{
  "aliases": [
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  "body": "\n## Statement\n\nLet $a \u003c b$ and let $f, g \\colon [a, b] \\to \\mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$.\nThen there is $c \\in (a, b)$ with\n$$(f(b) - f(a)) \\, g'(c) = (g(b) - g(a)) \\, f'(c).$$\n\n## Remarks\n\nApply [[Rolle's theorem|0194DY35BG]] to $h(x) = (f(b) - f(a)) g(x) - (g(b) - g(a)) f(x)$, which has $h(a) = h(b)$.\nWith $g(x) = x$ it is the [[mean value theorem|0BARQY52N5]].\nIt is the main step in the proof of [[L'Hôpital's rule|0JWT4PPQ7B]]. Mathlib states it as `exists_ratio_deriv_eq_ratio_slope`.\n\n## Explanations\n\n### L1 human\n\nThink of $(g(t), f(t))$ as a point moving along a curve in the plane.\nAt some time, the direction of motion is parallel to the chord joining the start and the end.\n",
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    "name": "Cauchy mean value theorem",
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  "name": "Cauchy mean value theorem",
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