Local extremum

Extreme Value Theorem

Let be a metric space and let be a continuous function. Then for every compact subspace of the function has an absolute maximum and absolute minimum on , i.e. there exists such that for all . #m/thm/anal

Proof

#missing/proof

Euclidean spaces

The Extreme Value Theorem is stated as follows1

Let be a non-empty closed and bounded subset of and let be a continuous function. Then is bounded and there exist and such that . #m/thm/calculus

As a consequence of this, it is possible to determine the absolute extrema of a function on such a domain by narrowing our view to the specific set of circumstances in which an absolute extrema can occur. These are

  1. At critical points, i.e. and .
  2. At extrema along the boundary .
  3. At extrema along the boundary of the boundary
    &c.
Example

In the case of a rectangular domain in

this involves checking for local extrema in

and then the extrema on the boundary

and then the extrema on the boundary of the boundary , i.e. the corners

and then determining which of these values are indeed and .

#state/tidy | #SemBr | #lang/en

Footnotes

  1. 2022. MATH1011: Multivariable Calculus, p. 59 ↩