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  "body": "\n## Statement\n\nLet $I$ be an open interval and let $f \\colon I \\to \\mathbb{R}$ be differentiable.\nThen $f$ is [[convex|0VV0HSW1ZS]] if and only if $f'$ is increasing on $I$.\n\n## Remarks\n\nIf $f$ is convex, the [[slope inequalities|0BY5FTHEEE]] give $f'(x) \\le f'(y)$ for $x \u003c y$.\nIf $f'$ is increasing, the [[mean value theorem|0BARQY52N5]] on $[x, y]$ and $[y, z]$ gives the slope inequalities.\nThen the graph lies above each tangent line: $f(y) \\ge f(x) + f'(x)(y - x)$.\nMathlib states the converse direction as `MonotoneOn.convexOn_of_deriv`.\n\n## Explanations\n\n### L1 human\n\nA differentiable function is convex when its slope keeps growing.\nThe graph then turns to the left as we move along it to the right.\n",
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