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  "body": "\n## Statement\n\nLet $I$ be an open interval, let $f \\colon I \\to \\mathbb{R}$, and let $a \\in I$.\nAssume that $f$ has a [[local maximum|00999TYPXB]] or a [[local minimum|0NS8BP96QX]] at $a$, and that $f$ is differentiable at $a$.\nThen $f'(a) = 0$.\n\n## Remarks\n\nAt a local maximum, the difference quotient $(f(a + h) - f(a))/h$ is $\\le 0$ for small $h \u003e 0$ and $\\ge 0$ for small $h \u003c 0$. So the limit is both $\\le 0$ and $\\ge 0$.\nThe point must be interior. On $[0, 1]$, the function $x$ has a maximum at $1$, where its derivative is $1$.\nMathlib states it as `IsLocalExtr.deriv_eq_zero`.\n\n## Explanations\n\n### L1 human\n\nAt the top of a smooth hill, the tangent line is horizontal.\nIf the slope were positive or negative, moving a little would give a higher value.\n",
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