Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 3 / 19D / b / Solution

Evaluating the Lagrange interpolation polynomial coefficients at five, and their derivatives at one, givesBoth rules are exact for every polynomial of degree at most two. For $f(x)=x^3$, the first gives $95$, while the true value is $125$; the second gives $4$, while the true derivative is $3$. Thus the respective approximation errors, estimate minus true value, are $-30$ and $1$. There is no contradiction: a cubic lies outside the exactness class. The first formula extrapolates beyond the interpolation nodes, so its error can be substantial.