Math 6620 • Analysis of Numerical Methods II

Suppose \(a < b\) and \(f \in C^3([a,b])\). Consider the interpolation problem: Find a polynomial \(p(x)\)
of degree two or less which satisfies the conditions \(p(a) = f(a)\), \(p'(a) = f'(a)\), and \(p(b) = f(b)\).
\begin{enumerate}

\item
Prove that this problem has a solution and that the solution is unique. (You do not need to
find the solution.)

\item
Prove that the error in using \(p(x)\) to approximate \(f(x)\) for \(x \in (a,b)\) is given by the formula
\begin{equation}
\tag{1}
f(x)-p(x)=\frac{f^{(3)}(\xi)}{3!}(x-a)^2(x-b),
\end{equation}
for some \(\xi \in (a,b)\).


\end{enumerate}