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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 $(a, b)$.\nThen there is $c \\in (a, b)$ with\n$$f'(c) = \\frac{f(b) - f(a)}{b - a}.$$\n\n## Remarks\n\nApply [[Rolle's theorem|0194DY35BG]] to $g(x) = f(x) - \\frac{f(b) - f(a)}{b - a} (x - a)$, which has $g(a) = g(b)$.\nIt turns bounds on $f'$ into bounds on $f$. It gives [[zero derivative implies constant|003QDZ4T16]] and the [[monotonicity tests|07J5X2T1ZX]].\nMathlib states it as `exists_deriv_eq_slope`.\n\n## History\n\nAugustin-Louis Cauchy stated and proved the theorem in its modern form in 1823.\n\n## Explanations\n\n### L1 human\n\nSomewhere between $a$ and $b$, the tangent line is parallel to the chord from $(a, f(a))$ to $(b, f(b))$.\nIf a car travels 100 km in one hour, at some moment its speed is exactly 100 km per hour.\n",
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