Realvalued function/Extrema/Introduction/Section

Definition

Let ${}M$ denote a set, and

$f\colon M\longrightarrow \mathbb {R}$

a function. We say that ${}f$ attains in a point ${}x\in M$ its maximum, if

$f(x)\geq f(x'){\text{ holds for all }}x'\in M,$

and that ${}f$ attains in ${}x$ its minimum, if

$f(x)\leq f(x'){\text{ holds for all }}x'\in M.$

The common name for a maximum or a minimum is extremum. In the preceding definition, we also talk about the global maximum, since the property refers to all elements of the domain of the definition. When we are only interested in the behavior on an open, maybe small, neighborhood, then the concept of a local maximum is relevant.

Definition

Suppose that ${}D\subseteq \mathbb {R}$ is a subset and let

$f\colon D\longrightarrow \mathbb {R}$

denote a function. We say that ${}f$ attains a local maximum in a point ${}x\in D$, if there exists some ${}\epsilon >0$, such that for all ${}x'\in D$ fulfilling ${}\vert {x-x'}\vert \leq \epsilon$, the estimate

${}f(x)\geq f(x')\,$

holds. We say that ${}f$ attains a local minimum in a point ${}x\in D$, if there exists some ${}\epsilon >0$, such that for all ${}x'\in D$ fulfilling ${}\vert {x-x'}\vert \leq \epsilon$, the estimate

${}f(x)\leq f(x')\,$

holds.

If ${}f(x)>f(x')$ holds for all ${}x'\neq x$, then we talk about an isolated maximum.