Definition:Maximum Value of Real Function

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Absolute Maximum

Let $f: \R \to \R$ be a real function.

Let $f$ be bounded above by a supremum $B$.

It may or may not be the case that $\exists x \in \R: \map f x = B$.

If such a value exists, it is called the (absolute) maximum of $f$ on $S$, and that this (absolute) maximum is attained at $x$.

Local Maximum

Let $f$ be a real function defined on an open interval $\openint a b$.

Let $\xi \in \openint a b$.

Then $f$ has a local maximum at $\xi$ if and only if:

$\exists \openint c d \subseteq \openint a b: \forall x \in \openint c d: \map f x \le \map f \xi$

That is, if and only if there is some subinterval on which $f$ attains a maximum within that interval.

Also see

  • Results about the maximum value of a real function can be found here.

Linguistic Note

The plural form of maximum is maxima.