= Endpoint derivative signs of Lagrange cardinal polynomials
{title2=$(-1)^iL_i^{(k)}(1)>0$}
At decreasing distinct <interpolation nodes> $1\ge t_0>\cdots>t_n\ge-1$, the <Lagrange cardinal polynomial> $L_i(x)=\prod_{j\ne i}(x-t_j)/\prod_{j\ne i}(t_i-t_j)$ has denominator sign $(-1)^i$. For $1\le k\le n$, its numerator's $k$th <derivative> at one is $k!$ times a sum of products of $n-k$ nonnegative factors $1-t_j$. At most one factor vanishes, and at least one product avoids it, so that sum is strictly positive. The same alternating nodal signs therefore maximize every endpoint derivative simultaneously.
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