Category:Minimum Value of Real Function

From ProofWiki
Jump to navigation Jump to search

This category contains results about Minimum Value of Real Function.
Definitions specific to this category can be found in Definitions/Minimum Value of Real Function.

Absolute Minimum

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

Let $f$ be bounded below by an infimum $B$.

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 minimum value of $f$ on $S$, and this minimum is attained at $x$.

Local Minimum

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.

This category currently contains no pages or media.