The process of examining a function for the presence of stationary points and finding them is one of the important elements when plotting a function. You can find the stationary points of a function if you have a certain set of mathematical knowledge.

You will need

• - the function to be investigated for the presence of stationary points;
• - definition of stationary points: stationary points of a function are points (argument values) at which the derivative of the first order function vanishes.

Instruction

• Using the table of derivatives and formulas for differentiating functions, it is necessary to find the derivative of the function. This step is the most difficult and responsible in the course of the task. If you make a mistake on this stage, further calculations will not make sense.
• Check if the derivative of a function depends on the argument. If the found derivative does not depend on the argument, that is, it is a number (for example, f "(x) \u003d 5), then in this case the function does not have stationary points. Such a solution is possible only if the function under study is a linear function of the first order (to For example, f(x) = 5x+1) If the derivative of the function depends on the argument, then proceed to the last step.
• Compose the equation f "(x) \u003d 0 and solve it. The equation may not have solutions - in this case, the function has no stationary points. If the equation has solutions, then these found values ​​of the argument will be the stationary points of the function. On this At the next stage, the solution of the equation should be checked by the argument substitution method.

§ 3 STATIONARY POINTS AND DIFFERENTIAL CALCULUS 369

it can be seen that, generally speaking, there are two circles of the considered family tangent to the line l: their centers are located along different sides segment P Q. One of the points of contact gives the absolute maximum of the value j, while the other - only a "relative" maximum: this means that the values ​​of j at this point are greater than the values ​​in some neighborhood of the considered point. The larger of the two maxima - the absolute maximum - is given by the point of contact, which is located in the acute angle formed by the line l and the continuation of the segment P Q, and the smaller - by the point of contact, which is located in the obtuse angle formed by these lines. (The point of intersection of the line l with the continuation of the segment P Q gives the minimum value of the angle j, namely j = 0.)

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

Generalizing the considered problem, we can replace the line l by some curve C and look for points R on the curve C, of ​​which the given segment P Q, which does not intersect C, is visible at the largest 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 the point R.

§ 3. Stationary points and differential calculus

1. Extreme and stationary points. In the foregoing reasoning, we did not use the technical methods of differential calculus at all.

It is hard not to admit that our elementary methods are simpler and more direct than those 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 common methods, although, on the other hand, general principle, clarifying the meaning of the special procedures applied, of course, should always play a leading role. Such is precisely the significance of the methods of differential calculus in dealing with 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 undoubtedly determined by individual traits problems under consideration and applied methods.

In his historical development differential calculus has been affected to a very large extent by the 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 study in detail the derivative f0 (x) of the function f(x) and its geometric meaning. There we will see that, in short, 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 points of maximum or minimum of a smooth curve y = f(x), the tangent to the curve must necessarily be horizontal, i.e., the slope must be equal to zero. Thus, we obtain the condition f0 (x) = 0 for the extremum points.

To be clear about what the vanishing of the derivative f0 (x) means, 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; we 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 reached at point D, the smallest - at point A. At point B there is a maximum - in the sense that at all points of some neighborhood of point B, the value of f(x) is less than 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, point C has a relative minimum, and point A has an absolute minimum. Finally, as regards the point E, it has neither a maximum nor a minimum, although the equality f0 (x) = 0 still holds at it. It follows from this that the vanishing of the derivative f0 (x) is necessary, but by no means 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 f0 (x) = 0 certainly takes place, but not at any point where f0 (x) = 0, there must be an extremum. Those points at which the derivative f0 (x) vanishes - regardless of whether they have an extremum - are called stationary. Further analysis leads to more or less

§ 3 STATIONARY POINTS AND DIFFERENTIAL CALCULUS 371

complex 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 minima of functions of several variables. saddle points. There are extreme problems that cannot be expressed in terms of a function f(x) of one variable. The simplest related example is the problem of finding the extrema of the function z = f(x, y) in two independent variables.

We can always think of the function f(x, y) as the height z of the surface above the x, y plane, and we will interpret this picture, say, as a mountain landscape. The maximum of the function f(x, y) corresponds to mountain top, at least - to 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, besides the tops of the 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 explore them more closely. 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 range; suppose that we need to go from C to D. Consider first those paths leading from C to D, which 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 highest point will be in

lowest possible position. The highest point E on this path is the point of a mountain pass in our landscape; it can also be called a saddle point. It is clear that there is neither a maximum nor a minimum at the point E, since, arbitrarily close to E, there are points on the surface that are higher than E and those that are lower than E. In the previous argument, it would be possible not to restrict ourselves to considering only those paths , which arise when the surface is crossed by planes, and consider any paths connecting C and D. The characteristic given by us to the point E would not change from this.

In the same way, if we wished to go from vertex A to vertex B, then every path we could take would have a lowest point; considering even only plane sections, we would find such a path AB for which the smallest point would be located the highest, and we would again get the same point E. Thus, this saddle point E has the property to deliver the highest minimum or the lowest maximum: here there is a "maximum" or "minimum" - abbreviated minimax. The tangent plane at point E is horizontal; indeed, since E is the lowest point of the path AB, the tangent to AB at E is horizontal, and similarly, since E is the highest point of the path CD, the tangent to CD at 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 values ​​of the function.

Another way to represent the function f(x, y) geometrically is to draw level lines - the same ones used in cartography to indicate heights on the ground (see p. 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, the level lines are the same as the curves of the family f(x, y) = c. Through ordinary

Rice. 194. Stacy onary points in a doubly connected domain

§ 3 STATIONARY POINTS AND DIFFERENTIAL CALCULUS 373

the point of the plane passes exactly one level line; maximum and minimum points are surrounded closed lines level, two (or more) level lines intersect at saddle points. On fig. 193 level lines are drawn corresponding to the landscape depicted in fig. 192.

In this case, the remarkable property of the saddle point E becomes especially clear: any path connecting A and B and not passing through E lies partially 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. On this point we can only give a brief indication here and confine ourselves to considering one example.

Consider a mountain 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 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, es-

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

ing C and C0 , which would not rise to a greater height than it is inevitable. Each 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). (We should mention the trivial case, which is an exception, when some horizontal plane touches the annular mountain range along a closed curve.) In the case of a domain bounded by p closed curves, generally speaking, there must be at least p − 1 minimax points. Relations of the same kind, as established by Marston Morse, also hold for multidimensional regions,

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

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

from different points of a closed curve, there are (at least) two stationary values: a minimum and a maximum. When passing to three dimensions, no new facts are revealed if we restrict 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. Consider the surface of the torus C. Whatever the point P is, of course, there are always points on the torus C that give the largest 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. Imagine on the torus one of the "meridian" circles L (Fig. 195) and on this circle L we find the point Q closest to P. Then, moving the circle L along the torus, we find its position so 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 farthest 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

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

Definitions:

extremum name the maximum or minimum value of a function on a given set.

extreme 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.

Low 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 higher point). In the neighborhood of x = 8, it again has a maximum value (again, let's clarify: it is in this neighborhood that there is no point above). At these points, the increase is replaced by a decrease. They are maximum points:

xmax = 3, xmax = 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 is replaced by 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 extremum:

Sufficient condition for an extremum:

On the segment, the function y = f(x) can reach its minimum or maximum value either at critical points or at the ends of the segment .

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

Stationary points of a function. A necessary condition for a local extremum of a function

The first sufficient condition for a local extremum

The second and third sufficient conditions for a local extremum

The smallest and largest values ​​of a function on a segment

Convex functions and inflection points

1. Stationary points of a function. A necessary condition for a local extremum of a function

Definition 1 . Let the function be defined on
. Dot is called the stationary point of the function
, if
differentiated at a point and
.

Theorem 1 (necessary condition for a local extremum of a function) . Let the function
determined on
and has at the point
local extreme. Then one of the following conditions is met:

Thus, in order to find points that are suspicious of an extremum, it is necessary to find stationary points of the function and points at which the derivative of the function does not exist, but which belong to the domain of the function.

Example . Let
. Find points for it that are suspicious for an extremum. To solve the problem, first of all, we find the domain of the function:
. We now find the derivative of the function:

Points where the derivative does not exist:
. Stationary function points:

Because and
, and
belong to the domain of the function definition, then both of them will be suspicious for an extremum. But in order to conclude whether there really will be an extremum, it is necessary to apply sufficient conditions for the extremum.

2. First sufficient condition for a local extremum

Theorem 1 (first sufficient condition for a local extremum) . Let the function
determined on
and is differentiated on this interval everywhere, except possibly at the point
, but at this point function
is continuous. If there exist such right and left semi-neighborhoods of a point , in each of which
retains a certain sign, then

1) function
has a local extremum at the point , if
takes values ​​of different signs in the corresponding semi-neighborhoods;

2) function
does not have a local extremum at the point , if to the right and to the left of the point
has the same sign.

Proof . 1) Assume that in a semi-neighborhood
derivative
, and in

.

Thus at the point function
has a local extremum, namely, a local maximum, which was to be proved.

2) Suppose that to the left and to the right of the point the derivative retains its sign, for example,
. Then on
and
function
strictly monotonically increasing, that is:

Thus the extremum at the point function
does not, which was to be proved.

Remark 1 . If the derivative
when passing through a point changes sign from "+" to "-", then at the point function
has a local maximum, and if the sign changes from "-" to "+", then a local minimum.

Remark 2 . An important condition is the continuity of the function
at the point . If this condition is not satisfied, then Theorem 1 may not hold.

Example . The function is considered (Fig. 1):

This function is defined on and is continuous everywhere except for the point
, where it has a removable discontinuity. When passing through a point

changes sign from "-" to "+", but the function does not have a local minimum at this point, but has a local maximum by definition. Indeed, near the point
it is possible to construct such a neighborhood that for all arguments from this neighborhood the values ​​of the function will be less than the value
. Theorem 1 did not work because at the point
the function had a break.

Remark 3 . The first sufficient local extremum condition cannot be used when the derivative of the function
changes its sign in each left and each right semi-neighbourhood of the point .

Example . The function being considered is:

Insofar as
, then
, and therefore
, but
. In this way:

,

those. at the point
function
has a local minimum by definition. Let's see if the first sufficient condition for a local extremum works here.

For
:

For the first term on the right side of the resulting formula, we have:

,

and therefore in a small neighborhood of the point
the sign of the derivative is determined by the sign of the second term, that is:

,

which means that in any neighborhood of the point

will take both positive and negative values. Indeed, consider an arbitrary neighborhood of the point
:
. When

,

then

(Fig. 2), and changes its sign here infinitely many times. Thus, the first sufficient condition for a local extremum cannot be used in the above example.