Math 6620 • Analysis of Numerical Methods II

Suppose \(f(x) \in C^{n+1}([a,b])\), and consider the interpolation problem: For \(a \le x_0 < x_1 <\)
\(x_2 < \cdots < x_{n-1} < x_n \le b\), find a polynomial \(p(x)\) of degree at most \(n\) for which \(p(x_j) = f(x_j)\),
\(j = 0,1,\ldots,n\).

\begin{enumerate}

\item
Show that this problem has at most one solution.


\item
Show that this problem has a solution.


\item
Show, for any \(x \in [a,b]\), that
\[
f(x)-p(x)=\frac{f^{n+1}(\xi)}{(n+1)!}(x-x_0)(x-x_1)\cdots(x-x_n),
\]
for some \(\xi \in [a,b]\).


\end{enumerate}