Graduate work in Unified State Examination form for 11th graders it 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 solving typical problems 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 will tell you in detail how to find the intervals of increasing and decreasing derivatives of a function analytically and graphically. 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 the tasks in the 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 Unified State Exam options, constantly updated taking into account the latest changes and innovations.
Very important information about the behavior of the function provide intervals of increasing and decreasing. Finding them is part of the process of examining the function and plotting the graph. In addition, the extremum points at which there is a change from increasing to decreasing or from decreasing to increasing are given Special attention when finding the largest and smallest values of a function on a certain interval.
In this article we will give the necessary definitions, formulate a sufficient criterion for the increase and decrease of a function on an interval and sufficient conditions for the existence of an extremum, and apply this entire theory to solving examples and problems.
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Increasing and decreasing function on an interval.
Definition of an increasing function.
The function y=f(x) increases on the interval X if for any and 
inequality holds. In other words - higher value the argument corresponds to the larger value of the function.
Definition of a decreasing function.
The 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, at x=a and x=b, then these points are included in the increasing or decreasing interval. 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 is 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 function y=f(x) if the inequality is true for all x in its neighborhood. The value of the function at the maximum point is called maximum of the function and denote .
The point is called minimum point function y=f(x) if the inequality is true for all x in its neighborhood. 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 function values corresponding to the extremum points are called extrema of the function.
Do not confuse the extrema of a function with the largest and lowest value functions.
In the first picture highest value function on the segment is achieved at the maximum point and is equal to the maximum of the function, and in the second figure - the maximum value of the function is achieved at the point x=b, which is not the 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) is positive for any x from the interval X, then the function increases by X;
- if the derivative of the function y=f(x) is negative for any x from the interval X, then the function decreases on 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 domain of 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. 
Thus, 
And 
.
At the point The x=2 function is defined and continuous, so it should be added to both the increasing and decreasing intervals. At the point x=0 the function is not defined, so we do not include this point in the required intervals.
We present a graph of the function to compare the results obtained with it.
Answer:
The function increases with 
, decreases on the interval (0;2] .
Sufficient conditions for the extremum of a function.
To find the maxima and minima of a function, you can use any of the three signs of extremum, of course, if the function satisfies their conditions. The most common and convenient is the first of them.
The first sufficient condition for an extremum.
Let the function y=f(x) be differentiable in the -neighborhood of the point and continuous at the point itself.
In other words:
Algorithm for finding extremum points based on the first sign of extremum of a function.
- We find the domain of definition of the function.
- We find the derivative of the function on the domain of definition.
- We determine the zeros of the numerator, the zeros of the denominator of the derivative and the points of the domain of definition in which the derivative does not exist (all listed points are called points of possible extremum, passing through these points, the derivative can just change its sign).
- These points divide the domain of definition of the function into intervals in which the derivative retains its sign. We determine the signs of the derivative on each of the intervals (for example, by calculating the value of the derivative of a function at any point in a particular interval).
- We select points at which the function is continuous and, passing through which, the derivative changes sign - these are the extremum points.
There are too many words, let’s better look at a few examples of finding extremum points and extrema of a function using the first sufficient condition for the extremum of a function.
Example.
Find the extrema of the function.
Solution.
The domain of a function is the entire set of real numbers except x=2.
Finding the derivative: 
The zeros of the numerator are the points x=-1 and x=5, the denominator goes to zero at x=2. Mark these points on the number axis 
We determine the signs of the derivative at each interval; to do this, we calculate the value of the derivative at any of the points of each interval, for example, at the points x=-2, x=0, x=3 and x=6.
Therefore, on the interval the derivative is positive (in the figure we put a plus sign over this interval). Likewise 
Therefore, we put a minus above the second interval, a minus above the third, and a plus above the fourth.
It remains to select points at which the function is continuous and its derivative changes sign. These are the extremum points.
At the point x=-1 the function is continuous and the derivative changes sign from plus to minus, therefore, according to the first sign of extremum, x=-1 is the maximum point, the maximum of the function corresponds to it 
.
At the point x=5 the function is continuous and the derivative changes sign from minus to plus, therefore, x=-1 is the minimum point, the minimum of the function corresponds to it 
.
Graphic illustration.
Answer:
PLEASE NOTE: the first sufficient criterion for an extremum does not require differentiability of the function at the point itself.
Example.
Find extremum points and extrema of the function 
.
Solution.
The domain of a function is the entire set of real numbers. The function itself can be written as: 
Let's find the derivative of the function: 
At the point x=0 the derivative does not exist, since the values of the one-sided limits do not coincide when the argument tends to zero: 
At the same time, the original function is continuous at the point x=0 (see the section on studying the function for continuity): 
Let's find the value of the argument at which the derivative goes to zero: 
Let's mark all the obtained points on the number line and determine the sign of the derivative on each of the intervals. To do this, we calculate the values of the derivative at arbitrary points of each interval, for example, at x=-6, x=-4, x=-1, x=1, x=4, x=6.
That is, 
Thus, according to the first sign of an extremum, the minimum points are 
, the maximum points are 
.
We calculate the corresponding minima of the function 
We calculate the corresponding maxima of the function 
Graphic illustration.
Answer:
.
The second sign of an extremum of a function.
As you can see, this sign of an extremum of a function requires the existence of a derivative at least to the second order at the point.
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 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 function
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:
\ \; .
Let us determine the sign of the function values at the ends of the segment.
f(0) = 3, f(0) > 0
f(10) = , f(10) < 0.
Since the function decreases on the segment and the sign of the function values changes, then there is one zero of the function on this segment.
Answer: the function f(x) increases on the intervals: (-∞; 0]; ;
on the interval the function has one function zero.
2. Extremum points of the function: maximum points and minimum points. Necessary and sufficient conditions for the existence of an extremum of a function. Rule for studying a function for extremum .
Definition 1:The points at which the derivative is equal to zero are called critical or stationary.
Definition 2. A point is called a minimum (maximum) point of a function if the value of the function at this point is less (greater than) the nearest values of the function.
It should be kept in mind that the maximum and minimum in in this case are local.
In Fig. 1. Local maxima and minima are shown.
The maximum and minimum functions are combined common name: extremum of the function.Theorem 1.(a necessary sign of the existence of an extremum of a function). If a function differentiable at a point has a maximum or minimum at this point, then its derivative at vanishes, .
Theorem 2.(a sufficient sign of the existence of an extremum of the function). If continuous function has a derivative at all points of some interval containing critical point(except maybe this point itself), and if the derivative, when the argument passes from left to right through the critical point, changes sign from plus to minus, then the function at this point has a maximum, and when the sign changes from minus to plus, it has a minimum.
