In July 2020, NASA launches an expedition to Mars. The spacecraft will deliver to Mars an electronic carrier with the names of all registered members of the expedition.

Registration of participants is open. Get your ticket to Mars at this link.

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One of these code options should be copied and pasted into your web page code, preferably between the tags and or right after the tag. According to the first option, MathJax loads faster and slows down the page less. But the second option automatically tracks and loads the latest versions of MathJax. If you insert the first code, then it will need to be updated periodically. If you paste the second code, then the pages will load more slowly, but you will not need to constantly monitor MathJax updates.

The easiest way to connect MathJax is in Blogger or WordPress: in the site control panel, add a widget designed to insert third-party JavaScript code, copy the first or second version of the load code above into it, and place the widget closer to the beginning of the template (by the way, this is not necessary at all , since the MathJax script is loaded asynchronously). That's all. Now learn the MathML, LaTeX, and ASCIIMathML markup syntax and you're ready to embed math formulas into your web pages.

Another New Year's Eve... frosty weather and snowflakes on the window glass... All this prompted me to write again about... fractals, and what Wolfram Alpha knows about it. On this occasion, there is an interesting article in which there are examples of two-dimensional fractal structures. Here we will consider more complex examples of three-dimensional fractals.

A fractal can be visually represented (described) as a geometric figure or body (meaning that both are a set, in this case, a set of points), the details of which have the same shape as the original figure itself. That is, it is a self-similar structure, considering the details of which, when magnified, we will see the same shape as without magnification. Whereas in the case of an ordinary geometric figure (not a fractal), when zoomed in, we will see details that have a simpler shape than the original figure itself. For example, at a sufficiently high magnification, part of an ellipse looks like a straight line segment. This does not happen with fractals: with any increase in them, we will again see the same complex shape, which with each increase will be repeated again and again.

Benoit Mandelbrot, the founder of the science of fractals, in his article Fractals and Art for Science wrote: "Fractals are geometric shapes that are as complex in their details as they are in their overall form. That is, if part of the fractal will be enlarged to the size of the whole, it will look like the whole, or exactly, or perhaps with a slight deformation.

Definition . Extremum points of a function of two variables are the minimum and maximum points of this function. The values ​​of the function itself at the extremum points are called extrema of a function of two variables .

Definition . Dot P(x 0 , y 0 ) is called z = z(x, y) , if the value of the function at this point is greater than at the points of its neighborhood. The value of the function at the maximum point is called the maximum of a function of two variables .

Definition . Dot P(x 0 , y 0 ) is called maximum point of a function of two variables z = z(x, y) , if the value of the function at this point is greater than at the points of its neighborhood. Function value at the maximum point is called the maximum of a function of two variables .

Theorem (a necessary criterion for the extremum of a function of two variables). If point P(x 0 , y 0 ) - extremum point of a function of two variables z = z(x, y) , then the first partial derivatives of the function (with respect to "x" and "y") at this point are equal to zero or do not exist:

Definition . The point at which the first partial derivatives of a function of two variables are equal to zero is called stationary points .

Definition . The point at which the first partial derivatives of a function of two variables are equal to zero or do not exist is called critical points .

As in the case of a function of one variable, the necessary condition for the existence of an extremum of a function of two variables is not sufficient. There are quite a few functions in which the first partial derivative of the function is equal to zero or does not exist, but there are no extrema at the corresponding points. Every extremum point is a critical point, but not every critical point is an extremum .

A sufficient criterion for the existence of an extremum of a function of two variables. At the point P there is an extremum of a function of two variables if, in the vicinity of this point, the total increment of the function does not change sign. Since at the critical point the first total differential is equal to zero, then the increment of the function determines the second total differential

The best understanding of the application of the total differential will come from studying and practicing steps 3 and 4 of the algorithm for finding extrema of a function of two variables, which follows the second point of this lesson.

Local nature of extrema of a function of two variables. The maximum of a function of two variables in any part of the domain of definition of the function is not necessarily the maximum in the entire domain of definition, just as the minimum in any section is not a minimum in the entire domain of definition. Let us consider the height of the waves in the area of ​​the coastal area of ​​the sea (the area is smaller than the area). Then in this section we can fix (at least visually) the highest wave height. But in another section, where the wind causes a greater wave height, we fix the minimum wave height. This means that the maximum wave height in the first section may be less than the minimum wave height in the second section. Therefore, as in the case of an extremum of a function of one variable, it is necessary to clarify this concept and speak of extrema as local extrema of a function of two variables.

Of greatest interest is the algorithm for finding the extrema of a function of two variables, since, firstly, it differs from the algorithm for finding the extrema of a function of one variable, and secondly, by analogy with it, one can compose an algorithm for finding a function of three variables. In particular, it will be necessary to calculate the determinants .

So, the algorithm for finding the extrema of a function of two variables .

Given a function of two variables .

Step 2. We compose a system of equations from the equalities of these derivatives to zero (their equality to zero is a necessary sign of the existence of an extremum):

The solutions of this system of equations are points of a possible extremum - critical points.

Step 3. Let be the critical point found at step 2. To make sure that there is an extremum of the function of two variables at it, we find the second-order partial derivatives

as partial derivatives of the first-order partial derivatives found in step 1.

Step 4. We assign the second-order partial derivatives found in step 3 to letter designations:

Step 4. Find the determinant:

, i.e., there is no extremum at the found critical point,

and , i.e. at the found critical point there is a minimum of the function of two variables,

and , i.e., at the critical point found, there is a maximum of a function of two variables.

Let us turn to the graph of the function y \u003d x 3 - 3x 2. Consider the neighborhood of the point x = 0, i.e. some interval containing this point. It is logical that there is such a neighborhood of the point x \u003d 0 that the function y \u003d x 3 - 3x 2 takes the largest value in this neighborhood at the point x \u003d 0. For example, on the interval (-1; 1) the largest value equal to 0, the function takes at the point x = 0. The point x = 0 is called the maximum point of this function.

Similarly, the point x \u003d 2 is called the minimum point of the function x 3 - 3x 2, since at this point the value of the function is not greater than its value at another point in the vicinity of the point x \u003d 2, for example, the neighborhood (1.5; 2.5).

Thus, the point x 0 is called the maximum point of the function f (x) if there is a neighborhood of the point x 0 - such that the inequality f (x) ≤ f (x 0) is satisfied for all x from this neighborhood.

For example, the point x 0 \u003d 0 is the maximum point of the function f (x) \u003d 1 - x 2, since f (0) \u003d 1 and the inequality f (x) ≤ 1 is true for all values ​​of x.

The minimum point of the function f (x) is called the point x 0 if there is such a neighborhood of the point x 0 that the inequality f (x) ≥ f (x 0) is satisfied for all x from this neighborhood.

For example, the point x 0 \u003d 2 is the minimum point of the function f (x) \u003d 3 + (x - 2) 2, since f (2) \u003d 3 and f (x) ≥ 3 for all x.

Extreme points are called minimum points and maximum points.

Let us turn to the function f(x), which is defined in some neighborhood of the point x 0 and has a derivative at this point.

If x 0 is an extremum point of a differentiable function f (x), then f "(x 0) \u003d 0. This statement is called Fermat's theorem.

Fermat's theorem has a clear geometric meaning: at the extremum point, the tangent is parallel to the x-axis and therefore its slope
f "(x 0) is zero.

For example, the function f (x) \u003d 1 - 3x 2 has a maximum at the point x 0 \u003d 0, its derivative f "(x) \u003d -2x, f "(0) \u003d 0.

The function f (x) \u003d (x - 2) 2 + 3 has a minimum at the point x 0 \u003d 2, f "(x) \u003d 2 (x - 2), f "(2) \u003d 0.

Note that if f "(x 0) \u003d 0, then this is not enough to assert that x 0 is necessarily the extremum point of the function f (x).

For example, if f (x) \u003d x 3, then f "(0) \u003d 0. However, the point x \u003d 0 is not an extremum point, since the function x 3 increases on the entire real axis.

So, the extremum points of a differentiable function must be sought only among the roots of the equation
f "(x) \u003d 0, but the root of this equation is not always an extremum point.

Stationary points are points at which the derivative of a function is equal to zero.

Thus, in order for the point x 0 to be an extremum point, it is necessary that it be a stationary point.

Consider sufficient conditions for a stationary point to be an extremum point, i.e. conditions under which a stationary point is a minimum or maximum point of a function.

If the derivative to the left of the stationary point is positive, and to the right it is negative, i.e. derivative changes sign "+" to sign "-" when passing through this point, then this stationary point is the maximum point.

Indeed, in this case, to the left of the stationary point, the function increases, and to the right, it decreases, i.e. this point is the maximum point.

If the derivative changes sign "-" to sign "+" when passing through a stationary point, then this stationary point is a minimum point.

If the derivative does not change sign when passing through a stationary point, i.e. the derivative is positive or negative to the left and to the right of the stationary point, then this point is not an extremum point.

Let's consider one of the tasks. Find the extremum points of the function f (x) \u003d x 4 - 4x 3.

Solution.

1) Find the derivative: f "(x) \u003d 4x 3 - 12x 2 \u003d 4x 2 (x - 3).

2) Find stationary points: 4x 2 (x - 3) \u003d 0, x 1 \u003d 0, x 2 \u003d 3.

3) Using the interval method, we establish that the derivative f "(x) \u003d 4x 2 (x - 3) is positive for x\u003e 3, negative for x< 0 и при 0 < х < 3.

4) Since when passing through the point x 1 \u003d 0, the sign of the derivative does not change, this point is not an extremum point.

5) The derivative changes the sign "-" to the sign "+" when passing through the point x 2 \u003d 3. Therefore, x 2 \u003d 3 is the minimum point.

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A function and the study of its features occupies one of the key chapters in modern mathematics. The main component of any function is graphs depicting not only its properties, but also the parameters of the derivative of this function. Let's take a look at this tricky topic. So what is the best way to find the maximum and minimum points of a function?

Any variable that depends in some way on the values ​​of another quantity can be called a function. For example, the function f(x 2) is quadratic and determines the values ​​for the entire set x. Let's say that x = 9, then the value of our function will be equal to 9 2 = 81.

Functions come in a variety of types: logical, vector, logarithmic, trigonometric, numeric, and others. Such outstanding minds as Lacroix, Lagrange, Leibniz and Bernoulli were engaged in their study. Their writings serve as a bulwark in modern ways of studying functions. Before finding the minimum points, it is very important to understand the very meaning of the function and its derivative.

All functions are dependent on their variables, which means that they can change their value at any time. On the graph, this will be depicted as a curve that either descends or rises along the y-axis (this is the whole set of numbers "y" along the vertical of the graph). And so definition of a point of a maximum and a minimum of function just is connected with these "oscillations". Let us explain what this relationship is.

The derivative of any function is drawn on a graph in order to study its main characteristics and calculate how quickly the function changes (ie changes its value depending on the variable "x"). At the moment when the function increases, the graph of its derivative will also increase, but at any second the function may begin to decrease, and then the graph of the derivative will decrease. Those points at which the derivative goes from minus to plus are called minimum points. In order to know how to find minimum points, you should better understand

Definition and functions implies several concepts from In general, the very definition of the derivative can be expressed as follows: this is the value that shows the rate of change of the function.

The mathematical way of defining it for many students seems complicated, but in fact everything is much simpler. It is only necessary to follow the standard plan for finding the derivative of any function. The following describes how you can find the minimum point of a function without applying the rules of differentiation and without memorizing the table of derivatives.

In the school curriculum of mathematics, it is possible to find the minimum point of a function in two ways. We have already analyzed the first method using the graph, but how to determine the numerical value of the derivative? To do this, you will need to learn several formulas that describe the properties of the derivative and help convert variables like "x" into numbers. The following method is universal, so it can be applied to almost all kinds of functions (both geometric and logarithmic).

The most basic component in the study of a function and its derivative is the knowledge of the rules of differentiation. Only with their help it is possible to transform cumbersome expressions and large complex functions. Let's get acquainted with them, there are a lot of them, but they are all very simple due to the regular properties of both power and logarithmic functions.

We have already discussed how to find the minimum points, however, there is the concept of maximum points of a function. If the minimum denotes those points at which the function goes from minus to plus, then the maximum points are those points on the x-axis at which the derivative of the function changes from plus to the opposite - minus.

You can find it using the method described above, only it should be taken into account that they denote those areas where the function begins to decrease, that is, the derivative will be less than zero.

In mathematics, it is customary to generalize both concepts, replacing them with the phrase "points of extrema". When the task asks to determine these points, this means that it is necessary to calculate the derivative of this function and find the minimum and maximum points.

The extremum point of a function is the point in the function's domain where the value of the function takes on a minimum or maximum value. The function values ​​at these points are called extrema (minimum and maximum) of the function.

Definition . Dot x 1 function scope f(x) is called maximum point of the function, if the value of the function at this point is greater than the values ​​of the function at points close enough to it, located to the right and left of it (that is, the inequality f(x 0 ) > f(x 0 + Δ x) x 1 max.

Definition . Dot x 2 function scopes f(x) is called the minimum point of the function if the value of the function at this point is less than the values ​​of the function at points close enough to it, located to the right and left of it (that is, the inequality f(x 0 ) < f(x 0 + Δ x) ). In this case, the function is said to have at the point x 2 minimum.

Let's say the point x 1 - maximum point of the function f(x) . Then in the interval up to x 1 the function is increasing, so the derivative of the function is greater than zero ( f "(x) > 0 ), and in the interval after x 1 the function is decreasing, therefore, the derivative of the function is less than zero ( f "(x) < 0 ). Тогда в точке x 1

Let us also assume that the point x 2 - minimum point of the function f(x) . Then in the interval up to x 2 the function is decreasing, and the derivative of the function is less than zero ( f "(x) < 0 ), а в интервале после x 2 the function is increasing, and the derivative of the function is greater than zero ( f "(x) > 0 ). In this case also at the point x 2 the derivative of the function is zero or does not exist.

Fermat's theorem (a necessary criterion for the existence of an extremum of a function). If point x 0 - function extremum point f(x) , then at this point the derivative of the function is equal to zero ( f "(x) = 0 ) or does not exist.

Definition . The points at which the derivative of a function is equal to zero or does not exist are called critical points .

Example 1. Consider the function .

At the point x= 0 the derivative of the function is equal to zero, therefore, the point x= 0 is the critical point. However, as can be seen on the graph of the function, it increases in the entire domain of definition, so the point x= 0 is not an extremum point of this function.

Thus, the conditions that the derivative of a function at a point is equal to zero or does not exist are necessary conditions for an extremum, but not sufficient, since other examples of functions can be given for which these conditions are satisfied, but the function does not have an extremum at the corresponding point. Therefore, it is necessary to have sufficient signs to judge whether there is an extremum at a particular critical point and which one - a maximum or a minimum.

Theorem (the first sufficient criterion for the existence of an extremum of a function). Critical point x 0 f(x) , if the derivative of the function changes sign when passing through this point, and if the sign changes from "plus" to "minus", then the maximum point, and if from "minus" to "plus", then the minimum point.

If near the point x 0 , to the left and to the right of it, the derivative retains its sign, this means that the function either only decreases or only increases in some neighborhood of the point x 0 . In this case, at the point x 0 there is no extremum.

So, to determine the extremum points of the function, you need to do the following :

Example 2. Find extrema of a function .

Solution. Let's find the derivative of the function:

Equate the derivative to zero to find the critical points:

.

Since for any values ​​\u200b\u200bof "x" the denominator is not equal to zero, then we equate the numerator to zero:

Got one critical point x= 3 . We determine the sign of the derivative in the intervals delimited by this point:

in the range from minus infinity to 3 - minus sign, that is, the function decreases,

in the range from 3 to plus infinity - a plus sign, that is, the function increases.

That is, point x= 3 is the minimum point.

Find the value of the function at the minimum point:

Thus, the extremum point of the function is found: (3; 0) , and it is the minimum point.

Theorem (the second sufficient criterion for the existence of an extremum of a function). Critical point x 0 is the extremum point of the function f(x) , if the second derivative of the function at this point is not equal to zero ( f ""(x) ≠ 0 ), moreover, if the second derivative is greater than zero ( f ""(x) > 0 ), then the maximum point, and if the second derivative is less than zero ( f ""(x) < 0 ), то точкой минимума.

Remark 1. If at a point x 0 both the first and second derivatives vanish, then at this point it is impossible to judge the presence of an extremum on the basis of the second sufficient sign. In this case, you need to use the first sufficient criterion for the extremum of the function.

Remark 2. The second sufficient criterion for the extremum of a function is also inapplicable when the first derivative does not exist at the stationary point (then the second derivative does not exist either). In this case, it is also necessary to use the first sufficient criterion for the extremum of the function.

From the above definitions it follows that the extremum of a function is of a local nature - this is the largest and smallest value of the function compared to the nearest values.

Suppose you consider your earnings in a time span of one year. If in May you earned 45,000 rubles, and in April 42,000 rubles and in June 39,000 rubles, then the May earnings are the maximum of the earnings function compared to the nearest values. But in October you earned 71,000 rubles, in September 75,000 rubles, and in November 74,000 rubles, so the October earnings are the minimum of the earnings function compared to nearby values. And you can easily see that the maximum among the values ​​of April-May-June is less than the minimum of September-October-November.

Generally speaking, a function may have several extrema on an interval, and it may turn out that any minimum of the function is greater than any maximum. So, for the function shown in the figure above, .

That is, one should not think that the maximum and minimum of the function are, respectively, its maximum and minimum values ​​on the entire segment under consideration. At the point of maximum, the function has the greatest value only in comparison with those values ​​that it has at all points sufficiently close to the maximum point, and at the minimum point, the smallest value only in comparison with those values ​​that it has at all points sufficiently close to the minimum point.

Therefore, we can refine the concept of extremum points of a function given above and call the minimum points local minimum points, and the maximum points - local maximum points.

Example 3

Solution. The function is defined and continuous on the whole number line. Its derivative also exists on the entire number line. Therefore, in this case, only those at which , i.e., serve as critical points. , whence and . Critical points and divide the entire domain of the function into three intervals of monotonicity: . We select one control point in each of them and find the sign of the derivative at this point.

For the interval, the reference point can be : we find . Taking a point in the interval, we get , and taking a point in the interval, we have . So, in the intervals and , and in the interval . According to the first sufficient sign of an extremum, there is no extremum at the point (since the derivative retains its sign in the interval ), and the function has a minimum at the point (since the derivative changes sign from minus to plus when passing through this point). Find the corresponding values ​​of the function: , and . In the interval, the function decreases, since in this interval , and in the interval it increases, since in this interval.

To clarify the construction of the graph, we find the points of intersection of it with the coordinate axes. When we obtain an equation whose roots and , i.e., two points (0; 0) and (4; 0) of the graph of the function are found. Using all the information received, we build a graph (see at the beginning of the example).

For self-checking during calculations, you can use the online derivatives calculator.

Example 4. Find the extrema of the function and build its graph.

The domain of the function is the entire number line, except for the point, i.e. .

To shorten the study, we can use the fact that this function is even, since . Therefore, its graph is symmetrical about the axis Oy and the study can only be performed for the interval .

Finding the derivative and critical points of the function:

1) ;

2) ,

but the function suffers a break at this point, so it cannot be an extremum point.

Thus, the given function has two critical points: and . Taking into account the parity of the function, we check only the point by the second sufficient sign of the extremum. To do this, we find the second derivative and determine its sign at : we get . Since and , then is the minimum point of the function, while .

To get a more complete picture of the graph of the function, let's find out its behavior on the boundaries of the domain of definition:

(here the symbol indicates the desire x to zero on the right, and x remains positive; similarly means aspiration x to zero on the left, and x remains negative). Thus, if , then . Next, we find

,

those. if , then .

The graph of the function has no points of intersection with the axes. The picture is at the beginning of the example.

For self-checking during calculations, you can use the online derivatives calculator.

Example 8. Find the extrema of the function .

Solution. Find the domain of the function. Since the inequality must hold, we obtain from .

Let's find the first derivative of the function.