Definition:Minimum Value of Real Function/Local
< Definition:Minimum Value of Real Function(Redirected from Definition:Local Minimum)
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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 minimum at $\xi$ if and only if:
- $\exists \openint c d \subseteq \openint a b: \forall x \in \openint c d: \map f x \ge \map f \xi$
That is, if and only if there is some subinterval on which $f$ attains a minimum within that interval.
Strict Local Minimum
$f$ has a strict local minimum 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 minimum, 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 minimum.
Also known as
A local minimum is also known as a relative minimum.
Collectively, local maxima and local minima can be referred to as turning points.
Also see
- Results about local minima can be found here.
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): minimum (plural minima)
- 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): minimum (plural minima)
- 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): local minimum (minima)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): stationary point (in one variable)