Point M is called the inner for some set G, if it belongs to this set with some of its neighborhood. Point N is called the boundary for the set G, if there are points in any complete surroundings, as belonging to G and not belonging to it.

The combination of all boundary points of the set G is called the border.

The set G will be called an area if all of its points is internal (open set). The set G with the attached boundary r is called a closed area. The area is called limited if it is entirely contained inside the circle of a sufficiently large radius.

The smallest I. the greatest value Functions in this area are called absolute extremums of the function in this area.

Weierstrass theorem: The function, continuous in a limited and closed area, reaches its smallest and its largest values \u200b\u200bin this area.

Corollary. The absolute extremum function in this area is achieved either at a critical point of the function belonging to this area, or to find the greatest and smallest function values \u200b\u200bin a closed domain, it is necessary to find all its critical points in this area, calculate the values \u200b\u200bof the function at these points (including the boundary) and By comparing the numbers obtained, choose the largest and smallest of them.

Example 4.1. Find the absolute extremum function (the largest and smallest)
in a triangular region with vertices
,
,
(Fig.1).

;
,

that is, the point O (0, 0) is a critical point belonging to the region D. Z (0,0) \u003d 0.

We explore the border:

a) OA: Y \u003d 0
; z (x, 0) \u003d 0; z (0, 0) \u003d 0; z (1, 0) \u003d 0,

b) s: x \u003d 0
z (0, y) \u003d 0; z (0, 0) \u003d 0; z (0, 2) \u003d 0,

c) av:;
,

Example 4.2. Find the largest and smallest values \u200b\u200bof the function in a closed area limited by the coordinate axes and direct
.

1) We will find critical points lying in the area:

,
,

.

Explore the border. Because The border consists of a segment of the OA axis oh, the segment of the OS axis and the segment of AB, then we define the greatest and smallest values \u200b\u200bof the function Z on each of these segments.

, z (0, 2) \u003d - 3, z (0, 0) \u003d 5, z (0, 4) \u003d 5.

M 3 (5/3.7 / 3), Z (5/3, 7/3) \u003d - 10/3.

Among all the found values, we select z Naib \u003d z (4, 0) \u003d 13; z Nym \u003d z (1, 2) \u003d - 4.

5. Conditional extremum. Lagrange multiplier method

Consider the task specific to the functions of several variables when its extremum is not looking for on the entire definition area, but on the set that satisfies a certain condition.

Let the function consider
, arguments and which satisfies the condition
, called the communication equation.

Point
called a point of a conditional maximum (minimum) if there is such a neighborhood of this point, which is for all points
from this neighborhood satisfying condition
The inequality is performed
or
.

Figure 2 shows the point of the conditional maximum
. Obviously, it is not a point of unconditional extremum function
(Figure 2 is a point
).

The easiest way to find a conditional extremum functions of two variables is to reduce the task of finding the extremum of the function of one variable. Suppose the communication equation
managed to be resolved relative to one of the variables, for example, to express through :
. Substitting the resulting expression into the function of two variables, we get

those. The function of one variable. Its extremum will be a conditional extremum function
.

Example 5.1.Find a maximum point and a minimum function
given that
.

Decision. Express from the equation
variable through the variable and substitute the expression
in function . Receive
or
. This feature has the only minimum
. The corresponding function of the function
. In this way,
- Point of conditional extremum (minimum).

In the considered example, the communication equation
it turned out linear, so it was easily resolved relative to one of the variables. However, in more complex cases, this fails.

To find a conditional extremum in general, the method of Lagrange multipliers is used. Consider the function of three variables. This feature is called Lagrange function, and - Lagrange multiplier. The following theorem is true.

Theorem.If point
is a point of conditional extremum function
given that
then there is a value such that point
is a point of extremum function
.

Thus, to find a conditional extremum function
given that
requires System Solution

P extraward from these equations coincides with the communication equation. The first two system equations can be rewritten in the form, i.e. At the point of conditional extremum gradients of functions
and
collinear. In fig. 3 shows the geometrical meaning of Lagrange conditions. Line
dotted, level line
functions
solid. From fig. It follows that at the point of the conditional extremum line level function
concerns line
.

Example 5.2.. Find Extremum Points Functions
given that
Using Lagrange multiplier method.

Decision. We make a function of Lagrange. Equating to zero its private derivatives, we obtain the system of equations:

Her sole solution. Thus, the point of the conditional extremum can only be point (3; 1). It is easy to make sure that at this point the function
it has a conditional minimum. In the event that the number of variables is more than two, it also betes to consider several equations of communication. Accordingly, in this case there will be several Lagrange multipliers.

The task of finding a conditional extremum is used in solving such economic tasks as finding the optimal distribution of resources, the choice of the optimal portfolio of securities, etc.

Lagrange method

The method of bringing the quadratic form to the sum of the squares specified in 1759 J. Lagrang (J. Lagrange). Let Dana.

from plern x 0 , X. 1 , ..., x n. with coefficients from the field k. Characteristics need to bring this form to canonical. see

with the help of a nondegenerate linear transformation of variables. L. m. Consists of the following. We can assume that not all form coefficients (1) are zero. Therefore, two cases are possible.

1) with some g,diagonal then

where shape F 1 (x). Do not contain a variable x g.2) if all but that

where F 2 (x) shape contains two variables x G. and x h. Forms under the signs of squares in (4) are linearly independent. The use of transformations of the form (3) and (4) shape (1) after a finite number of steps is given to the sum of squares of linearly independent linear forms. With the help of private derivatives of formula (3) and (4) can be written as

Lit.: G and n t m a x e r f. R., The theory of matrices, 2 ed., M., 1966; K U R O W A. G., course of higher algebra, 11 ed., M., 1975; Alexandrov P. S., lectures on analytical geometry ..., M., 1968. I. V. Proskuryakov.

Mathematical encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985.

Watch what is "Lagrange Method" in other dictionaries:

Lagrange method - Lagrange Method - Method for solving a number of classes of mathematical programming problems by finding a saddle point (x *, λ *) Lagrange functions., What is achieved by equating zero of private derivatives of this function by ... ... Economics and Mathematical Dictionary

Lagrange method - Method for solving a number of classes of mathematical programming tasks by finding a saddle point (x *,? *) Lagrange functions., What is achieved by equating zero of private derivatives of this function by xi and? I. See Lagrangian. )