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  "body": "\n## Definition\n\nLet $I$ be an open interval, let $f \\colon I \\to \\mathbb{R}$ be continuous, and let $c \\in I$.\nThe point $c$ is an inflection point of $f$ if there is $\\delta \u003e 0$ with one of the following properties.\nThe function $f$ is [[convex|0VV0HSW1ZS]] on $(c - \\delta, c]$ and [[concave|0EF2VG8XKQ]] on $[c, c + \\delta)$.\nOr $f$ is concave on $(c - \\delta, c]$ and convex on $[c, c + \\delta)$.\n\n## Remarks\n\nIf $f$ is twice differentiable at an inflection point $c$, then $f''(c) = 0$.\nThe converse fails: $f(x) = x^4$ has $f''(0) = 0$, but $0$ is not an inflection point.\nThe function $x^3$ has an inflection point at $0$. So does $\\sin x$, at each multiple of $\\pi$.\nSome texts also ask the graph to have a tangent line at $c$.\n\n## Explanations\n\n### L1 human\n\nAt an inflection point, the graph changes from bending up to bending down, or the reverse.\nThe graph crosses its tangent line there.\n",
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