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  "body": "\n## Statement\n\nLet $k \\ge 0$ and let $a \\ne x$ be real numbers. Let $J$ be the closed interval with endpoints $a$ and $x$.\nLet $f$ be a real function with $f^{(k)}$ continuous on $J$ and $f^{(k+1)}$ existing on the interior of $J$.\nThen there is $\\xi$ strictly between $a$ and $x$ with\n$$f(x) = P_k(x) + \\frac{f^{(k+1)}(\\xi)}{(k + 1)!} (x - a)^{k+1},$$\nwhere $P_k$ is the [[Taylor polynomial|0HFSFVMWQB]] of degree $k$ of $f$ at $a$.\n\n## Remarks\n\nFor $k = 0$ this is the [[mean value theorem|0BARQY52N5]].\nIf $|f^{(k+1)}| \\le M$ between $a$ and $x$, then $|f(x) - P_k(x)| \\le M |x - a|^{k+1} / (k + 1)!$.\nMathlib states it as `taylor_mean_remainder_lagrange`.\n\n## Explanations\n\n### L1 human\n\nThe error of the Taylor polynomial looks like the next term of the polynomial.\nOnly the derivative is taken at an unknown point $\\xi$ instead of at $a$.\n",
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  "name": "Taylor's theorem with the Lagrange remainder",
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