Definition:Minimum Value of Real Function/Local

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

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.

Also see

• Definition:Absolute Minimum

• Definition:Local Maximum
• Definition:Absolute Maximum

• Derivative at Maximum or Minimum

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)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): minimum (plural minima)
• 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)