# Minima and maxima

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In mathematics, **minima** and **maxima**, known collectively as **extrema**, are the *smallest value* (minimum) or *largest value* (maximum), that a function takes in a point either within a given neighbourhood (local extremum) or on the whole function domain (global extremum).

## Definition

### Minimum

A real-valued function *f* is said to have a **local minimum** at the point *x*^{*}, if there exists some ε > 0, such that *f*(*x*^{*}) ≤ *f*(*x*) whenever |*x* − *x*^{*}| < ε. The value of the function at this point is called **minimum** of the function.

The definition of a **local maximum** is similar, only with the ≥ sign in place of ≤.