How to find the increase interval. Increasing and decreasing intervals

Increasing and decreasing intervals provide very important information about the behavior of a function. Finding them is part of the function exploration and plotting process. In addition, extremum points, at which there is a change from increase to decrease or from decrease to increase, are given special attention when finding the largest and smallest values ​​of the 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 whole 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 the inequality is satisfied. In other words, a larger value of the argument corresponds to a larger value of the function.

Decreasing function definition.

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

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

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

Extremum points, function extrema.

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

The point is called minimum point function y=f(x) if the inequality is true for all x from its neighborhood. The value of the function at the minimum point is called function minimum 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 function extrema.

Do not confuse function extremes with the maximum and minimum values ​​of the function.

In the first figure, the maximum 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 maximum value of the function is reached at the point x=b, which is not the maximum point.

Sufficient conditions for increasing and decreasing functions.

On the basis of sufficient conditions (signs) for the increase and decrease of the function, the intervals of increase and decrease of the function are found.

Here are the formulations of the signs of increasing and decreasing functions on the 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 is decreasing on X .

Thus, to determine the intervals of increase and decrease of a function, it is necessary:

Consider an example of finding the intervals of increasing and decreasing functions to clarify the algorithm.

Example.

Find the intervals of increase and decrease of the function .

Solution.

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

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

To determine the intervals of increase and decrease of a function by a sufficient criterion, we solve the inequalities and on the domain of definition. Let us use a generalization of the interval method. The only real root of the numerator is x = 2 , and the denominator vanishes 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. By pluses and minuses, we conditionally denote the intervals on which the derivative is positive or negative. The arrows below schematically show the increase or decrease of the function on the corresponding interval.

In this way, and .

At the point x=2 the function is defined and continuous, so it must be added to both the ascending and descending intervals. At the point x=0, the function is not defined, so this point is not included in the required intervals.

We present the graph of the function to compare the obtained results with it.

Answer:

The function increases at , 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 extremum signs, 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 a -neighborhood of the point and be continuous at the point itself.

In other words:

Algorithm for finding extremum points by the first sign of the function extremum.

• Finding the scope 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 where the derivative does not exist (all the listed points are called points of possible extremum, passing through these points, the derivative just can change its sign).
• These points divide the domain 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 the function at any point of a single interval).
• We select points at which the function is continuous and, passing through which, the derivative changes sign - they are the extremum points.

Too many words, let's consider a few examples of finding extremum points and extremums of a function using the first sufficient condition for the extremum of a function.

Example.

Find the extrema of the function .

Solution.

The scope of the function is the entire set of real numbers, except for x=2 .

We find 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 line

We determine the signs of the derivative on each interval, for 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, the derivative is positive on the interval (in the figure we put a plus sign over this interval). Similarly

Therefore, we put a minus over the second interval, a minus over the third, and a plus over the fourth.

It remains to choose the 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 the extremum, x=-1 is the maximum point, it corresponds to the maximum of the function .

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, it corresponds to the minimum of the function .

Graphic illustration.

Answer:

PLEASE NOTE: the first sufficient sign of an extremum does not require the function to be differentiable at the point itself.

Example.

Find extreme points and extrema of a function .

Solution.

The domain of the 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 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 investigating a function for continuity):

Find the values ​​of the argument at which the derivative vanishes:

We mark all the obtained points on the real 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, when 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 the extremum of the function.

As you can see, this sign of the extremum of the function requires the existence of a derivative at least up to the second order at the point .

Let a rectangular coordinate system be given on some plane. The graph of some function , (X is the domain of definition) is the set of points of this plane with coordinates, where .

To build a graph, you need to draw a set of points on the plane, the coordinates of which (x; y) are related by the relation .

Most often, the graph of a function is some curve.

The easiest way to plot a graph is to plot by points.

A table is compiled in which the value of the argument is in one cell, and the value of the function from this argument is in the opposite cell. Then the obtained points are marked on the plane, and a curve is drawn through them.

An example of plotting a function graph by points:

Let's build a table.

Now we are building a graph.

But in this way it is not always possible to build a fairly accurate graph - for accuracy, you need to take a lot of points. Therefore, various methods of studying the function are used.

The full scheme of the study of function is introduced in higher educational institutions. One of the points of the study of the function is to find the intervals of increase (decrease) of the function.

A function is called increasing (decreasing) on ​​some interval if , for any x 2 and x 1 from this interval, such that x 2 >x 1 .

For example, a function whose graph is shown in the following figure, at intervals increases, and on the interval (-5; 3) decreases. That is, in between the graph is going up. And on the interval (-5; 3) "downhill".

Another point in the study of a function is the study of a function for periodicity.

A function is called periodic if there exists a number T such that .

The number T is called the period of the function. For example, the function is periodic, here the period is 2P, so

Examples of graphs of periodic functions:

The period of the first function is 3, and the second is 4.

A function is called even if an example of an even function y=x 2 .

A function is called odd if Example of an odd function y=x 3 .

The graph of an even function is symmetrical about the y-axis (axial symmetry).

The graph of an odd function is symmetrical about the origin (central symmetry).

Examples of even (left) and odd (right) function graphs.

derivative. If the derivative of a function is positive for any point in the interval, then the function is increasing; if it is negative, it is decreasing.

To find the intervals of increase and decrease of a function, you need to find the domain of its definition, the 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. Find the domain 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 \u003d (8 x - 2 x²) / x ^ 4 \u003d 2 (4 - x) / x³.

3. Solve the inequalities y’ > 0 and y’ 0;
(4 - x)/x³

4. The left side of the inequality has one real x = 4 and turns into at x = 0. Therefore, the value x = 4 is included in the interval and in the interval of decreasing, 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 into at x = 0. Therefore, the value x = 4 is included in the interval and in the interval of decreasing, and the point 0 is not included.
So, the required function increases on the interval x ∈ (-∞; 0) ∪ .

Sources:

• how to find decreasing intervals on a function

The function is a strict dependence of one number on another, or the value of the function (y) on the argument (x). Each process (not only in mathematics) can be described by its function, which will have characteristic features: intervals of decrease and increase, points of minima and maxima, and so on.

You will need

• - paper;
• - a pen.

Instruction

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 a function, which is often required in algebraic problems of this kind.

Instruction

The first step in solving the problem of determining the intervals in which the function monotonically increases or decreases is the calculation of this function. To do this, find out all the values ​​of the arguments (values ​​on the x-axis) for which you can find the value of the function. Mark the points where the gaps are observed. Find the derivative of the function. Having defined the expression that represents the derivative, equate it to zero. After that, you should find the roots of the resulting . Not about the area of ​​\u200b\u200ballowable.

The points at which the function or at which its derivative is equal to zero are the boundaries of the intervals monotony. These ranges, as well as the points separating them, should be sequentially entered in 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 a table.

The line denoting the derivative of the function f'(x) is written corresponding to the values ​​of the arguments: "+" - if the derivative is positive, "-" - negative or "0" - equal to zero. On the next line, note the monotonicity of the original expression itself. The up arrow corresponds to the increase, the down arrow corresponds to the decrease. Check the features. These are the points where the derivative is zero. An extremum can be either a high point or a low point. If the previous section of the function was increasing, and the current one is decreasing, this is the maximum point. In the case when the function was decreasing up to a given point, and now it is increasing, this is the minimum point. Enter in the table the values ​​of the function at the extremum points.

Sources:

• what is the definition of monotonicity

The study of the behavior of a function that has a complex dependence on the argument is carried out using the derivative. By the nature of the change in the derivative, one can find critical points and areas of growth or decrease of the function.

Function extremes

Definition 2

A point $x_0$ is called a point of maximum of the function $f(x)$ if there exists a neighborhood of this point such that for all $x$ from this neighborhood the inequality $f(x)\le f(x_0)$ is satisfied.

Definition 3

A point $x_0$ is called a point of maximum of the function $f(x)$ if there exists a neighborhood of this point such that for all $x$ from this neighborhood the inequality $f(x)\ge f(x_0)$ is satisfied.

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 an extremum, one can formulate theorems on sufficient and necessary conditions for its existence.

Theorem 2

Sufficient extremum condition

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 $f"(x)$ exist and keep a constant sign. Then:

1) If on the interval $(a,x_0)$ the derivative $f"\left(x\right)>0$, and on the interval $(x_0,b)$ the derivative $f"\left(x\right)

2) If the derivative $f"\left(x\right)0$ is on the interval $(a,x_0)$, 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 extremum points

The rule for examining a function for an extremum

2) Find the derivative $f"(x)$;

7) Draw conclusions about the presence of maxima and minima on each interval, using Theorem 2.

Function Ascending and Decreasing

Let us first introduce the definitions of increasing and decreasing functions.

Definition 5

A function $y=f(x)$ defined on an interval $X$ is called increasing if for any points $x_1,x_2\in X$ for $x_1

Definition 6

A function $y=f(x)$ defined on an interval $X$ is called decreasing if for any points $x_1,x_2\in X$ for $x_1f(x_2)$.

Examining a Function for Increasing and Decreasing

You can investigate functions for increasing and decreasing using the derivative.

In order to examine a function for intervals of increase and decrease, you must do the following:

1) Find the domain of the function $f(x)$;

2) Find the derivative $f"(x)$;

3) Find the points where the equality $f"\left(x\right)=0$;

4) Find points where $f"(x)$ does not exist;

5) Mark on the coordinate line all the found points and the domain of the given function;

6) Determine the sign of the derivative $f"(x)$ on each resulting interval;

7) Conclude: on the intervals where $f"\left(x\right)0$ the function increases.

Examples of problems for the study of functions for increasing, decreasing and the presence of extremum points

Example 1

Investigate the function for increasing and decreasing, and the presence of points of maxima and minima: $f(x)=(2x)^3-15x^2+36x+1$

Since the first 6 points are the same, we will draw 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:

\ \ .

- Extremum points of a function of one variable. Sufficient conditions for an extremum

Let the function f(x), defined and continuous in the interval , be not monotone in it. There are such parts [ , ] of the interval , in which the maximum and minimum values ​​are reached by the function at the internal point, i.e. between i.

It is said that the function f(x) has a maximum (or minimum) at a point if this point can be surrounded by such a neighborhood (x 0 - , x 0 +) contained in the interval where the function is given, that the inequality is satisfied for all its points.

f(x)< f(x 0)(или f(x)>f(x0))

In other words, the point x 0 gives the function f (x) a maximum (minimum) if the value f (x 0) turns out to be the largest (smallest) of the values ​​taken by the function in some (at least small) neighborhood of this point. Note that the very definition of the maximum (minimum) assumes that the function is given on both sides of the point x 0 .

If there is such a neighborhood within which (for x=x 0) the strict inequality

f(x) f(x0)

then they say that the function has its own maximum (minimum) at the point x 0, otherwise it has an improper one.

If the function has maxima at points x 0 and x 1, then, applying the second Weierstrass theorem to the interval, we see that the function reaches its smallest value in this interval at some point x 2 between x 0 and x 1 and has a minimum there. Likewise, between two lows there is bound to be a high. In the simplest (and, in practice, the most important) case, when a function generally has only a finite number of maxima and minima, they simply alternate.

Note that to designate a maximum or minimum, there is also a term that unites them - extremum.

The concepts of maximum (max f(x)) and minimum (min f(x)) are local properties of the function and take place at a certain point x 0 . The concepts of maximum (sup f(x)) and minimum (inf f(x)) values ​​refer to a finite segment and are global properties of a function on a segment.

Figure 1 shows that at points x 1 and x 3 there are local maxima, and at points x 2 and x 4 - local minima. However, the function reaches its lowest value at the point x=a, and the highest value at the point x=b.

Let us pose the problem of finding all values ​​of the argument that provide the function with an extremum. When solving it, the derivative will play the main role.

Suppose first that for the function f(x) in the interval (a,b) there is a finite derivative. If at the point x 0 the function has an extremum, then, applying to the interval (x 0 -, x 0 +), which was discussed above, Fermat's theorem, we conclude that f (x) \u003d 0 this is the necessary condition for the extremum. The extremum should be sought only at those points where the derivative is equal to zero.

It should not be thought, however, that each point at which the derivative is equal to zero delivers an extremum to the function: the just indicated necessary condition is not sufficient.