Definition:Maximum Value of Real Function/Local
Jump to navigation
Jump to search
Definition
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.
Strict Local Maximum
$f$ has a strict local maximum at $\xi$ if and only if:
- $\exists \openint c d \subseteq \openint a b: \forall x \in \openint c d: \map f x < \map f \xi$
Warning
In the definition of a local maximum, note the requirement for the intervals to be open.
A closed interval of course includes the value of $f$ at its end points and so every closed interval attains a maximum.
Also known as
A local maximum is also known as a relative maximum.
Collectively, local maxima and local minima can be referred to as turning points.
Also see
- Results about local maxima can be found here.
Linguistic Note
The plural form of maximum is maxima.
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 11.1$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): maximum (plural maxima)
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): turning point
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): maximum (plural maxima)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): turning point
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): greatest value
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): stationary point (in one variable)