Among the stationary points of the function there is a point. Critical points on the graph of a function

Definitions:

Extremum call the maximum or minimum value of a function on a given set.

Extremum point is the point at which the maximum or minimum value of the function is reached.

Maximum point is the point at which the maximum value of the function is reached.

Minimum point is the point at which the minimum value of the function is reached.

Explanation.

In the figure, in the vicinity of the point x = 3, the function reaches its maximum value (that is, in the vicinity of this particular point there is no point higher). In the neighborhood of x = 8, it again has a maximum value (let us clarify again: it is in this neighborhood that there is no point higher). At these points, the increase gives way to a decrease. They are the maximum points:

x max = 3, x max = 8.

In the vicinity of the point x = 5, the minimum value of the function is reached (that is, in the vicinity of x = 5 there is no point below). At this point the decrease gives way to an increase. It is the minimum point:

The maximum and minimum points are extremum points of the function, and the values ​​of the function at these points are its extremes.

Critical and stationary points of the function:

Necessary condition for an extremum:

Sufficient condition for an extremum:

On a segment the function y = f(x) can reach the smallest or largest value either at critical points or at the ends of the segment .

Algorithm for studying a continuous functiony = f(x) for monotonicity and extrema:

§ 3 STATIONARY POINTS AND DIFFERENTIAL CALCULUS 369

it is clear that, generally speaking, there are two circles of the family under consideration, tangent to the straight line l: their centers are located along different sides segment P Q. One of the points of tangency gives the absolute maximum of the value of j, while the other gives only a “relative” maximum: this means that the values ​​of j at this point are greater than the values ​​in some neighborhood of the point in question. The larger of the two maximums - the absolute maximum - is given by the point of contact that is located at sharp corner, formed by the straight line l and the continuation of the segment P Q, and the smaller one - by the point of tangency that is located at the obtuse angle formed by these straight lines. (The point of intersection of the straight line l with the continuation of the segment P Q gives the minimum value of the angle j, namely j = 0.)

Rice. 190. From which point l is the segment P Q visible at the greatest angle?

Generalizing the problem considered, we can replace straight line l with some curve C and look for points R on curve C from which a given segment P Q, not intersecting C, is visible at the greatest or smallest angle. In this problem, as in the previous one, the circle passing through P, Q and R must touch the curve C at point R.

§ 3. Stationary points and differential calculus

1. Extreme and stationary points. In the previous discussions we did not use the technical methods of differential calculus at all.

It is difficult not to admit that our elementary methods are simpler and more direct than the methods of analysis. In general, when dealing with a particular scientific problem, it is better to proceed from its individual

features rather than relying solely on general methods, although, on the other hand, general principle, which clarifies the meaning of the special procedures applied, of course, should always play a leading role. This is precisely the significance of the methods of differential calculus when considering extremal problems. Observed in modern science the desire for generality represents only one side of the matter, since what is truly vital in mathematics is, without any doubt, determined by the individual characteristics of the problems considered and the methods used.

In his historical development differential calculus has been influenced to a very large extent by individual problems associated with finding the largest and lowest values quantities The connection between extremal problems and differential calculus can be understood as follows. In Chapter VIII we will undertake a detailed study of the derivative f0 (x) of the function f(x) and its geometric meaning. There we will see that, briefly speaking, the derivative f0 (x) is the slope of the tangent to the curve y = f(x) at the point (x, y). It is geometrically obvious that at the maximum or minimum points of a smooth curve y = f(x), the tangent to the curve must certainly be horizontal, i.e., the slope must be zero. Thus, we obtain the condition f0 (x) = 0 for the extremum points.

To clearly understand what it means for the derivative f0 (x) to vanish, consider the curve shown in Fig. 191. We see here five points A, B, C, D, E, at which the tangent to the curve is horizontal; Let us denote the corresponding values ​​of f(x) at these points by a, b, c, d, e. Highest value f(x) (within the area shown in the drawing) is achieved at point D, the smallest at point A. At point B there is a maximum - in the sense that at all points in a certain neighborhood of point B the value of f(x) is less than b, although at points close to D, the value of f(x) is still greater than b. For this reason, it is customary to say that at point B there is a relative maximum of the function f(x), while at point D there is an absolute maximum. Similarly, at point C there is a relative minimum, and at point A there is an absolute minimum. Finally, as for point E, there is neither a maximum nor a minimum at it, although the equality f0 (x) = 0 is still true there. It follows that the vanishing of the derivative f0 (x) is necessary, but not sufficient condition for the appearance of an extremum of a smooth function f(x); in other words, at every point where there is an extremum (absolute or relative), the equality f0 (x) = 0 certainly holds, but not at every point where f0 (x) = 0, there must be an extremum. Those points at which the derivative f0 (x) vanishes, regardless of whether there is an extremum at them, are called stationary. Further analysis leads to more or less

§ 3 STATIONARY POINTS AND DIFFERENTIAL CALCULUS 371

difficult conditions, concerning the higher derivatives of the function f(x) and completely characterizing the maxima, minima and other stationary points.

Rice. 191. Stationary points of a function

2. Maxima and minimum of functions of several variables. Saddle points. There are extreme problems that cannot be expressed using the concept of a function f(x) of one variable. The simplest example relevant here is the problem of finding the extrema of the function z = f(x, y) of two independent variables.

We can always imagine the function f(x, y) as the height z of the surface above the x, y plane, and we will interpret this picture as, say, a mountain landscape. The maximum of the function f(x, y) corresponds to mountain peak, minimum - the bottom of a pit or lake. In both cases, unless the surface is smooth, the tangent plane to the surface is necessarily horizontal. But, in addition to the tops of mountains and the lowest points in the pits, there may be other points at which the tangent plane is horizontal: these are “saddle” points corresponding to mountain passes. Let's examine them more carefully. Suppose (Fig. 192) that there are two peaks A and B in a mountain range and two points C and D on different slopes of the ridge; Let us assume that from C we need to go to D. Let us first consider those paths leading from C to D that are obtained by intersecting the surface with planes passing through C and D. Each such path has a highest point. When the position of the cutting plane changes, the path also changes, and it will be possible to find a path for which the highest point will be at

lowest possible position. The highest point E on this route is the point of a mountain pass in our landscape; it can also be called a saddle point. It is clear that at point E there is neither a maximum nor a minimum, since no matter how close to E there are points on the surface that are above E and those that are below E. In the previous reasoning, one could not limit ourselves to considering only those paths , which arise when planes intersect a surface, and consider any paths connecting C and D. The characteristic we gave to point E would not change from this.

In the same way, if we wanted to go from peak A to peak B, then every path we could choose would have a lowest point; even considering only plane sections, we would find a path AB for which the smallest point would be located the highest, and again the previous point E would be obtained. Thus, this saddle point E has the property of delivering the highest minimum or the lowest maximum: here there is a “maximinum” or “minimaximum” - minimax for short. The tangent plane at point E is horizontal; indeed, since E is the lowest point of the path AB, then the tangent to AB in E is horizontal, and similarly, since E is the highest point of the path CD, then the tangent to CD in E is horizontal. Therefore, the tangent plane necessarily passing through these two tangent lines is horizontal. So we find three various types points with horizontal tangent planes: maximum points, minimum points and, finally, saddle points; Accordingly, there are three different types of stationary function values.

Another way to represent the function f(x, y) geometrically is to draw level lines - the same ones that are used in cartography to indicate heights on the ground (see page 308). A level line is a curve in the x, y plane along which the function f(x, y) has the same value; in other words, level lines are the same as curves of the family f(x, y) = c. Through ordinary

Rice. 194. Staci unary points in a doubly connected region

§ 3 STATIONARY POINTS AND DIFFERENTIAL CALCULUS 373

exactly one level line passes a point on the plane; maximum and minimum points are surrounded by closed level lines; two (or more) level lines intersect at saddle points. In Fig. 193 level lines are drawn corresponding to the landscape shown in Fig. 192.

In this case, the remarkable property of the saddle point E becomes especially clear: every path connecting A and B and not passing through E partially lies in the region where f(x, y)< f(E), тогда как путь AEB на рис. 192 имеет минимум как раз в точке E. Таким же образом мы убеждаемся, что значение f(x, y) в точке E представляет собой наименьший максимум на путях, связывающих C и D.

3. Minimax points and topology. There is a deep connection between general theory stationary points and topological ideas. In this regard, we can only give a brief indication here and limit ourselves to considering one example.

Consider a mountainous landscape on a ring-shaped island B with two coastal contours C and C0; if we denote, as before, the height above sea level by u = f(x, y), and let us assume that f(x, y) = 0 on the contours C and C0 and f(x, y) > 0

inside, then there must be at least one mountain pass on the island: in Fig. 194 such a pass is located at the point where two level lines intersect. The validity of the stated statement becomes clear when

Should we set ourselves the task of finding such a path, connecting

common C and C0, which would not rise to a greater height than it is inevitable. Every the path from C to C0 has the highest

highest point, and if we choose a path for which the highest point is the lowest, then the highest point thus obtained will be the saddle point of the function u = f(x, y). (It is necessary to stipulate the trivial case, which represents an exception, when a certain horizontal plane touches a ring-shaped mountain range along a closed curve.) In the case of a region bounded by p closed curves, in general there must exist at least p − 1 minimax points. Similar relationships, as established by Marston Morse, also take place for multidimensional regions,

but the variety of topological possibilities and types of stationary points in this case is much greater. These relationships form the basis modern theory stationary points.

4. Distance of a point from the surface. For point P distances

from various points For a closed curve, there are (at least) two stationary values: a minimum and a maximum. When moving to three dimensions, no new facts are discovered if we limit ourselves to considering a surface C that is topologically equivalent to a sphere (such as an ellipsoid). But if the surface is of genus 1 or higher, then the situation is different. Let us consider the surface of the torus C. Whatever the point P, there are, of course, always points on the torus C that give the greatest and smallest distance from P, and the corresponding segments are perpendicular to the surface itself. But we will now establish that in this case there are also minimax points. Let us imagine one of the “meridian” circles L on the torus (Fig. 195) and on this circle L we will find the point Q closest to P. Then, moving the circle L along the torus, we find its position such that the distance P Q becomes: a) minimal - then we get a point on C closest to P; b) maximum - then you get a stationary minimax point. In the same way, we could find the point on L that is furthest away from P, and then look for the position of L at which the greatest distance found would be: c) maximum (we get the point on C furthest from P), d) minimum. So we get four different stationary values ​​for the distance of the torus point C from the point P.

Rice. 195–196. Distance from point to surface

Exercise. Repeat the same reasoning for another type L0 of a closed curve on C, which also cannot be contracted to a point (Fig. 196).

In the previous discussions we did not use the technical methods of differential calculus at all.

It is difficult not to admit that our elementary methods are simpler and more direct than the methods of analysis. In general, when dealing with a particular scientific problem, it is better to proceed from its individual characteristics than to rely solely on general methods, although, on the other hand, the general principle, which clarifies the meaning of the special procedures applied, of course, should always play a guiding role. This is precisely the significance of the methods of differential calculus when considering extremal problems. The desire for generality observed in modern science represents only one side of the matter, since what is truly vital in mathematics is, without any doubt, determined by the individual characteristics of the problems considered and the methods used.

In its historical development, differential calculus was to a very significant extent influenced by individual problems associated with finding the largest and smallest values ​​of quantities. The connection between extremal problems and differential calculus can be understood as follows. In Chapter VIII we will engage in a detailed study of the derivative f"(x) of the function f(x) and its geometric meaning. There we will see that, briefly speaking, the derivative f"(x) is the slope of the tangent to the curve y = f(x) at point (x, y). It is geometrically obvious that at the maximum or minimum points of a smooth curve y = f(x) the tangent to the curve must certainly be horizontal, i.e., the slope must be zero. Thus, we obtain the condition for extremum points f"(x) = 0.

To clearly understand what it means for the derivative f"(x) to vanish, consider the curve shown in Fig. 191. We see here five points A, B, C, D, ?, at which the tangent to the curve is horizontal ; let us denote the corresponding values ​​of f(x) at these points by a, b, c, d, e. The largest value of f(x) (within the area shown in the drawing) is achieved at point D, the smallest at point A. At point B there is a maximum - in the sense that at all points some neighborhood points B, the value of f(x) is less than b, although at points close to D, the value of f(x) is still greater than b. For this reason, it is customary to say that at point B there is relative maximum of function f(x), whereas at point D - absolute maximum. In the same way, at point C there is relative minimum, and at point A - absolute minimum. Finally, as for point E, there is neither a maximum nor a minimum in it, although the equality is still realized in it f"(x) = Q, It follows that the vanishing of the derivative f"(x) is necessary, but not at all sufficient condition for the appearance of an extremum of a smooth function f(x); in other words, at any point where there is an extremum (absolute or relative), the equality certainly takes place f"(x) = 0, but not at every point where f"(x) = 0, must be an extremum. Those points at which the derivative f"(x) vanishes, regardless of whether there is an extremum at them, are called stationary. Further analysis leads to more or less complex conditions concerning the higher derivatives of the function f(x) and completely characterizing the maxima, minima and other stationary points.

Critical points– these are the points at which the derivative of a function is equal to zero or does not exist. If the derivative is equal to 0 then the function at this point takes local minimum or maximum. On the graph at such points the function has a horizontal asymptote, that is, the tangent is parallel to the Ox axis.

Such points are called stationary. If you see a “hump” or “hole” on the graph of a continuous function, remember that the maximum or minimum is reached at a critical point. Let's take the following task as an example.

Example 1. Find the critical points of the function y=2x^3-3x^2+5.
Solution. The algorithm for finding critical points is as follows:

So the function has two critical points.

Next, if you need to study a function, then we determine the sign of the derivative to the left and to the right of the critical point. If the derivative changes sign from “-” to “+” when passing through the critical point, then the function takes local minimum. If from “+” to “-” should local maximum.

Second type of critical points these are the zeros of the denominator of fractional and irrational functions

Logarithmic and trigonometric functions that are not defined at these points


Third type of critical points have piecewise continuous functions and modules.
For example, any module-function has a minimum or maximum at the break point.

For example module y = | x -5 | at point x = 5 has a minimum (critical point).
The derivative does not exist in it, but on the right and left takes the value 1 and -1, respectively.

Try to determine the critical points of functions

1)
2)
3)
4)
5)

If the answer is y you get the value
1) x=4;
2) x=-1;x=1;
3) x=9;
4) x=Pi*k;
5) x=1.
then you already know how to find critical points and be able to cope with a simple test or tests.