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  "body": "\n## Statement\n\nLet $n \\ge 1$, let $I \\subseteq \\R$ be an open interval, and let $f \\colon I \\to \\R$ be $n$ times differentiable.\nLet $x_0, \\dots, x_n \\in I$ be pairwise distinct, and let $m$ and $M$ be the smallest and largest of them.\nThen there is $\\xi \\in (m, M)$ such that\n$$f[x_0, \\dots, x_n] = \\frac{f^{(n)}(\\xi)}{n!},$$\nwhere $f[x_0, \\dots, x_n]$ is the [[divided difference|N5ZHV4HWDC]].\n\n## Remarks\n\nLet $p$ be the [[interpolation polynomial|SS1YMS2NPC]] of $f$ at the nodes.\nThen $g = f - p$ has $n + 1$ distinct zeros in $[m, M]$.\nApplying [[Rolle's theorem|0194DY35BG]] $n$ times gives a point $\\xi \\in (m, M)$ with $g^{(n)}(\\xi) = 0$.\nSince $p^{(n)}$ is the constant $n! \\, f[x_0, \\dots, x_n]$, the claim follows.\n\nFor $n = 1$ this is the [[mean value theorem|0BARQY52N5]].\nExample: for $f(x) = x^3$ and the nodes $0, 1, 2$, one has $f[0, 1] = 1$, $f[1, 2] = 7$ and $f[0, 1, 2] = 3$.\nSince $f''(\\xi)/2 = 3\\xi$, the theorem holds with $\\xi = 1$.\n\n## Explanations\n\n### L1 human\n\nThe ordinary mean value theorem says that the slope of a chord equals the slope of the tangent somewhere in between.\nThis theorem says the same thing one level up: a higher divided difference equals a scaled higher derivative at some point between the nodes.\n\n### L2 human\n\nIt turns statements about derivatives into statements about divided differences and back.\nFor example, if $f^{(n)} \\ge 0$ on $I$, then every $n$-th divided difference of $f$ is nonnegative.\nCombined with the [[Newton form|9W77FPTM9E]] of the interpolation polynomial, it gives the [[interpolation error formula|HASA2XSG66]].\n",
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