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  "body": "\n## Summary\n\nIn mathematics, a **smooth maximum** of an indexed family *x*\u003csub\u003e1\u003c/sub\u003e, ..., *x*\u003csub\u003e*n*\u003c/sub\u003e of numbers is a smooth approximation to the maximum function $\\max(x_1,\\ldots,x_n),$ meaning a parametric family of functions $m_\\alpha(x_1,\\ldots,x_n)$ such that for every $\\alpha$, the function $m_\\alpha$ is smooth, and the family converges to the maximum function $m_\\alpha \\to \\max$ as $\\alpha\\to\\infty$. The concept of **smooth minimum** is similarly defined. In many cases, a single family approximates both: maximum as the parameter goes to positive infinity, minimum as the parameter goes to negative infinity; in symbols, $m_\\alpha \\to \\max$ as $\\alpha \\to \\infty$ and $m_\\alpha \\to \\min$ as $\\alpha \\to -\\infty$. The term can also be used loosely for a specific smooth function that behaves similarly to a maximum, without necessarily being part of a parametrized family.\n",
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