k.$$

[Consider the polynomials $(x-x_1)^k (x-x_2)^{k+1}$, $(x-x_1)^{k+1}(x-x_2)^k$ and argue by induction.]

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