91Ó°ÊÓ

Is the following function a natural cubic spline? $$ S(x)=\left\\{\begin{array}{ll} x^{3}-1 & x \in\left[-1, \frac{1}{2}\right] \\ 3 x^{3}-1 & x \in\left[\frac{1}{2}, 1\right] \end{array}\right. $$

Find the Lagrange and Newton forms of the interpolating polynomial for these data: \begin{tabular}{c||r|r|r} \(x\) & \(-2\) & 0 & 1 \\ \hline\(f(x)\) & 0 & 1 & \(-1\) \end{tabular} Write both polynomials in the form \(a+b x+c x^{2}\) to verify that they are identical as functions.