Maxima and minima

In mathematics, an extreme value (or extreme; plural: Extrema ) the generic term for local and global maximum and minimum. A local maximum is the value of the function at a point, the function does not assume higher values ​​in the vicinity thereof. The corresponding point is a local maximizer / minimizer or extreme point called ( maximizer / minimizer ), the combination of location and value extreme point.

A global maximum is also called the absolute maximum for a local maximum and the relative maximum term is needed. Local and global minima are defined analogously.

The solution of an extreme value problem for a simple representation see curve sketching, called the extremal solution.

  • 3.1 General definition of local extrema of real-valued functions on topological spaces
  • 3.2 Discrete Optimization
  • 3.3 Calculus of Variations

One-dimensional case

Formal definition

It is a subset of the real numbers ( for example, a period), and a function.

Has on the site

  • A local minimum, if there is an interval that includes, such that for all;
  • A global minimum if and only if for all;
  • A local maximum if there is an interval that includes, such that for all;
  • A global maximum if and only if for all.

If the function has a maximum at the point, it is called the point high point, she has there a minimum so called the nadir point. If either a high or a low point before, it is called an extreme point.

Existence of extrema

Is a continuous function and a compact set, so takes on its global maximum and its global minimum. These can also be accepted at the boundary points or.

This statement follows from the theorem of Heine- Borel, but is often also named after K. Weierstrass or B. Bolzano.

Determination of extreme points of differentiable functions

It was open, and a differentiable function.

Necessary criterion

If at one point a local extremum and there is differentiable, then there is the first derivative equal to zero:

Adequate criteria

  • Is twice differentiable, and is in addition also so has a local extremum. , And so it is a local minimum, and is at a local maximum.
  • More generally, is - times differentiable, and suppose
  • If the first derivative changes sign at, then there is an extremum. In a sign change from positive to negative is a maximum, with a change of sign from minus to plus to a minimum.
  • For continuous functions at intervals applies: between two local minima of a function is always a local maximum between two local maxima is always a local minimum.
  • For differentiable functions on intervals apply: Are there two places, so that the first derivative in the interval has only the zero point, and is, and so has in a local minimum. Does the analogous condition with and so has in a local maximum.

However, there are features that further help for any of those criteria.

Examples

  • The first derivative has only one zero. The second derivative is positive there, so takes on 0 at a local minimum, ie.
  • The first derivative has only one zero. The second derivative is also 0 there One can now proceed in several ways: And the third derivative is 0 there, the fourth derivative, however, is higher with the first derivative, which is not 0. Because this derivative has a positive value and the previous derivation is odd, by applies (1), that the function there have a local minimum.
  • The first derivative has a change in sign from minus to plus, so it has at a local minimum at 0.
  • It is so in the interval has a local minimum. Since the first derivative in this interval has only the zero point, the local minimum has to be accepted there.
  • The function is defined by for and has the following properties: She's at a global minimum.
  • It is infinitely differentiable.
  • All derivatives are equal to 0 at
  • The first derivative does not change sign at 0
  • The other two criteria above are not applicable.

Example of use

In practice, extreme value calculations can be used to calculate the greatest or smallest possible specifications, such as the following example shows (see also optimization problem ):

  • How should a rectangular area, which has a maximum area at a certain extent?

Solution:

The scope of which is constant, the surface to be maximized, the length and width:

1) Insert and manipulates 2)

Form derivative functions

There is only one local maximum, which is at the same time in the present example ( without proof ) and the global maximum, as the second derivative is independent of the variables is always less than zero.

To find an extreme value, the first derivative must be set equal to zero ( as it explains the slope of the original function and this slope at extreme values ​​is zero. If the second derivative of the function equal to zero, a minimum or maximum is present).

Insertion in 1)

It follows that the largest possible surface area of a rectangle is to be made ​​with predetermined amount when both side lengths are equal (which corresponds to a square ). Conversely, however, can also say that a rectangle with a given area has the least amount when

Behave - ie in a square!

Multi-dimensional case

It was and is a function. It should also be an interior point of. A local minimum / maximum in is provided when a surrounding environment exists in which no dot assumes a smaller or larger value function.

Analogous to the one dimensional case, the disappearance of the derivative

A necessary condition for the point assumes an extremum. Sufficiently in this case is the definiteness of the Hessian matrix: it is positive definite, there is a local minimum; if it is negative definite, there is a local maximum; it is indefinite, there is no extreme point, but a saddle point. If she is only semi-definite, no decision is possible on the basis of the Hessian matrix.

Other extreme values

General definition of local extrema of real-valued functions on topological spaces

Be a topological space and continuous. A point is a local maximum and a local maximum point of from if there is an environment of so

Applies to all. Local minima are defined analogously.

Discrete Optimization

For discrete optimization problems of the above- defined notion of local extremum is not suitable as a local extremum is present at every point in this sense. For extrema of a function a different environment term is therefore used: One uses a neighborhood function which assigns to each point the set of its neighbors,

Stands for the power set of.

Then has a local maximum at a point, if for all neighbors. Local minima are defined analogously.

Calculus of variations

Extreme values ​​of functions whose arguments are themselves functions that are the subject of the calculus of variations.

Maximum principle
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