Monotone
Very important property function is its monotonicity. Knowing this property of various special functions, it is possible to determine the behavior of various physical, economic, social and many other processes.
Highlight the following types monotony of functions:
1) function increases, if on a certain interval, if for any two points and this interval such that . Those. a larger argument value corresponds to a larger function value;
2) function decreases, if on a certain interval, if for any two points and this interval such that . Those. a larger argument value corresponds to a smaller function value;
3) function non-decreasing, if on a certain interval, if for any two points and this interval such that ;
4) function does not increase, if on a certain interval, if for any two points and this interval such that .
2. For the first two cases, the term “strict monotonicity” is also used.
3. The last two cases are specific and are usually specified as a composition of several functions.
4. Separately, we note that the increase and decrease of the graph of a function should be considered from left to right and nothing else.
2. Even/odd.
The function is called odd, if when the sign of the argument changes, it changes its value to the opposite. The formula for this looks like this 
. This means that after substituting “minus x” values into the function in place of all x’s, the function will change its sign. The graph of such a function is symmetrical about the origin.
Examples of odd functions are etc.
For example, the graph actually has symmetry about the origin:
The function is called even, if when the sign of the argument changes, it does not change its value. The formula for this looks like this. This means that after substituting “minus x” values into the function in place of all x’s, the function will not change as a result. The graph of such a function is symmetrical about the axis.
Examples of even functions are etc.
For example, let’s show the symmetry of the graph about the axis:
If a function does not belong to any of the specified types, then it is called neither even nor odd or function general view . Such functions have no symmetry.
Such a function, for example, is the one we recently reviewed linear function with schedule:
3. A special property of functions is periodicity.
The fact is that periodic functions, which are considered in the standard school curriculum, are only trigonometric functions. We have already talked about them in detail when studying the relevant topic.
Periodic function is a function that does not change its values when a certain constant non-zero number is added to the argument.
This minimum number is called period of the function and are designated by the letter .
The formula for this looks like this: 
.
Let's look at this property using the example of a sine graph:
Let us remember that the period of the functions and is , and the period and is .
As we already know, for trigonometric functions with a complex argument there may be a non-standard period. We are talking about functions of the form:
Their period is equal. And about the functions:
Their period is equal.
As you can see, to calculate a new period, the standard period is simply divided by the factor in the argument. It does not depend on other modifications of the function.
Limitation.
Function y=f(x) is called bounded from below on the set X⊂D(f) if there is a number a such that for any xϵX the inequality f(x) holds< a.
Function y=f(x) is called bounded from above on the set X⊂D(f) if there is a number a such that for any хϵХ the inequality f(x) holds< a.
If the interval X is not specified, then the function is considered to be limited over the entire domain of definition. A function that is bounded both above and below is called bounded.
The limitation of the function is easy to read from the graph. You can draw some line y=a, and if the function is higher than this line, then it is bounded from below.
If below, then accordingly above. Below is a graph of a function bounded below. Guys, try to draw a graph of a limited function yourself.
Topic: Properties of functions: intervals of increasing and decreasing; greatest and smallest value; extremum points (local maximum and minimum), convexity of the function.
Intervals of increasing and decreasing.
Based on sufficient conditions (signs) for the increase and decrease of a function, intervals of increase and decrease of the function are found.
Here are the formulations of the signs of increasing and decreasing functions on an interval:
· if the derivative of the function y=f(x) positive for anyone x from the interval X, then the function increases by X;
· if the derivative of the function y=f(x) negative for anyone x from the interval X, then the function decreases by X.
Thus, to determine the intervals of increase and decrease of a function, it is necessary:
· find the domain of definition of the function;
· find the derivative of the function;
· solve inequalities on the domain of definition;
Increasing, decreasing and extrema of a function
Finding the intervals of increase, decrease and extrema of a function is both an independent task and an essential part of other tasks, in particular, full function study. Initial information about the increase, decrease and extrema of the function is given in theoretical chapter on derivative, which I highly recommend for preliminary study (or repetition)– also for the reason that the following material is based on the very essentially derivative, being a harmonious continuation of this article. Although, if time is short, then a purely formal practice of examples from today’s lesson is also possible.
And today there is a spirit of rare unanimity in the air, and I can directly feel that everyone present is burning with desire learn to explore a function using its derivative. Therefore, reasonable, good, eternal terminology immediately appears on your monitor screens.
For what? One of the reasons is the most practical: so that it is clear what is generally required of you in a particular task!
Monotonicity of the function. Extremum points and extrema of a function
Let's consider some function. To put it simply, we assume that she continuous on the entire number line:
Just in case, let’s immediately get rid of possible illusions, especially for those readers who have recently become acquainted with intervals of constant sign of the function. Now we NOT INTERESTED, how the graph of the function is located relative to the axis (above, below, where the axis intersects). To be convincing, mentally erase the axes and leave one graph. Because that’s where the interest lies.
Function increases on an interval if for any two points of this interval connected by the relation , the inequality is true. That is, a larger value of the argument corresponds to a larger value of the function, and its graph goes “from bottom to top”. The demonstration function grows over the interval.
Likewise, the function decreases on an interval if for any two points of a given interval such that , the inequality is true. That is, a larger value of the argument corresponds to a smaller value of the function, and its graph goes “from top to bottom”. Our function decreases on intervals 
.
If a function increases or decreases over an interval, then it is called strictly monotonous at this interval. What is monotony? Take it literally – monotony.
You can also define non-decreasing function (relaxed condition in the first definition) and non-increasing function (softened condition in the 2nd definition). A non-decreasing or non-increasing function on an interval is called a monotonic function on a given interval (strict monotony - special case“just” monotony).
The theory also considers other approaches to determining the increase/decrease of a function, including on half-intervals, segments, but in order not to pour oil-oil-oil on your head, we will agree to operate with open intervals with categorical definitions - this is clearer, and for solving many practical problems quite enough.
Thus, in my articles the wording “monotonicity of a function” will almost always be hidden intervals strict monotony(strictly increasing or strictly decreasing function).
Neighborhood of a point. Words after which students run away wherever they can and hide in horror in the corners. ...Although after the post Cauchy limits They’re probably no longer hiding, but just shuddering slightly =) Don’t worry, there won’t be any proofs of the theorems now mathematical analysis– I needed the surroundings to formulate definitions more strictly extremum points. Let's remember:
Neighborhood of a point an interval that contains a given point is called, and for convenience the interval is often assumed to be symmetrical. For example, a point and its standard neighborhood:
Actually, the definitions:
The point is called strict maximum point, If exists her neighborhood, for all values of which, except for the point itself, the inequality . In our specific example this is the point.
The point is called strict minimum point, If exists her neighborhood, for all values of which, except for the point itself, the inequality . In the drawing there is point “a”.
Note : the requirement of neighborhood symmetry is not at all necessary. In addition, it is important the very fact of existence surroundings (even tiny, even microscopic), satisfying specified conditions
The points are called strictly extremum points or simply extremum points functions. That is, it is a generalized term for maximum points and minimum points.
How do we understand the word “extreme”? Yes, just as directly as monotony. Extreme points of roller coasters.
As in the case of monotonicity, loose postulates exist and are even more common in theory (which, of course, the strict cases considered fall under!):
The point is called maximum point, If exists its surroundings are such that for all
The point is called minimum point, If exists its surroundings are such that for all values of this neighborhood, the inequality holds.
Note that according to the last two definitions, any point of a constant function (or a “flat section” of a function) is considered both a maximum and a minimum point! The function, by the way, is both non-increasing and non-decreasing, that is, monotonic. However, we will leave these considerations to theorists, since in practice we almost always contemplate traditional “hills” and “hollows” (see drawing) with a unique “king of the hill” or “princess of the swamp”. As a variety, it occurs tip, directed up or down, for example, the minimum of the function at the point.
Oh, and speaking of royalty:
– the meaning is called maximum functions;
– the meaning is called minimum functions.
Common name - extremes functions.
Please be careful with your words!
Extremum points– these are “X” values.
Extremes– “game” meanings.
! Note : sometimes the listed terms refer to the “X-Y” points that lie directly on the GRAPH OF the function ITSELF.
How many extrema can a function have?
None, 1, 2, 3, ... etc. to infinity. For example, sine has infinitely many minima and maxima.
IMPORTANT! The term "maximum of function" not identical the term “maximum value of a function”. It is easy to notice that the value is maximum only in a local neighborhood, and at the top left there are “cooler comrades”. Likewise, “minimum of a function” is not the same as “minimum value of a function,” and in the drawing we see that the value is minimum only in a certain area. In this regard, extremum points are also called local extremum points, and the extrema – local extremes . They walk and wander nearby and global brethren. So, any parabola has at its vertex global minimum or global maximum. Further, I will not distinguish between types of extremes, and the explanation is voiced more for general educational purposes - the additional adjectives “local”/“global” should not take you by surprise.
Let's summarize our small excursion into theory with a test shot: what does the task “find the monotonicity intervals and extremum points of the function” mean?
The wording encourages you to find:
– intervals of increasing/decreasing function (non-decreasing, non-increasing appears much less often);
– maximum and/or minimum points (if any exist). Well, to avoid failure, it’s better to find the minimums/maximums themselves ;-)
How to determine all this? Using the derivative function!
How to find intervals of increasing, decreasing,
extremum points and extrema of the function?
Many rules, in fact, are already known and understood from lesson about the meaning of a derivative.
Tangent derivative 
brings the cheerful news that function is increasing throughout domain of definition.
With cotangent and its derivative 
the situation is exactly the opposite.
The arcsine increases over the interval - the derivative here is positive: 
.
When the function is defined, but not differentiable. However, at the critical point there is a right-handed derivative and a right-handed tangent, and at the other edge there are their left-handed counterparts.
I think it won’t be too difficult for you to carry out similar reasoning for the arc cosine and its derivative.
All of the above cases, many of which are tabular derivatives, I remind you, follow directly from derivative definitions.
Why explore a function using its derivative?
To better understand what the graph of this function looks like: where it goes “bottom up”, where “top down”, where it reaches minimums and maximums (if it reaches at all). Not all functions are so simple - in most cases we have no idea at all about the graph of a particular function.
It's time to move on to more meaningful examples and consider algorithm for finding intervals of monotonicity and extrema of a function:
Example 1
Find intervals of increase/decrease and extrema of the function
Solution:
1) The first step is to find domain of a function, and also take note of the break points (if they exist). IN in this case the function is continuous on the entire number line, and this action to a certain extent formally. But in a number of cases, serious passions flare up here, so let’s treat the paragraph without disdain.
2) The second point of the algorithm is due to
a necessary condition for an extremum:
If there is an extremum at a point, then either the value does not exist.
Confused by the ending? Extremum of the “modulus x” function .
The condition is necessary, but not enough, and the converse is not always true. So, it does not yet follow from the equality that the function reaches a maximum or minimum at point . A classic example has already been highlighted above - this is a cubic parabola and its critical point.
But be that as it may, the necessary condition for an extremum dictates the need to find suspicious points. To do this, find the derivative and solve the equation:
At the beginning of the first article about function graphs I told you how to quickly build a parabola using an example 
: “...we take the first derivative and equate it to zero: ...So, the solution to our equation: - it is at this point that the vertex of the parabola is located...”. Now, I think, everyone understands why the vertex of the parabola is located exactly at this point =) In general, we should start with a similar example here, but it is too simple (even for a teapot). In addition, there is an analogue at the very end of the lesson about derivative of a function. Therefore, let's increase the degree:
Example 2
Find intervals of monotonicity and extrema of the function
This is an example for independent decision. A complete solution and an approximate final sample of the problem at the end of the lesson.
The long-awaited moment of meeting with fractional-rational functions has arrived:
Example 3
Explore a function using the first derivative
Pay attention to how variably one and the same task can be reformulated.
Solution:
1) The function suffers infinite discontinuities at points.
2) We detect critical points. Let's find the first derivative and equate it to zero: 
Let's solve the equation. A fraction is zero when its numerator is zero:
Thus, we get three critical points: 
3) We plot ALL detected points on the number line and interval method we define the signs of the DERIVATIVE:
I remind you that you need to take some point in the interval and calculate the value of the derivative at it 
and determine its sign. It’s more profitable not to even count, but to “estimate” verbally. Let's take, for example, a point belonging to the interval and perform the substitution: 
. 
Two “pluses” and one “minus” give a “minus”, therefore, which means that the derivative is negative over the entire interval.
The action, as you understand, needs to be carried out for each of the six intervals. By the way, note that the numerator factor and denominator are strictly positive for any point in any interval, which greatly simplifies the task.
So, the derivative told us that the FUNCTION ITSELF increases by 
and decreases by . It is convenient to connect intervals of the same type with the join icon.
At the point the function reaches its maximum:
At the point the function reaches a minimum: 
Think about why you don't have to recalculate the second value ;-)
When passing through a point, the derivative does not change sign, so the function has NO EXTREMUM there - it both decreased and remained decreasing.
! Let's repeat important point : points are not considered critical - they contain a function not determined. Accordingly, here In principle there can be no extremes(even if the derivative changes sign).
Answer: function increases by 
and decreases by At the point the maximum of the function is reached: 
, and at the point – the minimum: .
Knowledge of monotonicity intervals and extrema, coupled with established asymptotes already gives a very good idea of appearance function graphics. A person of average training level is able to verbally determine that the graph of a function has two vertical asymptotes and oblique asymptote. Here is our hero: 
Try once again to correlate the results of the study with the graph of this function.
There is no extremum at the critical point, but there is graph inflection(which, as a rule, happens in similar cases).
Example 4
Find the extrema of the function
Example 5
Find monotonicity intervals, maxima and minima of the function
…it’s almost like some kind of “X in a cube” holiday today....
Soooo, who in the gallery offered to drink for this? =)
Each task has its own substantive nuances and technical details, which are commented out at the end of the lesson.
The final work in the form of the Unified State Exam for 11th graders necessarily contains tasks on calculating limits, intervals of decreasing and increasing derivatives of a function, searching for extremum points and constructing graphs. Good knowledge of this topic allows you to correctly answer several exam questions and not experience difficulties in further professional training.
Fundamentals of differential calculus - one of the main topics of mathematics modern school. She studies the use of the derivative to study the dependencies of variables - it is through the derivative that one can analyze the increase and decrease of a function without resorting to a drawing.
Comprehensive preparation of graduates for passing the Unified State Exam on educational portal“Shkolkovo” will help you deeply understand the principles of differentiation - understand the theory in detail, study examples of solutions typical tasks and try your hand at independent work. We will help you close gaps in knowledge - clarify your understanding of the lexical concepts of the topic and the dependencies of quantities. Students will be able to review how to find intervals of monotonicity, which means the derivative of a function rises or decreases on a certain segment when boundary points are and are not included in the intervals found.
Before you begin directly solving thematic problems, we recommend that you first go to the “Theoretical Background” section and repeat the definitions of concepts, rules and tabular formulas. Here you can read how to find and write down each interval of increasing and decreasing function on the derivative graph.
All information offered is presented in the most accessible form for understanding, practically from scratch. The website provides materials for perception and assimilation in several various forms– reading, video viewing and direct training under the guidance of experienced teachers. Professional teachers They will tell you in detail how to find the intervals of increasing and decreasing derivatives of a function using analytical and graphical methods. During the webinars, you will be able to ask any question you are interested in, both on theory and on solving specific problems.
Having remembered the main points of the topic, look at examples of increasing the derivative of a function, similar to tasks exam options. To consolidate what you have learned, take a look at the “Catalog” - here you will find practical exercises for independent work. The tasks in the section are selected at different levels of difficulty, taking into account the development of skills. For example, each of them is accompanied by solution algorithms and correct answers.
By choosing the "Constructor" section, students will be able to practice studying the increase and decrease of the derivative of a function on real options Unified State Examination, constantly updated taking into account latest changes and innovations.
Extrema of the function
Definition 2
A point $x_0$ is called a maximum point of a function $f(x)$ if there is a neighborhood of this point such that for all $x$ in this neighborhood the inequality $f(x)\le f(x_0)$ holds.
Definition 3
A point $x_0$ is called a maximum point of a function $f(x)$ if there is a neighborhood of this point such that for all $x$ in this neighborhood the inequality $f(x)\ge f(x_0)$ holds.
The concept of an extremum of a function is closely related to the concept of a critical point of a function. Let us introduce its definition.
Definition 4
$x_0$ is called a critical point of the function $f(x)$ if:
1) $x_0$ - internal point of the domain of definition;
2) $f"\left(x_0\right)=0$ or does not exist.
For the concept of extremum, we can formulate theorems on sufficient and necessary conditions his existence.
Theorem 2
Sufficient condition for an extremum
Let the point $x_0$ be critical for the function $y=f(x)$ and lie in the interval $(a,b)$. Let the derivative $f"(x)$ exist on each interval $\left(a,x_0\right)\ and\ (x_0,b)$ and preserve permanent sign. Then:
1) If on the interval $(a,x_0)$ the derivative is $f"\left(x\right)>0$, and on the interval $(x_0,b)$ the derivative is $f"\left(x\right)
2) If on the interval $(a,x_0)$ the derivative $f"\left(x\right)0$, then the point $x_0$ is the minimum point for this function.
3) If both on the interval $(a,x_0)$ and on the interval $(x_0,b)$ the derivative $f"\left(x\right) >0$ or the derivative $f"\left(x\right)
This theorem is illustrated in Figure 1.
Figure 1. Sufficient condition for the existence of extrema
Examples of extremes (Fig. 2).
Figure 2. Examples of extreme points
Rule for studying a function for extremum
2) Find the derivative $f"(x)$;
7) Draw conclusions about the presence of maxima and minima on each interval, using Theorem 2.
Increasing and decreasing functions
Let us first introduce the definitions of increasing and decreasing functions.
Definition 5
A function $y=f(x)$ defined on the interval $X$ is said to be increasing if for any points $x_1,x_2\in X$ at $x_1
Definition 6
A function $y=f(x)$ defined on the interval $X$ is said to be decreasing if for any points $x_1,x_2\in X$ for $x_1f(x_2)$.
Studying a function for increasing and decreasing
You can study increasing and decreasing functions using the derivative.
In order to examine a function for intervals of increasing and decreasing, you must do the following:
1) Find the domain of definition of the function $f(x)$;
2) Find the derivative $f"(x)$;
3) Find the points at which the equality $f"\left(x\right)=0$ holds;
4) Find the points at which $f"(x)$ does not exist;
5) Mark on the coordinate line all the points found and the domain of definition of this function;
6) Determine the sign of the derivative $f"(x)$ on each resulting interval;
7) Draw a conclusion: on intervals where $f"\left(x\right)0$ the function increases.
Examples of problems for studying functions for increasing, decreasing and the presence of extrema points
Example 1
Examine the function for increasing and decreasing, and the presence of maximum and minimum points: $f(x)=(2x)^3-15x^2+36x+1$
Since the first 6 points are the same, let’s carry them out first.
1) Domain of definition - all real numbers;
2) $f"\left(x\right)=6x^2-30x+36$;
3) $f"\left(x\right)=0$;
\ \ \
4) $f"(x)$ exists at all points of the domain of definition;
5) Coordinate line:
Figure 3.
6) Determine the sign of the derivative $f"(x)$ on each interval:
\ \}