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  "body": "\n## Definition\n\nLet $D \\subseteq \\mathbb{R}^n$, let $f \\colon D \\to \\mathbb{R}$, and let $a \\in D$.\nThe function $f$ has a local maximum at $a$ if there is $\\varepsilon \u003e 0$ such that $f(x) \\le f(a)$ for all $x \\in D$ with $\\|x - a\\| \u003c \\varepsilon$.\nThe function $f$ has a local minimum at $a$ if there is $\\varepsilon \u003e 0$ such that $f(x) \\ge f(a)$ for all $x \\in D$ with $\\|x - a\\| \u003c \\varepsilon$.\nA local extremum of $f$ is a local maximum or a local minimum of $f$.\nThe extremum is strict if the inequality is strict for all such $x \\ne a$.\n\n## Remarks\n\nA global maximum of $f$ is a point $a$ with $f(x) \\le f(a)$ for all $x \\in D$. Each global maximum is a local maximum.\nThe function $f(x, y) = x^2 + y^2$ has a strict local minimum at $(0, 0)$, which is also a global minimum.\nIn two or more variables, a local minimum that is the only critical point need not be a global minimum. For $f(x, y) = x^2 + y^2 (1 - x)^3$, the only critical point $(0, 0)$ is a local minimum, but $f(2, 3) = -5 \u003c f(0, 0)$.\nAt an interior local extremum of a differentiable function, the [[gradient is zero|0F405477F8]].\n\n## Explanations\n\n### L1 human\n\nA local maximum is a hilltop: it is the highest point in a small disc around it, but a higher hill can exist far away.\nA local minimum is the bottom of a valley in the same sense.\n",
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