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  "body": "\n## Statement\n\nLet $f$ be a real function differentiable on an open interval around $c$, with $f'(c) = 0$.\nAssume that the [[second derivative|0Y9D2FJRQ4]] $f''(c)$ exists.\nIf $f''(c) \u003e 0$, then $f$ has a strict [[local minimum|0NS8BP96QX]] at $c$. If $f''(c) \u003c 0$, then $f$ has a strict local maximum at $c$.\n\n## Remarks\n\nIf $f''(c) \u003e 0$, then $f'(x)/(x - c) \\to f''(c)$. So $f'$ is negative just left of $c$ and positive just right of $c$, and the [[first derivative test|01A2EGXJEM]] applies.\nIf $f''(c) = 0$, the test gives no answer. The functions $x^4$, $-x^4$, and $x^3$ at $0$ show all three cases.\nMathlib states the minimum case as `isLocalMin_of_deriv_deriv_pos`.\n\n## Explanations\n\n### L1 human\n\nAt a stationary point where the graph bends upward, the point is the bottom of a valley.\nWhere it bends downward, it is a hilltop.\n",
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