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  "body": "\n## Statement\n\nLet $I$ be an open interval and let $f \\colon I \\to \\mathbb{R}$ be twice differentiable.\nIf $f''(x) \\ge 0$ for all $x \\in I$, then $f$ is [[convex|0VV0HSW1ZS]].\n\n## Remarks\n\nThe derivative $f'$ has a nonnegative derivative, so $f'$ is [[increasing|026BJBQNGJ]]. Then [[convexity follows|005JT76HCK]].\nIf $f'' \u003e 0$ on $I$, then $f$ is strictly convex. The converse fails: $x^4$ is strictly convex, but its second derivative at $0$ is $0$.\nMathlib states it as `convexOn_of_deriv2_nonneg`.\n\n## Explanations\n\n### L1 human\n\nA nonnegative [[second derivative|0Y9D2FJRQ4]] means that the slope never decreases.\nSo the graph bends upward everywhere.\n",
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