Category:Minimum Value of Real Function
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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.
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