Definition of an increasing function.

Function y=f(x) increases over the interval X, if for any and inequality holds. In other words, a larger value of the argument corresponds to a larger value of the function.

Definition of a decreasing function.

Function y=f(x) decreases on the interval X, if for any and inequality holds . In other words, a larger value of the argument corresponds to a smaller value of the function.

NOTE: if the function is defined and continuous at the ends of the increasing or decreasing interval (a;b), that is, when x=a And x=b, then these points are included in the interval of increasing or decreasing. This does not contradict the definitions of an increasing and decreasing function on the interval X.

For example, from the properties of basic elementary functions we know that y=sinx defined and continuous for all real values ​​of the argument. Therefore, from the increase in the sine function on the interval, we can assert that it increases on the interval.

Extremum points, extrema of a function.

The point is called maximum point functions y=f(x), if for everyone x from its neighborhood the inequality is valid. The value of the function at the maximum point is called maximum of the function and denote .

The point is called minimum point functions y=f(x), if for everyone x from its neighborhood the inequality is valid. The value of the function at the minimum point is called minimum function and denote .

The neighborhood of a point is understood as the interval , where is a sufficiently small positive number.

The minimum and maximum points are called extremum points, and the values ​​of the function corresponding to the extremum points are called extrema of the function.

Do not confuse the extrema of a function with the largest and smallest values ​​of the function.

In the first figure, the largest value of the function on the segment is reached at the maximum point and is equal to the maximum of the function, and in the second figure - the highest value of the function is achieved at the point x=b, which is not a maximum point.

Sufficient conditions for increasing and decreasing functions.

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:

Let's consider an example of finding the intervals of increasing and decreasing functions to explain the algorithm.

Example.

Find the intervals of increasing and decreasing function.

Solution.

The first step is to find the definition of the function. In our example, the expression in the denominator should not go to zero, therefore, .

Let's move on to finding the derivative of the function:

To determine the intervals of increase and decrease of a function based on a sufficient criterion, we solve inequalities on the domain of definition. Let's use a generalization of the interval method. The only real root of the numerator is x = 2, and the denominator goes to zero at x=0. These points divide the domain of definition into intervals in which the derivative of the function retains its sign. Let's mark these points on the number line. We conventionally denote by pluses and minuses the intervals at which the derivative is positive or negative. The arrows below schematically show the increase or decrease of the function on the corresponding interval.

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 for its 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 on each interval $\left(a,x_0\right)\ and\ (x_0,b)$ the derivative $f"(x)$ exists and maintains a constant 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 go through them 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:

\\, i.e. sine function is limited. The function is odd: sin(−x)=−sin x for all x ∈ R. The graph of the function is symmetrical with respect to the origin. Periodic function 2 π : sin(x+2 π· k) = sin x, where k ∈ Z for all x ∈ R. sin x = 0 for x = π·k, k ∈ Z. sin x > 0 (positive) for all x ∈ ( 2π k, π+2π·k), k ∈ Z. sin x< 0 (отрицательная) для всех x ∈ (π+2π·k, 2π+2π k), k ∈ Z.

Cosine function

The domain of definition of a function is the set R of all real numbers. The set of function values ​​is the segment [-1; 1], i.e. cosine function is limited. The function is even: cos(−x)=cos x for all x ∈ R. The function is periodic with the smallest positive period 2 π :cos(x+2 π· k) = cos x, where k∈ Z for all x ∈ R. cos x = 0at cos x > 0 for all cos x< 0для всех The function increases from −1 to 1 on the intervals: The function decreases from −1 to 1 on the intervals: The largest value of the function sin x = 1 at points: The smallest value of the function sin x = −1 at points:

Tangent function

Multiple Function Values- the entire number line, i.e. tangent - function unlimited.

Odd function: tg(−x)=−tg x
The graph of the function is symmetrical about the OY axis.

The function is periodic with the smallest positive period π , i.e. tg(x+ π· k) = tan x, k ∈ Z for all x from the domain of definition.

Cotangent function

Multiple Function Values- the entire number line, i.e. cotangent - function unlimited.

Odd function: ctg(−x)=−ctg x for all x from the domain of definition.
The graph of the function is symmetrical about the OY axis.

The function is periodic with the smallest positive period π , i.e. ctg(x+ π· k)=ctg x, k ∈ Z for all x from the domain of definition.

21)) A set of numbers, each of which is provided with its own number n (n = 1, 2, 3, ...), is called a number sequence.

The individual numbers of a sequence are called its terms and are usually denoted as follows: first term a 1, second a 2 , .... n th member a n etc. The entire number sequence is designated

a 1 , a 2 , a 3 , ... , a n, ... or ( a n}.

22) Arithmetic progression. A numerical sequence, each member of which, starting from the second, is equal to the previous one added to a constant number for this sequence d,called arithmetic progression. Number d called progression difference. Any member of an arithmetic progression is calculated using the formula:

a n = a 1 + d (n – 1) .

Sum of the first n terms of an arithmetic progression calculated as:

Geometric progression. A numerical sequence, each member of which, starting from the second, is equal to the previous one, multiplied by a number constant for this sequence q, called geometric

progression. Number q called denominator of progression. Any member of the geometric progression is calculated using the formula:

bn=b 1 qn- 1 .

Sum of the first n terms of a geometric progression calculated as:

An infinitely decreasing geometric progression is an infinite geometric progression whose denominator satisfies the condition.

With unlimited increase the amount the first terms of an infinitely decreasing geometric progression tends to a number called the sum of an infinitely decreasing geometric progression.

) The derivative of the function f(x), f′(x) , is itself a function. This means that we can find its derivative. Let's call f′(x) the derivative of the function f(x) of the first order. The derivative of the derivative of the function f(x) is called the second-order derivative (or second derivative).

Geometric meaning of derivative. The derivative at point x 0 is equal to the slope of the tangent to the graph of the function y = f(x) at this point.

Equation of the tangent to the graph of a function: y = f(a) + f "(a)(x – a) y = f(a) + f "(a)(x – a)

Physical meaning of derivative. If a point moves along the x axis and its coordinate changes according to the law x(t), then the instantaneous speed of the point is:

24)) Derivative of the sum (difference) of functions

The derivative of an algebraic sum of functions is expressed by the following theorem.

Derivative of the sum (difference) of two differentiable functions is equal to the sum (difference) of the derivatives of these functions:

The derivative of a finite algebraic sum of differentiable functions is equal to the same algebraic sum of derivatives of terms. For example,

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 monotonicity is a special case of “simply” monotonicity).

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, now there will be no proofs of theorems of mathematical analysis - I needed the surroundings to formulate the 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 everyone values ​​of which, except for the point itself, the inequality . In our specific example, this is a dot.

The point is called strict minimum point, If exists her neighborhood, for everyone 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 neighborhood (whether tiny or microscopic) that satisfies the specified conditions

The points are called strictly extremum points or just 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 everyone
The point is called minimum point, If exists its surroundings are such that for everyone 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. ad infinitum. 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, the “minimum of a function” is not the same as the “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 short excursion into the 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 breakpoints (if they exist). In this case, the function is continuous on the entire number line, and this action is to a certain extent formal. 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 you to solve on your own. 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 an important point: points are not considered critical - they contain a function not defined. 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 ​​the appearance of the function graph. A person of average training is able to verbally determine that the graph of a function has two vertical asymptotes and an 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 subtleties, which are commented on at the end of the lesson.