The function increases over the interval. Sufficient signs of increasing and decreasing functions
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Definition of an increasing function.
Function y=f(x) increases over 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.
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 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 smallest values of the function.
In the first picture highest value functions on an interval 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. 
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 necessary definitions, we formulate a sufficient sign of increasing and decreasing 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, a larger argument value corresponds to a larger function value.
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 the main 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 smallest values of the function.
In the first figure, the greatest value of the 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 greatest 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.
Graduation work V 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.
Basics 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 guidance 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 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.
Derivative. If the derivative of a function is positive for any point in the interval, then the function increases; if it is negative, it decreases.
To find the intervals of increase and decrease of a function, you need to find its domain of definition, derivative, solve inequalities of the form F’(x) > 0 and F’(x)
Solution.
3. Solve the inequalities y’ > 0 and y’ 0;
(4 - x)/x³
Solution.
1. Let's find the domain of definition of the function. Obviously, the expression in the denominator must always be different from zero. Therefore, 0 is excluded from the domain of definition: the function is defined for x ∈ (-∞; 0)∪(0; +∞).
2. Calculate the derivative of the function:
y'(x) = ((3 x² + 2 x - 4)' x² – (3 x² + 2 x - 4) (x²)')/x^4 = ((6 x + 2) x² – (3 x² + 2 x - 4) 2 x)/x^4 = (6 x³ + 2 x² – 6 x³ – 4 x² + 8 x)/x^ 4 = (8 x – 2 x²)/x^4 = 2 (4 - x)/x³.
3. Solve the inequalities y’ > 0 and y’ 0;
(4 - x)/x³
4. Left side inequality has one real x = 4 and turns to at x = 0. Therefore, the value x = 4 is included in both the interval and the decreasing interval, and the point 0 is not included.
So, the required function increases on the interval x ∈ (-∞; 0) ∪ .
4. The left side of the inequality has one real x = 4 and turns to at x = 0. Therefore, the value x = 4 is included in both the interval and the decreasing interval, and point 0 is not included.
So, the required function increases on the interval x ∈ (-∞; 0) ∪ .
Sources:
- how to find decreasing intervals on a function
A function represents a strict dependence of one number on another, or the value of a function (y) on an argument (x). Each process (not only in mathematics) can be described by its own function, which will have characteristics: intervals of decreasing and increasing, points of minimums and maximums, and so on.
You will need
- - paper;
- - pen.
Instructions
Example 2.
Find the intervals of decreasing f(x)=sinx +x.
The derivative of this function will be equal to: f’(x)=cosx+1.
Solving the inequality cosx+1
Interval monotony a function can be called an interval in which the function either only increases or only decreases. A number of specific actions will help to find such ranges for the function, which is often required in algebraic problems of this kind.
Instructions
The first step in solving the problem of determining the intervals in which a function monotonically increases or decreases is to calculate this function. To do this, find out all the argument values (values along the x-axis) for which you can find the value of the function. Mark the points where discontinuities are observed. Find the derivative of the function. Once you have determined the expression that represents the derivative, set it equal to zero. After this, you should find the roots of the resulting . Not about the area of permissible.
The points at which the function or at which its derivative is equal to zero represent the boundaries of the intervals monotony. These ranges, as well as the points separating them, should be sequentially entered into the table. Find the sign of the derivative of the function in the resulting intervals. To do this, substitute any argument from the interval into the expression corresponding to the derivative. If the result is positive, the function in this range increases, otherwise it decreases. The results are entered into the table.
In the line denoting the derivative of the function f’(x), the corresponding values of the arguments are written: “+” - if the derivative is positive, “-” - negative or “0” - equal to zero. In the next line, note the monotony of the original expression itself. An up arrow corresponds to an increase, and a down arrow corresponds to a decrease. Check the functions. These are the points at which the derivative is zero. An extremum can be either a maximum point or a minimum point. If the previous section of the function increased and the current one decreased, this is the maximum point. In the case when the function was decreasing before a given point, and now it is increasing, this is the minimum point. Enter the values of the function at the extremum points into the table.
Sources:
- what is the definition of monotony
The behavior of a function that has a complex dependence on an argument is studied using the derivative. By the nature of the change in the derivative, you can find critical points and areas of growth or decrease of the function.
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 the function does not belong to one of 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. Special property 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. It's about 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 хϵХ 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. Schedule limited function Guys, try to draw it 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;