A cubic gives equality in the pointwise interpolation remainder

Given: The interval $[0,3]$, the function $f(t)=t^3$, nodes $x_0=0,x_1=1,x_2=2$, and query $x=3$.

Cardinal-polynomial interpolation: $I_{2}f$ is the polynomial interpolant to the three nodal values.

Formal real polynomials, evaluation, degree, leading coefficient, and monic polynomials: a formal polynomial is a finite coefficient list and its evaluation is the corresponding finite sum.

Existence and uniqueness of the cardinal-polynomial interpolant: the polynomial of degree at most $2$ matching the three distinct nodal values is uniquely $I_{2}f$.

Higher derivatives and the classes $C^k$ and $C^{\mathrm{\infty }}$: $C^3$ means that the derivatives through order $3$ exist and are continuous.

For a natural $n\ge 1$ the function $x\mapsto x^n$ is differentiable everywhere with derivative $\iota (n)x^{n-1}$; for $n=0$ it is the constant $1$, with derivative $0$; for a natural $n\ge 1$ the function $x\mapsto x^{-n}$ is differentiable at every $x\mathrm{\ne }0$ with derivative $-\iota (n)x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term: powers have their usual derivatives, and polynomial functions are differentiable term by term.

Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$: scalar multiples and sums differentiate by the corresponding linear rules.

Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function: polynomial functions are continuous.

Pointwise Lagrange interpolation remainder: for $f\in C^3([0,3])$ and three distinct nodes, at every non-node query there is an intermediate $\xi$ with the stated cubic remainder formula.

The factorial $n!$ and the falling factorial $n^{\underset{‾}{k}}$, defined by recursion in $\mathbb{N}$ and The canonical natural $\iota (n)=n\cdot 1_F$ of a field: the real factorial is the canonical real image of the natural factorial, so $3{!}_{\mathbb{R}}=1\cdot 2\cdot 3=6$.