Extreme value

Minima and maxima of the function cos (3π x ) / x in the range 0.1≤ x ≤1.1

In mathematics , extreme value (or extremum ; plural: extrema ) is the generic term for a local or global maximum or minimum . A local maximum or local minimum is the value of the function at a point when the function does not assume any larger or smaller values in a sufficiently small environment ; the associated point is called the local maximizer or local minimizer , maximum point or minimum point or, in summary, also called extreme point , the combination of point and value extreme point . ${\displaystyle x}$ ${\displaystyle x}$

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

The solution of an extreme value problem , for a simple representation see the discussion of curves , is called the extreme solution .

One-dimensional case

Formal definition

Let it be a subset of the real numbers (e.g. an interval ) and a function . ${\displaystyle U\subseteq \mathbb {R} }$ ${\displaystyle f\colon U\to \mathbb {R} }$

${\displaystyle f}$ has at the point ${\displaystyle x_{0}\in U}$

• a local minimum if there is an interval that contains such that holds for all ; ${\displaystyle I=(a,b)}$ ${\displaystyle x_{0}}$ ${\displaystyle f(x_{0})\leq f(x)}$ ${\displaystyle x\in I\cap U}$
• a global minimum if applies to all ; ${\displaystyle f(x_{0})\leq f(x)}$ ${\displaystyle x\in U}$
• a local maximum if there is an interval that contains such that holds for all ; ${\displaystyle I=(a,b)}$ ${\displaystyle x_{0}}$ ${\displaystyle f(x_{0})\geq f(x)}$ ${\displaystyle x\in I\cap U}$
• a global maximum if applies to everyone . ${\displaystyle f(x_{0})\geq f(x)}$ ${\displaystyle x\in U}$

If the function has a maximum at this point , the point is called the high point , if it has a minimum there, the point is called the low point . If there is a high or a low point, one speaks of an extreme point . ${\displaystyle x_{0}}$ ${\displaystyle (x_{0},f(x_{0}))}$

Existence of extremes

Are real numbers and is a continuous function , then assume a global maximum and a global minimum. These can also be accepted in the peripheral areas or . ${\displaystyle a\leq b}$ ${\displaystyle f\colon [a,b]\to \mathbb {R} }$ ${\displaystyle f}$ ${\displaystyle a}$ ${\displaystyle b}$

This statement follows from Heine-Borel's theorem , but is also often named after K. Weierstrass or B. Bolzano or referred to as the theorem of maximum and minimum.

Determination of extreme points of differentiable functions

It is open and has a differentiable function. ${\displaystyle U\subseteq \mathbb {R} }$ ${\displaystyle f\colon U\to \mathbb {R} }$

Necessary criterion

If there is a local extremum at a point and is differentiable there, the first derivative is zero there: ${\displaystyle f}$ ${\displaystyle x_{0}\in U}$

${\displaystyle f'(x_{0})=0\,}$.

Sufficient criteria

• If it is twice differentiable, and it is also true , then has a local extremum at the point . If and , it is a local minimum, for on the other hand it is a local maximum. ${\displaystyle f}$ ${\displaystyle f'(x_{0})=0\,}$ ${\displaystyle f''(x_{0})\neq 0\,}$ ${\displaystyle f}$ ${\displaystyle x_{0}}$ ${\displaystyle f'(x_{0})=0\,}$ ${\displaystyle f''(x_{0})>0\,}$ ${\displaystyle f''(x_{0})<0\,}$

• More generally, on the other hand, and can be derived from according to the Taylor formula, the following applies: Can be derived n times and is included ${\displaystyle f}$ ${\displaystyle f}$

${\displaystyle f'(x_{0})=f''(x_{0})=\ldots =f^{(n-1)}(x_{0})=0\,\wedge f^{(n)}(x_{0})\neq 0,}$

so it follows:

(1) If is even and (or ), then at has a relative maximum (or minimum). ${\displaystyle n}$ ${\displaystyle f^{(n)}(x_{0})<0}$ ${\displaystyle f^{(n)}(x_{0})>0}$ ${\displaystyle f}$ ${\displaystyle x_{0}}$

(2) If , however odd, has in no local extreme (the function value , but one of the rise , which is a turning point ). ${\displaystyle n}$ ${\displaystyle f}$ ${\displaystyle x_{0}}$

Or to put it quite generally: If the first non-zero derivative of the function at the point where is is a derivative of even order, then at this point it has an extreme point, with a non-zero derivative for a minimum and a derivative for there is a maximum. (Compare functions of the form: , .) ${\displaystyle f^{(n)}}$ ${\displaystyle f}$ ${\displaystyle x_{0}}$ ${\displaystyle f'(x_{0})=0}$ ${\displaystyle f}$ ${\displaystyle f^{(n)}>0}$ ${\displaystyle f^{(n)}<0}$ ${\displaystyle f(x)=x^{n}}$ ${\displaystyle n\in \mathbb {N} }$

• If the first derivative has a sign change , then there is an extremum. A change in sign from plus to minus is a maximum, and a change in sign from minus to plus is a minimum. ${\displaystyle x_{0}}$

• For continuous functions on intervals the following applies: Between two local minima of a function there is always a local maximum, and between two local maxima there is always a local minimum.

• For differentiable functions on intervals the following applies: If there are two digits with , so that the first derivative in the interval has only the zero , and is as well , then with has a local minimum. Applies the analog condition with and so is at a local maximum. ${\displaystyle a,b}$ ${\displaystyle a<x_{0}<b}$ ${\displaystyle (a,b)}$ ${\displaystyle x_{0}}$ ${\displaystyle f(a)>f(x_{0})}$ ${\displaystyle f(b)>f(x_{0})}$ ${\displaystyle f}$ ${\displaystyle x_{0}}$ ${\displaystyle f(a)<f(x_{0})}$ ${\displaystyle f(b)<f(x_{0})}$ ${\displaystyle f}$ ${\displaystyle x_{0}}$

However, there are also functions in which none of the above. Criteria helps (see below).

Examples

• ${\displaystyle f(x)=x^{2}+3.}$The first derivative only has a zero at. The second derivative is positive there, so it assumes a local minimum at 0, namely . ${\displaystyle f'(x)=2x}$ ${\displaystyle x_{0}=0}$ ${\displaystyle f''(x)=2}$ ${\displaystyle f}$ ${\displaystyle f(0)=3}$

• ${\displaystyle f(x)=x^{4}+3.}$The first derivative only has a zero at. The second derivative there is also 0. You can now continue in different ways: ${\displaystyle f'(x)=4x^{3}}$ ${\displaystyle x_{0}=0}$ ${\displaystyle f''(x)=12x^{2}}$
  • The third derivative is also 0. The fourth derivative, on the other hand, is the first higher derivative that is not 0. Since this derivative has a positive value and is even, according to (1) it holds that the function has a local minimum there. ${\displaystyle f'''(x)=24x}$ ${\displaystyle f^{(4)}(x)=24}$
  • The first derivative has at 0 a sign change from minus to plus, and thus has at a local minimum. ${\displaystyle f}$ ${\displaystyle x_{0}=0}$
  • It is so has a local minimum in the interval . Since the first derivative only has the zero in this interval , the local minimum must be assumed there. ${\displaystyle f(-1)=f(1)=4>3=f(0)}$ ${\displaystyle f}$ ${\displaystyle (-1,1)}$ ${\displaystyle x_{0}=0}$

• The function defined by for and by has the following properties: ${\displaystyle f(x)=\mathrm {e} ^{-1/x^{2}}\sin ^{2}{\frac {1}{x^{2}}}}$ ${\displaystyle x\neq 0}$ ${\displaystyle f(0)=0}$
  • It has a global minimum. ${\displaystyle x=0}$
  • It can be differentiated any number of times.
  • All derivatives at are equal to 0. ${\displaystyle x=0}$
  • The first derivative has no sign change at 0.
  • The other two criteria mentioned above are also not applicable.

Application example

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

• What should a rectangular area look like that has a maximum area with a certain circumference?

Solution:

The circumference is constant, the area should be maximized, the length and width are: ${\displaystyle U}$ ${\displaystyle A}$ ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle 1)\qquad U=2(a+b)\Rightarrow b={\frac {U}{2}}-a}$

${\displaystyle 2)\qquad A=a\cdot b}$

1) in 2) insert and reshape

${\displaystyle A(a)=-a^{2}+{\frac {1}{2}}Ua}$

Form derivative functions

${\displaystyle A'(a)=-2a+{\frac {1}{2}}U}$

${\displaystyle A''(a)=-2\qquad \Rightarrow }$ High point of function

There is only one local maximum, which in the present example (without verification) is also the global maximum, since the second derivative is always less than zero, regardless of the variable.

In order to find an extreme value, the first derivative must be set to zero (since this describes the slope of the original function and this slope is zero for extreme values. If the second derivative of the function is not equal to zero, then there is a minimum or maximum).

${\displaystyle A'(a)=-2a+{\frac {1}{2}}U=0\Rightarrow }$

${\displaystyle a={\frac {1}{4}}U\ \Rightarrow \ U=4a}$

Insert in 1)

${\displaystyle 4a=2(a+b)\ \Rightarrow \ a=b}$

It follows from this that the largest possible area of ​​a rectangle with a given circumference can be achieved if both side lengths are the same (which corresponds to a square). Conversely, however, it can also be said that a rectangle with a given area has the smallest circumference if

${\displaystyle a:b=1:1}$

cautious - i.e. with a square.

Multi-dimensional case

It is and a function. Furthermore, let be an inner point of . A local minimum / maximum in is given if there is an environment around in which no point assumes a smaller or larger function value. ${\displaystyle U\subseteq \mathbb {R} ^{n}}$ ${\displaystyle f\colon U\to \mathbb {R} }$ ${\displaystyle x}$ ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle x}$

The disappearance of the gradient is analogous to the one-dimensional case

${\displaystyle Df(x)=\operatorname {grad} f(x)}$

a necessary condition for the point to assume an extreme. In this case, the definiteness of the Hessian matrix is sufficient : if it is positive definite, there is a local minimum; if it is negative, it is a local maximum; if it is indefinite, there is no extreme point, but a saddle point . If it is only semidefinite, no decision based on the Hessian matrix is ​​possible (see Peanos surface ). ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle D^{2}f(x)}$

Infinite dimensional case

definition

The concept of the maximum and the minimum transfers directly to the infinite-dimensional case. Is a vector space and a subset of this vector space as well as a functional. Then got on the spot ${\displaystyle X}$ ${\displaystyle D\subset X}$ ${\displaystyle f\colon D\to \mathbb {R} }$ ${\displaystyle f}$ ${\displaystyle {\tilde {x}}\in D}$

• a (global) minimum if for all ${\displaystyle f({\tilde {x}})\leq f(x)}$ ${\displaystyle x\in D}$
• a (global) maximum if for all ${\displaystyle f({\tilde {x}})\geq f(x)}$ ${\displaystyle x\in D}$

The addition “global” is usually left out if it is clear from the context what is meant. Is additionally provided with a topology provided, so a topological space , then is at the point ${\displaystyle X}$ ${\displaystyle f}$ ${\displaystyle {\tilde {x}}\in D}$

• a local minimum if there is a neighborhood of such that applies to all . ${\displaystyle U}$ ${\displaystyle {\tilde {x}}}$ ${\displaystyle f({\tilde {x}})\leq f(x)}$ ${\displaystyle x\in U\cap D}$
• a local maximum if there is a neighborhood of such that applies to all . ${\displaystyle U}$ ${\displaystyle {\tilde {x}}}$ ${\displaystyle f({\tilde {x}})\geq f(x)}$ ${\displaystyle x\in U\cap D}$

A point is called a (local) extremum if it is a (local) minimum or a (local) maximum. Every global minimum (maximum) is a local minimum (maximum).

Existence, uniqueness and geometry of extremes

existence

Corresponding to the existence statements for real functions, there are also statements for the existence of extreme places of functionals. If there is a normalized space , then: ${\displaystyle X}$

• A weakly sub-semicontinuous functional on a weakly following-compact set assumes its minimum there.

Since this version is often impractical for application and checking, this is weakened to the statement that every continuous quasi-convex functional assumes a minimum on a bounded, convex and closed subset of a reflexive Banach space . This statement also applies to all convex functionals, since these are always quasi-convex. In the finite-dimensional, the convexity of the subset can be dispensed with.

Uniqueness

Under certain circumstances, the optimum points are even clearly determined. This includes, for example, strict convexity .

geometry

If one restricts oneself to certain classes of functionals, one can make statements about the geometry of the set of extreme points.

• If the functional is quasi-convex on a convex set, then the set of minima is convex.
• If the functional is quasi-concave on a convex set, then the set of maxima is convex.
• If the functional is convex on a convex set, then every local minimum is a global minimum.
• If the functional is concave on a convex set, then every local maximum is a global maximum.

Other extreme values

Discrete optimization

In the case of discrete optimization problems , the concept of the local extremum defined above is not suitable, since a local extremum in this sense is present at every point. A different concept of environment is therefore used for the extremes of a function : A neighborhood function is used , which assigns the set of its neighbors to each point , ${\displaystyle f\colon D\to \mathbb {R} }$ ${\displaystyle N}$

${\displaystyle N\colon D\to {\mathcal {P}}(D);}$

stands for the power set of . ${\displaystyle {\mathcal {P}}(D)}$ ${\displaystyle D}$

${\displaystyle f}$then has a local maximum at a point if holds for all neighbors . Local minima are defined analogously. ${\displaystyle x_{0}\in D}$ ${\displaystyle f(x)\leq f(x_{0})}$ ${\displaystyle x\in N(x_{0})}$

Calculus of variations

Extreme values ​​of functions whose arguments are functions themselves, e.g. B. the contour of a raindrop with minimal air resistance, are the subject of the calculation of variations .

See also

• optimum
• arg max
• Maximum principle

Web links

Wiktionary: extreme value  - explanations of meanings, word origins, synonyms, translations

• Extreme values ​​(explained for students)

Individual evidence

1. ^ W. Gellert, H. Küstner, M. Hellwich, H. Kästner: Small encyclopedia of mathematics. Leipzig 1970, pp. 433-434.