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  "body": "\n## Statement\n\nLet $a \u003c b$ and let $f \\colon [a, b] \\to \\mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$, with $f(a) = f(b)$.\nThen there is $c \\in (a, b)$ with $f'(c) = 0$.\n\n## Remarks\n\nBy the [[extreme value theorem|0WS1A8E2P9]], $f$ has a maximum and a minimum on $[a, b]$. If both occur at the endpoints, $f$ is constant. Otherwise one occurs at an interior point $c$, and [[Fermat's theorem|0PCVBRV9VJ]] gives $f'(c) = 0$.\nDifferentiability inside the interval is needed. The function $|x|$ on $[-1, 1]$ has no point with derivative $0$.\nMathlib states it as `exists_deriv_eq_zero`.\n\n## History\n\nMichel Rolle proved a version for polynomials in 1691. Cauchy gave a proof of the general theorem in 1823.\n\n## Explanations\n\n### L1 human\n\nA smooth graph that starts and ends at the same height must turn around somewhere.\nAt the turning point, the tangent is horizontal.\n",
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