= Nodal derivative norm from Lagrange cardinal polynomials
{title2=$\sup_{|p(t_i)|\le1}|p^{(k)}(x)|=\sum_i|\ell_i^{(k)}(x)|$}
At distinct <interpolation nodes>, differentiating <Lagrange interpolation> gives $p^{(k)}(x)=\sum_i p(t_i)\ell_i^{(k)}(x)$. The <triangle inequality> gives the displayed upper bound, attained by prescribing each nodal value to be the sign of the corresponding <derivative> weight. For increasingly ordered real nodes, the <Markov interlacing lemma> and <monotonicity of polynomial critical points in their roots> imply that $(-1)^i\ell_i^{(k)}(x)$ has at most one sign change, ignoring zero weights. Thus an extremizer has alternating nodal signs with at most one reversal of that alternating pattern. This reduces a choice among $2^{n+1}$ sign patterns to $n+1$ candidate <polynomials>.
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