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This teaching material is for reference only and relates to a wide range of topics. The article provides an overview of the graphs of the basic elementary functions and addresses the most important issue - how to build a chart quickly and quickly. Studying higher mathematics without knowing the graphs of the basic elementary functions will be difficult, so it is very important to remember what the graphs of parabola, hyperbola, sine, cosine, etc. look like, remember some values \u200b\u200bof the functions. Also, we will talk about some properties of the main functions. I do not pretend to the completeness and scientific thoroughness of the materials, the emphasis will be placed, first of all, in practice - those things with which you have to face literally at every step, in any topic of higher mathematics. Charts for dummies? You could say that.
By popular demand of readers clickable table of contents:
In addition, there is an ultra-short summary on the topic.
- master 16 types of graphs, having studied SIX pages!
Seriously, six, even myself was surprised. This compendium contains improved graphics and is available for a nominal fee, a demo version can be viewed. It is convenient to print the file so that the graphs are always at hand. Thanks for supporting the project!
And right off we go:
How to build coordinate axes?
In practice, test papers are almost always executed by students in separate notebooks lined in a cage. Why check marking? After all, work, in principle, can be done on A4 sheets. A cell is needed just for high-quality and accurate design drawings.
Any drawing of a function graph starts with the coordinate axes.
Drawings are two-dimensional and three-dimensional.
We first consider the two-dimensional case cartesian rectangular coordinate system:
1) We draw coordinate axes. Axis is called abscissa axis
and the axis is ordinate axis
. We always try to draw them neat and not crooked. Arrows should also not resemble the beard of Papa Carlo.
2) We sign the axes in capital letters "X" and "igrek". Do not forget to sign the axis.
3) We set the scale along the axes: draw zero and two ones. When executing a drawing, the most convenient and frequently encountered scale is: 1 unit \u003d 2 cells (drawing on the left) - if possible, stick to it. However, from time to time it happens that the drawing does not fit on the notebook sheet - then we scale down: 1 unit \u003d 1 cell (drawing on the right). It’s rare, but it happens that the scale of a drawing has to be reduced (or increased) even more
DO NOT "scribble from the machine gun" ... -5, -4, -3, -1, 0, 1, 2, 3, 4, 5, .... For the coordinate plane is not a monument to Descartes, and the student is not a dove. We put zero and two axial units. Sometimes instead units it is convenient to “detect” other values, for example, “two” on the abscissa axis and “three” on the ordinate axis - and this system (0, 2 and 3) also uniquely sets the coordinate grid.
Estimated drawing dimensions are best estimated BEFORE drawing. So, for example, if the task requires you to draw a triangle with the vertices,,, then it is completely clear that the popular scale of 1 unit \u003d 2 cells will not work. Why? Let's look at the point - here we will have to measure fifteen centimeters down, and, obviously, the drawing will not fit (or barely fit) on the notebook sheet. Therefore, immediately select a smaller scale of 1 unit \u003d 1 cell.
By the way, about centimeters and notebook cells. Is it true that 30 tetrad cells contain 15 centimeters? Measure in a notebook for interest 15 centimeters with a ruler. In the USSR, perhaps this was true ... It is interesting to note that if you measure these same centimeters horizontally and vertically, then the results (in the cells) will be different! Strictly speaking, modern notebooks are not checkered, but rectangular. Perhaps this will seem nonsense, but, for example, drawing a circle with a pair of compasses in such a situation is very inconvenient. To be honest, at such moments you begin to think about the correctness of Comrade Stalin, who sent to the camps for hackwork at the factory, not to mention the domestic automotive industry, falling aircraft or exploding power plants.
Speaking of quality, or a brief recommendation on stationery. Today, most notebooks are on sale, without saying bad words, completely homogeneous. For the reason that they get wet, and not only from gel, but also from ballpoint pens! Save on paper. I recommend using notebooks of the Arkhangelsk Pulp and Paper Mill (18 sheets, a cage) or Pyaterochka for registration tests, although it’s more expensive. It is advisable to choose a gel pen, even the cheapest Chinese gel pen is much better than a ballpoint pen that smears or tears paper. The only "competitive" ballpoint pen in my memory is Erich Krause. She writes clearly, beautifully and steadily - that with a full core, that with almost empty.
Additionally: Vision of a rectangular coordinate system through the eyes of analytic geometry is covered in the article Linear (non) dependence of vectors. Basis of vectors, for detailed information on the coordinate quarters can be found in the second paragraph of the lesson Linear inequalities.
Three-dimensional case
Almost everything is the same here.
1) We draw coordinate axes. Standard: applicate axis
- directed up, axis - directed to the right, axis - left down strictly at an angle of 45 degrees.
2) We sign the axis.
3) We set the scale along the axes. Axis scale - half the size of other axes. Also note that in the right drawing I used a non-standard “serif” along the axis (this possibility has already been mentioned above). From my point of view, it’s more precise, faster and more aesthetic - you don’t need to look under the microscope for the middle of the cell and “sculpt” the unit right next to the origin.
When doing a three-dimensional drawing, again - give priority to scale
1 unit \u003d 2 cells (drawing on the left).
What are all these rules for? Rules exist in order to break them. What am I going to do now. The fact is that the subsequent drawings of the article will be made by me in Excel, and the coordinate axes will look incorrect from the point of view of proper design. I could draw all the graphs by hand, but to draw them really is terrible as a reluctance Excel will draw them much more accurately.
Graphs and basic properties of elementary functions
The linear function is given by the equation. The linear function graph is direct. In order to build a line it is enough to know two points.
Example 1
Build a function graph. Find two points. It is advantageous to choose zero as one of the points.
If, then
We take some other point, for example, 1.
If, then
When completing tasks, the coordinates of the points are usually summarized in a table:
And the values \u200b\u200bthemselves are calculated verbally or on a draft calculator.
Two points are found, execute the drawing:
When drawing, we always sign graphics.
It will not be superfluous to recall particular cases of a linear function:
Notice how I arranged the captions, signatures should not be misunderstood when studying a drawing. In this case, it was extremely undesirable to put a signature near the intersection point of the lines, or at the bottom right between the graphs.
1) A linear function of the form () is called direct proportionality. For instance, . The direct proportionality graph always passes through the origin. Thus, the construction of the line is simplified - just find one point.
2) The equation of view defines a straight line parallel to the axis, in particular, the axis itself is defined by the equation. The function graph is built immediately, without finding any points. That is, the record should be understood as follows: "the game is always equal to –4, for any value of x."
3) The equation of the form defines a straight line parallel to the axis, in particular, the axis itself is given by the equation. The function graph is also built immediately. The record should be understood as follows: "X always, for any value of the player, is equal to 1".
Some will ask, why remember Grade 6 ?! So it, maybe so, only over the years of practice I met a dozen students who were stumped by the task of constructing a schedule like or.
Building a straight line is the most common action when performing drawings.
The straight line is examined in detail in the course of analytic geometry, and those who wish can refer to the article Equation of a line on a plane.
Graph of a quadratic, cubic function, graph of a polynomial
Parabola. Quadratic function graph  () is a parabola. Consider the famous case:
Recall some properties of the function.
So, the solution to our equation: - it is at this point that the top of the parabola is located. Why this is so can be found in a theoretical article on a derivative and a lesson on function extrema. In the meantime, we calculate the corresponding value of the "game":
So the vertex is at the point
Now we find other points, while brazenly we use the symmetry of the parabola. It should be noted that the function  – not evenbut, nevertheless, no one canceled the symmetry of the parabola.
In what order to find the remaining points, I think it will be clear from the final table:
This construction algorithm can be figuratively called a “shuttle” or the “back and forth” principle with Anfisa Chekhova.
Let's execute the drawing:
From the graphs examined, another useful sign is recalled:
For a quadratic function  () the following is true:
If, then the branches of the parabola are directed up.
If, then the branches of the parabola are directed down.
In-depth knowledge of the curve can be obtained in the lesson Hyperbola and Parabola.
Cubic parabola is set by function. Here is a drawing familiar from school:
We list the main properties of the function
Function graph
It represents one of the branches of a parabola. Let's execute the drawing:
The main properties of the function:
In this case, the axis is vertical asymptote
for a plot of hyperbola at.
It will be a BIG mistake if, when drawing up a drawing by negligence, allow the intersection of the graph with the asymptote.
Also one-sided limits tell us that hyperbole not limited from above and not limited from below.
We study the function at infinity: that is, if we begin to go along the axis left (or right) to infinity, then the “games” will be a slender step infinitely close approach zero, and, accordingly, the branches of the hyperbola infinitely close approach the axis.
So the axis is horizontal asymptote
for the function graph, if “X” tends to plus or minus infinity.
Function is odd, and, therefore, the hyperbola is symmetric with respect to the origin. This fact is obvious from the drawing, in addition, it is easily verified analytically:  .
A graph of a function of the form () represents two branches of a hyperbola.
If, then the hyperbola is located in the first and third coordinate quarters (see picture above).
If, then the hyperbola is located in the second and fourth coordinate quarters.
The indicated regularity of the residence of the hyperbola is not difficult to analyze from the point of view of geometric transformations of graphs.
Example 3
Build the right branch of the hyperbola
We use the pointwise construction method, while it is advantageous to select the values \u200b\u200bso that they are completely divided:
Let's execute the drawing:
It will not be difficult to build the left branch of the hyperbola, the oddness of the function will help here. Roughly speaking, in the table of pointwise construction, mentally add minus to each number, put the corresponding points and draw the second branch.
Detailed geometric information about the line under consideration can be found in the article Hyperbola and Parabola.
Exponential function graph
In this section, I will immediately consider the exponential function, since in problems of higher mathematics in 95% of cases it is an exponent.
I remind you that this is an irrational number: it will be required when building a schedule, which, in fact, I will build without ceremony. Three points are probably enough:
Let us leave the function graph alone, about it later.
The main properties of the function:
Functions graphs look basically the same, etc.
I must say that the second case is less common in practice, but it occurs, so I found it necessary to include it in this article.
Graph of a logarithmic function
Consider a function with a natural logarithm.
Let's do a point drawing:
If you forgot what the logarithm is, please refer to the school books.
The main properties of the function:
Domain: 
Range of Values:.
The function is not limited from above:  , albeit slowly, but the branch of the logarithm goes up to infinity.
We study the behavior of the function near zero on the right:  . So the axis is vertical asymptote
for the function graph with "x" tending to zero on the right.
Be sure to know and remember the typical value of the logarithm: .
The logarithm chart looks basically the same at the base:,, (decimal logarithm based on base 10), etc. In this case, the larger the base, the more gentle the schedule will be.
We will not consider the case; I don’t remember something when the last time I built a schedule with such a reason. And the logarithm seems to be a very rare guest in higher mathematics problems.
In conclusion, I will say one more fact: Exponential function and logarithmic functionAre two mutually inverse functions. If you look closely at the graph of the logarithm, you can see that this is the same exponent, it just is located a little differently.
Graphs of trigonometric functions
What does trigonometric torment in school begin with? Right. With sine
We plot the function
This line is called sine wave.
I remind you that “pi” is an irrational number:, and in trigonometry from it ripples in the eyes.
The main properties of the function:
This function is periodic with a period. What does it mean? Let's look at the segment. To the left and to the right of it the exact same piece of the graph is endlessly repeated.
Domain:, that is, for any value of "X" there is a sine value.
Range of Values:. Function is limited:, that is, all the "games" are sitting strictly in the segment.
This does not happen: or, more precisely, it happens, but the indicated equations have no solution. |
Parabola. The graph of the quadratic function () is a parabola. Consider the canonical case:
Recall some properties of the function.
Scope - any real number (any value of "X"). What does it mean? Whatever point on the axis we choose - for each “X” there is a parabola point. Mathematically, it is written like this:. The scope of any function is standardly denoted by or. The letter denotes a set of real numbers or, more simply, “any X” (when the work is executed in a notebook, they write not a curly letter, but a bold letter R).
The range of values \u200b\u200bis the set of all values \u200b\u200bthat the рек player ’variable can take. In this case: - the set of all positive values, including zero. The range of values \u200b\u200bis standardly denoted by or.
Function is even. If the function is even, then its graph is symmetrical about the axis. This is a very useful property, which greatly simplifies the construction of the graph, which we will soon see. Analytically, the parity of a function is expressed by a condition. How to check any function for parity? It is necessary to substitute in the equation. In the case of a parabola, the check looks like this: that means the function is even.
Function not limited from above. Analytically, a property is written like this:. By the way, here is an example of the geometric meaning of the limit of the function: if we go along the axis (left or right) to infinity, then the parabola branches (“play” values) will go up unlimitedly to “plus infinity”.
At learning function limits it is desirable to understand the geometric meaning of the limit.
It is no coincidence that I painted the properties of the function in such detail, all of the above things are useful to know and remember when plotting functions, as well as when studying function graphs.
Example 2
Plot function 
.
In this example, we will look at an important technical question: How to quickly build a parabola? In practical tasks, the need to draw a parabola arises very often, in particular, when calculating the area of \u200b\u200ba figure using a certain integral. Therefore, it is advisable to learn how to draw quickly, with minimal loss of time. I propose the following construction algorithm.
First we find the top of the parabola. To do this, take the first derivative and equate it to zero:
If derivatives are bad, you should read the lesson. How to find a derivative?
So, the solution to our equation: - it is at this point that the top of the parabola is located. We calculate the corresponding value of "game":
So the vertex is at the point
Now we find other points, while brazenly we use the symmetry of the parabola. It should be noted that the function – not evenbut, nevertheless, no one canceled the symmetry of the parabola.
In what order to find the remaining points, I think it will be clear from the final table:
This construction algorithm can be figuratively called a “shuttle”. Perhaps not everyone understands the essence of the shuttle, then for comparison I remind you of the well-known TV show "tudy-syudy with Anfisa Chekhova."
Let's execute the drawing:
From the graphs examined, another useful sign is recalled:
For a quadratic function (), the following is true:
If, then the branches of the parabola are directed upwards.
If, then the branches of the parabola are directed downward.
Cubic parabola
Cubic parabola is set by function. Here is a drawing familiar from school:
We list the main properties of the function
Scope - any real number:.
The range of values \u200b\u200bis any real number:.
Function is odd. If the function is odd, then its graph is symmetrical with respect to the origin. Analytically, the oddness of a function is expressed by the condition 
. We perform a check for the cubic function, for this, substitute “minus X” instead of “X”:
, then the function is odd.
Function not limited. In the language of function limits, this can be written as:
It is also more efficient to build a cubic parabola with the help of Anfisa Chekhova's “shuttle” algorithm:
Surely, you noticed what else the oddness of the function is manifested in. If we found that 
, then when calculating it is no longer necessary to count anything, automatically write that. This feature holds true for any odd function.
Now let's talk a bit about polynomial graphs.
Graph of any third degree polynomial 
() basically has the following form:
In this example, the coefficient is at the highest degree, so the graph is reversed. In principle, the graphs of polynomials of the 5th, 7th, 9th, and other odd degrees have essentially the same form. The higher the degree, the more intermediate “zagibulin”.
The polynomials of the 4th, 6th and other even degrees have a graph of the following principle:
This knowledge is useful in examining function graphs.
Function graph
Let's execute the drawing:
The main properties of the function:
Domain: .
Range of Values:.
That is, the function graph is completely in the first coordinate quarter.
Function not limited from above. Or using the limit: 
When constructing the simplest graphs with roots, a pointwise method of construction is also appropriate, while it is advantageous to select such “X” values \u200b\u200bso that the root is extracted completely:
Parabola. The graph of the quadratic function () is a parabola. Consider the canonical case:
Recall some properties of the function.
Scope - any real number (any value of "X"). What does it mean? Whatever point on the axis we choose - for each “X” there is a parabola point. Mathematically, it is written like this:. The scope of any function is standardly denoted by or. The letter denotes a set of real numbers or, more simply, “any X” (when the work is executed in a notebook, they write not a curly letter, but a bold letter R).
The range of values \u200b\u200bis the set of all values \u200b\u200bthat the рек player ’variable can take. In this case: - the set of all positive values, including zero. The range of values \u200b\u200bis standardly denoted by or.
Function is even. If the function is even, then its graph is symmetrical about the axis. This is a very useful property, which greatly simplifies the construction of the graph, which we will soon see. Analytically, the parity of a function is expressed by a condition. How to check any function for parity? It is necessary to substitute in the equation. In the case of a parabola, the check looks like this: that means the function is even.
Function not limited from above. Analytically, a property is written like this:. By the way, here is an example of the geometric meaning of the limit of the function: if we go along the axis (left or right) to infinity, then the parabola branches (“play” values) will go up unlimitedly to “plus infinity”.
At learning function limits it is desirable to understand the geometric meaning of the limit.
It is no coincidence that I painted the properties of the function in such detail, all of the above things are useful to know and remember when plotting functions, as well as when studying function graphs.
Example 2
Build a function graph.
In this example, we will look at an important technical question: How to quickly build a parabola? In practical tasks, the need to draw a parabola arises very often, in particular, when calculating area of \u200b\u200ba figure using a certain integral. Therefore, it is advisable to learn how to draw quickly, with minimal loss of time. I propose the following construction algorithm.
First we find the top of the parabola. To do this, take the first derivative and equate it to zero:
If derivatives are bad, you should read the lesson. How to find a derivative?
So, the solution to our equation: - it is at this point that the top of the parabola is located. We calculate the corresponding value of "game":
So the vertex is at the point
Now we find other points, while brazenly we use the symmetry of the parabola. It should be noted that the function - not evenbut, nevertheless, no one canceled the symmetry of the parabola.
In what order to find the remaining points, I think it will be clear from the final table:
This construction algorithm can be figuratively called a “shuttle”. Perhaps not everyone understands the essence of the shuttle, then for comparison I remind you of the well-known TV show "tudy-syudy with Anfisa Chekhova."
Let's execute the drawing:
From the graphs examined, another useful sign is recalled:
For a quadratic function (), the following is true:
If, then the branches of the parabola are directed upwards.
If, then the branches of the parabola are directed downward.
Cubic parabola
Cubic parabola is set by function. Here is a drawing familiar from school:
We list the main properties of the function
Scope - any real number:.
The range of values \u200b\u200bis any real number:.
Function is odd. If the function is odd, then its graph is symmetrical with respect to the origin. Analytically, the oddness of a function is expressed by the condition 
. We perform a check for the cubic function, for this, substitute “minus X” instead of “X”:
, then the function is odd.
Function not limited. In the language of function limits, this can be written as:
It is also more efficient to build a cubic parabola with the help of Anfisa Chekhova's “shuttle” algorithm:
Surely, you noticed what else the oddness of the function is manifested in. If we found that 
, then when calculating it is no longer necessary to count anything, automatically write that. This feature holds true for any odd function.
Now let's talk a bit about polynomial graphs.
Graph of any third degree polynomial 
() basically has the following form:
In this example, the coefficient is at the highest degree, so the graph is reversed. In principle, the graphs of polynomials of the 5th, 7th, 9th, and other odd degrees have essentially the same form. The higher the degree, the more intermediate “zagibulin”.
The polynomials of the 4th, 6th and other even degrees have a graph of the following principle:
This knowledge is useful in examining function graphs.
Function graph
Let's execute the drawing:
The main properties of the function:
Domain: .
Range of Values:.
That is, the function graph is completely in the first coordinate quarter.
Function not limited from above. Or using the limit: 
When constructing the simplest graphs with roots, a pointwise method of construction is also appropriate, while it is advantageous to select such “X” values \u200b\u200bso that the root is extracted completely:
In fact, I want to parse more examples with roots, for example, but they are much less common. I focus on more common cases, and, as practice shows, something like having to be built much more often. If it becomes necessary to find out what the graphs look like with other roots, then I recommend that you look into the school textbook or math reference book.
Hyperbola graph
Again, remember the trivial "school" hyperbole.
Let's execute the drawing:
The main properties of the function:
Domain: .
Range of Values:.
The entry means: "any real number, excluding zero"
At a point, the function suffers an infinite break. Or using unilaterallimits:,. Let's talk a little about one-sided limits. The entry means that we infinitely close approaching the axis to zero left. How does the schedule behave? He goes down to minus infinity infinitely close approaching the axis. It is this fact that is written by the limit. Similarly, the entry means that we infinitely close approaching the axis to zero on right. In this case, the branch of the hyperbola goes up by plus infinity, infinitely close approaching the axis. Or in short:.
kind of
where In other words, the cubic function is defined by a polynomial of the third degree.
Analytical properties
Application
A cubic parabola is sometimes used to calculate the transition curve in transport, since its calculation is much simpler than constructing a clothoid.
see also
Write a review on the article "Cubic Function"
Notes
Literature
- L. S. Pontryagin, // "Quantum", 1984, No. 3.
- I. N. Bronstein, K. A. Semendyaev, "A Handbook of Mathematics", Publishing House "Science", M. 1967, p. 84
Excerpt characterizing the cubic function
“Well, there, for whatever ...”
At this time, Petya, whom no one paid attention to, approached his father and, all red, breaking, then in a harsh, then in a thin voice, said:
“Well now, daddy, I will definitely say - and mama too, as you want - I will definitely say that you will let me into military service, because I cannot ... that's all ...”
The countess with horror raised her eyes to the sky, threw up her hands and angrily turned to her husband.
- So I agreed! - she said.
But the count immediately recovered from his excitement.
“Well, well,” he said. - Here is a warrior yet! Leave nonsense: you need to learn.
- It's not stupid, dad. Obolensky Fedya is younger than me and also goes, and most importantly, all the same, I can’t learn anything now that ... - Petya stopped, blushed with sweat and said the same: - when the fatherland is in danger.
- Full, full, stupid ...
“But you yourself said that we would sacrifice everything.”
“Petya, I tell you, shut up,” the count shouted, looking back at his wife, who, turning pale, looked with fixed eyes at the smaller son.
“And I tell you.” So Pyotr Kirillovich will say ...
- I tell you - nonsense, milk has not dried up yet, but wants to go to military service! Well, well, I’m telling you, - and the count, having taken the papers with him, probably to read it again in his study before the rest, went out of the room.
- Pyotr Kirillovich, well, let's go smoke ...
Pierre was embarrassed and indecisive. Natasha’s unusually brilliant and lively eyes constantly, more than affectionately turning towards him, brought him into this state.
“No, I think I'm going home ...”
- How to go home, but you wanted to have an evening with us ... And it rarely began to happen. And this one is mine ... - said the Count good-naturedly, pointing to Natasha, - only with you she was cheerful ...
- Yes, I forgot ... I definitely need to go home ... Cases ... - Pierre said hastily.
“Well, goodbye,” said the Earl, completely leaving the room.
“Why are you leaving?” Why are you upset? Why? .. - Pierre Natasha asked, looking defiantly into his eyes.
“Because I love you! - he wanted to say, but he did not say this, he blushed to tears and lowered his eyes.
“Because it’s better for me to visit you less ... Because ... no, I just have things to do.”
- From what? no, tell me, - Natasha began decisively, and suddenly became silent. They both looked at each other, startled and confused. He tried to grin, but could not: his smile expressed suffering, and he silently kissed her hand and went out.
Pierre decided on his own not to be with the Rostovs anymore. Petya, after a decisive refusal he received, went into his room and there, shutting himself from everyone, wept bitterly. Everyone did as if they had not noticed anything when he came to tea silent and gloomy, with tearful eyes.
The next day, the emperor arrived. Several people in the yard Rostovs took leave to go look at the king. This morning Petya dressed for a long time, combed his hair, and arranged collars as he did with large ones. He frowned in front of the mirror, made gestures, shrugged, and finally, without telling anyone, put on his cap and left the house from the back porch, trying not to be noticed. Petya decided to go straight to the place where the sovereign was, and directly explain to some chamberlain (Petya thought that the sovereign was always surrounded by chamberlains) that he, Count Rostov, in spite of his youth, wants to serve his fatherland, that youth cannot be an obstacle for devotion and that he is ready ... Petya, while he was preparing, prepared many wonderful words that he would say to the chamberlain.
The function y \u003d x ^ 2 is called the quadratic function. The graph of a quadratic function is a parabola. A general view of the parabola is shown in the figure below.
Quadratic function
Fig 1. General view of the parabola
As can be seen from the graph, it is symmetric about the axis Oy. The axis Oy is called the axis of symmetry of the parabola. This means that if you draw on the graph a line parallel to the Ox axis above this axis. Then she will cross the parabola at two points. The distance from these points to the Oy axis will be the same.
The axis of symmetry divides the graph of the parabola into two parts. These parts are called parabola branches. And the point of the parabola that lies on the axis of symmetry is called the vertex of the parabola. That is, the axis of symmetry passes through the top of the parabola. The coordinates of this point (0; 0).
Basic properties of a quadratic function
1. For x \u003d 0, y \u003d 0, and y\u003e 0 for x0
2. The quadratic function reaches its minimum value at its peak. Ymin at x \u003d 0; It should also be noted that the maximum value of the function does not exist.
3. The function decreases in the interval (-∞; 0] and increases in the interval)
