Monotonic function is a function that changes in the same direction.

Function increases if the larger value of the argument corresponds to the larger value of the function. In other words, if as the value increases x meaning y also increases, then it is an increasing function.

Function decreases if the larger value of the argument corresponds to the smaller value of the function. In other words, if as the value increases x meaning y decreases, then it is a decreasing function.

If a function is increasing or decreasing on some interval, then it is called monotonic on this interval.

Function constant (non-monotonic) , if it does not decrease and does not increase.

Theorem(necessary criterion for monotonicity):

1. If a differentiable function f(x) increases in some interval, then its derivative on this interval is non-negative, i.e.

2. If a differentiable function f(x) decreases in some interval, then its derivative on this interval is nonpositive, .

3. If the function does not change, then its derivative is equal to zero, i.e. .

Theorem(sufficient sign of monotonicity):

Let f(x) be continuous on the interval (a;b) and have a derivative at all points, then:

1. If inside (a;b) is positive, then f(x) increases.

2. If inside (a;b) is negative, then f(x) is decreasing.

3. If , then f(x) is constant.

Investigation of a function for extrema.

Extremum- the maximum or minimum value of the function on a given set. The point at which the extremum is reached is called the extremum point. Accordingly, if the minimum is reached, the extremum point is called the minimum point, and if the maximum is reached, the maximum point.

1. Find the domain of the function and the intervals on which the function is continuous.

2. Find the derivative.

3. Find critical points, i.e. points at which the derivative of a function is zero or does not exist.

4. In each of the intervals into which the domain of definition is divided by critical points, determine the sign of the derivative and the nature of the change in the function.

5. For each critical point, determine whether it is the exact maximum, minimum, or is not an extremum point.

Record the result of the study of the function intervals of monotonicity and extremum.

The largest and smallest value of the function.

Scheme for finding the largest and smallest values ​​of a function continuous on a segment.

1. Find the derivative.

2. Find critical points on the given segment.

3. Calculate the value of the function at critical points and at the ends of the segment.

4. From the calculated values, choose the smallest and largest.

Convexity and concavity of a function.

An arc is called convex if it intersects any of its secants in no more than two points.

Lines formed by a convexity upwards are called convex, and those formed by a convexity downwards are called concave.

It is geometrically clear that a convex arc lies under any of its tangents, and a concave arc lies above the tangent.

Function inflection points.

An inflection point is a point on a line that separates a convex arc from a concave one.

At the inflection point, the tangent crosses the line; in the vicinity of this point, the line lies on both sides of the tangent.

The interval of decrease of the first derivative corresponds to the section of the convexity of the function graph, and the interval of increase corresponds to the section of concavity.

Theorem(about inflection points):

If the second derivative is negative everywhere in the interval, then the arc of the line y = f(x) corresponding to this interval is convex. If the second derivative is positive everywhere in the interval, then the arc of the line y = f(x) corresponding to this interval is concave.

Required sign of the inflection point:

If is the abscissa of the inflection point, then either , or does not exist.

Sufficient sign of the inflection point:

The point is the inflection point of the line y = f(x), if , a ;

When to the left of it lies a section of convexity, to the right - a section of concavity, and when to the left lies a section of concavity, and to the right - convexity.

Asymptotes.

Definition.

An asymptote of a graph of a function is a straight line that has the property that the distance from the point of the graph of the function to this line tends to zero with an unlimited distance from the origin of the graph point.

Types of asymptotes:

1. The line is called the vertical asymptote of the graph of the function y=f(x) if at least one of the direct values or equals or .

Numeric set X counts symmetrical relative to zero, if for any xЄ X meaning - X also belongs to the set X.

Function y = f(XX, counts even X xЄ X, f(X) = f(-X).

For an even function, the graph is symmetrical about the y-axis.

Function y = f(X), which is given on the set X, counts odd, if the following conditions are satisfied: a) the set X symmetrical about zero; b) for any xЄ X, f(X) = -f(-X).

For an odd function, the graph is symmetrical with respect to the origin.

Function at = f(x), xЄ X, is called periodical on X if there is a number T (T ≠ 0) (period functions) that the following conditions are met:

• X - T And X + T from many X for anyone XЄ X;
• for anyone XЄ X, f(X + T) = f(X - T) = f(X).

In case when T is the period of the function, then any number of the form mT, Where mЄ Z, m≠ 0, this is also the period of this function. The smallest of the positive periods of a given function (if it exists) is called its main period.

In case when T- the main period of the function, then to build its graph, you can build a part of the graph on any of the intervals of the length definition area T, and then make a parallel translation of this section of the graph along the O axis X to ± T, ±2 T, ....

Function y = f(X), bounded from below on the set X A, which for any XЄ X, A ≤ f(X). Graph of a function that is bounded from below on the set X, lies completely above the line at = A(this is a horizontal line).

Function at = f(x), limited from above on the set X(at the same time, it must be defined on this set), if there is a number IN, which for any XЄ X, f(X) ≤ IN. The graph of a function that is bounded from above on the set X is completely below the line at = IN(this is a horizontal line).

The function is considered limited on the set X(at the same time, it must be defined on this set) if it is bounded on this set from above and below, i.e., there are such numbers A And IN, which for any XЄ X the inequalities A ≤ f(x) ≤ B. Graph of a function that is bounded on a set X, is completely located between the straight lines at = A And at = IN(these are horizontal lines).

Function at = f (X) is considered bounded on the set X(at the same time, it must be defined on this set), if there is a number WITH> 0, which for any xЄ X, │f(X)│≤ WITH.

Function at = f(X), XЄ X, is called increasing (non-decreasing) on a subset M WITH X when for each X 1 and X 2 of M such that X 1 < X 2 , fair f(X 1) < f(X 2) (f(X 1) ≤ f(X 2)). Or the function y is called increasing on the set TO, if the larger value of the argument from this set corresponds to the larger value of the function.

Function at = f(X), XЄX is called decreasing (non-increasing) on a subset M WITH X when for each X 1 and X 2 of M such that X 1 < X 2 , fair f(X 1) > f(X 2) (f(X 1) ≥ f(X 2)). or function at is called decreasing on the set TO, if the larger value of the argument from this set corresponds to the smaller value of the function.

Function at = f(x), XЄ X, is called monotonous on a subset M WITH X, if it is decreasing (non-increasing) or increasing (non-decreasing) on M.

If the function at = f(X), XЄ X, is decreasing or increasing on a subset M WITH X, then such a function is called strictly monotonous on the set M.

Number M called the largest value of the function u on set TO, if this number is the value of the function at a certain value of x 0 set of argumentsTO, and for other values ​​of the argument from the set K, the values ​​of the function y are not greater than the numberM.

Number m called the smallest value functions y on the set TO if this number is the value of the function at a certain value X 0 arguments from set TO, and for other values ​​of the argument x from the set TO the value of the function y is not less than a number m.

Main properties of the function , with which it is better to start its study and research is the area of ​​​​its definition and significance. It should be remembered how the graphs of elementary functions are depicted. Only then can you move on to building more complex graphs. The topic "Functions" has wide applications in economics and other fields of knowledge. Functions are studied throughout the course of mathematics and continue to be studied in higher education institutions . There, functions are studied using the first and second derivatives.

Which does not change sign, that is, either always non-negative or always non-positive. If in addition the increment is non-zero, then the function is called strictly monotonous. A monotonic function is a function that varies in the same direction.

The function increases if the larger value of the argument corresponds to the larger value of the function. The function is decreasing if the larger value of the argument corresponds to the smaller value of the function.

Definitions

Let a function be given Then

. . . .

A (strictly) increasing or decreasing function is said to be (strictly) monotonic.

Other terminology

Sometimes increasing functions are called non-decreasing, and decreasing functions non-increasing. Strictly increasing functions are then simply called increasing, and strictly decreasing functions are simply decreasing.

Properties of monotonic functions

Function Monotonicity Conditions

The converse is generally not true. The derivative of a strictly monotonic function can vanish. However, the set of points where the derivative is not equal to zero must be dense on the interval. More precisely, we have

Similarly, strictly decreases on an interval if and only if the following two conditions are satisfied:

Examples

see also

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See what "Monotone function" is in other dictionaries:

Monotonic function- - a function f (x), which can be either increasing on a certain interval (that is, the larger any value of the argument on this interval, the greater the value of the function), or decreasing (in the opposite case). ... ...

A function that, when the argument increases, either always increases (or at least does not decrease), or always decreases (does not increase) ... Big Encyclopedic Dictionary

- (monotonie function) A function in which as the value of the argument grows, the value of the function always changes in the same direction. Therefore, if y=f(x), then either dy/dx 0 for all values ​​of x, in which case y is increasing... ... Economic dictionary

- (from the Greek monótonos monophonic) a function whose increments Δf(x) = f(x’) f(x) do not change sign for Δx = x’ x > 0, i.e. either always non-negative or always non-positive. Speaking not quite accurately, M. f. these are functions that change in ... ... Great Soviet Encyclopedia

A function that, when the argument increases, either always increases (or at least does not decrease), or always decreases (does not increase). * * * MONOTONE FUNCTION MONOTONE FUNCTION, a function that, when the argument increases, either always increases (or ... ... encyclopedic Dictionary

A function of one variable, defined on a certain subset of real numbers, the increment to n at does not change sign, i.e., it is either always non-negative or always non-positive. If strictly greater than (less than) zero, then the M. f. called… … Mathematical Encyclopedia

A function that, when the argument increases, either always increases (or at least does not decrease), or always decreases (does not increase) ... Natural science. encyclopedic Dictionary

This is a sequence whose elements do not decrease with increasing number, or, conversely, do not increase. Such sequences are often found in research and have a number of distinctive features and additional properties. ... ... Wikipedia

function- A team or group of people, and the tools or other resources they use to carry out one or more processes or activities. For example, customer support. This term also has another meaning: ... ... Technical Translator's Handbook

Function- 1. Dependent variable; 2. Correspondence y \u003d f (x) between variables, due to which each considered value of a certain quantity x (argument or independent variable) corresponds to a certain value ... ... Economic and Mathematical Dictionary

Def.: A function is called increasing over a certain interval if in this interval each larger value of the argument corresponds to a larger value of the function.

Def.: A function is called decreasing over a certain interval if in this interval each larger value of the argument corresponds to a smaller value of the function.

As increasing . Similarly, decreasing functions are called monotonic.

If the function is not monotonic, then its domain of definition can be divided into a finite number of intervals of monotonicity, which can alternate with intervals of constancy of the function.

The monotonicity of a function y = f(x) is characterized by the sign of its first derivative f ¤ (x), namely, if in some interval f ¤ (x) > 0, then the function increases in this interval if in some interval f ¤ (x)< 0, то функция убывает в этом промежутке.

Finding intervals of monotonicity of the function y = f(x) is reduced to finding intervals of constant sign of its first derivative f ¤ (x).

From here we obtain a rule for finding intervals of monotonicity of the function y = f(x)

1. Find zeros and discontinuity points f ¤ (x).

2. Determine by trial method the sign of f ¤ (x) in the intervals into which the points obtained in item 1 divide the domain of definition of the function f(x).

Example:

Find intervals of monotonicity of the function y \u003d - x 2 + 10x + 7

Let us find f ¤ (x). y¢ = -2x +10

The point at which y¢ = 0 is one and it divides the domain of the function into the following intervals: (– ∞,5) AND (5 ,+ ∞), in each of which y¢ retains a constant sign. Let us substitute specific values ​​of the function into these intervals and determine the sign of y¢ on the indicated intervals, then:

on the interval (–∞.5] y¢ > 0,

the function increases on the interval, and on the interval AND (3 ,+ ∞), in each of which y¢ retains a constant sign. Substitute in these intervals the specific values ​​of the function and determine the sign of y¢ on the indicated intervals, then.

Monotonic function is a function increment which does not change sign, that is, either always non-negative or always non-positive. If in addition the increment is non-zero, then the function is called strictly monotonous. A monotonic function is a function that varies in the same direction.

The function increases if the larger value of the argument corresponds to the larger value of the function. The function is decreasing if the larger value of the argument corresponds to the smaller value of the function.

Let a function be given Then

A (strictly) increasing or decreasing function is said to be (strictly) monotonic.

Definition of extremum

A function y = f(x) is called increasing (decreasing) in some interval if for x1< x2 выполняется неравенство (f(x1) < f(x2) (f(x1) >f(x2)).

If a differentiable function y = f(x) on a segment increases (decreases), then its derivative on this segment f "(x) > 0

(f "(x)< 0).

A point xо is called a point of local maximum (minimum) of the function f(x) if there exists a neighborhood of the point xо, for all points of which the inequality f(x) ≤ f(xо) (f(x) ≥ f(xо)) is true.

The points of maximum and minimum are called extreme points, and the values ​​of the function at these points are called its extreme points.

extremum points

Necessary conditions for an extremum. If the point xo is an extremum point of the function f (x), then either f "(xo) \u003d 0, or f (xo) does not exist. Such points are called critical, and the function itself is defined at the critical point. The extrema of the function should be sought among its critical points.

The first sufficient condition. Let xo be a critical point. If f "(x) changes sign from plus to minus when passing through the point xo, then the function has a maximum at the point xo, otherwise it has a minimum. If the derivative does not change sign when passing through the critical point, then there is no extremum at the point xo.

The second sufficient condition. Let the function f (x) have a derivative f "(x) in the vicinity of the point xo and a second derivative at the point xo itself. If f" (xo) = 0,> 0 (<0), то точка xоявляется точкой локального минимума (максимума) функции f(x). Если же=0, то нужно либо пользоваться первым достаточным условием, либо привлекать высшие производные.

On a segment, the function y = f(x) can reach its minimum or maximum value either at critical points or at the ends of the segment.

7. Intervals of convexity, concavity of a function .Inflection points.

Function Graph y=f(x) called convex on the interval (a;b), if it is located below any of its tangents on this interval. Function Graph y=f(x) called concave on the interval (a;b), if it is located above any of its tangents in this interval. The figure shows a curve convex on (a;b) and concave to (b;c). Examples. Consider a sufficient sign that allows you to determine whether the graph of a function in a given interval will be convex or concave. Theorem. Let y=f(x) differentiable by (a;b). If at all points of the interval (a;b) second derivative of the function y = f(x) negative, i.e. f""(x) < 0, то график функции на этом интервале выпуклый, если же f""(x) > 0 is concave. Proof. Assume for definiteness that f""(x) < 0 и докажем, что график функции будет выпуклым. Take on the function graph y = f(x) arbitrary point M 0 with abscissa x 0  (a; b) and draw through the point M 0 tangent. Her equation. We must show that the graph of the function on (a;b) lies below this tangent, i.e. with the same value x curve ordinate y = f(x) will be less than the ordinate of the tangent.

Function inflection point

This term has other meanings as well. inflection point.

Function inflection point inner point domains, such that is continuous at that point, there is a finite or definite-sign infinite derivative at that point, and is both the end of a strictly convex upward interval and the beginning of a strictly downward convex interval, or vice versa.

Unofficial

In this case, the point is inflection point function graph, that is, the graph of the function at the point "bends" through tangent to it at this point: for , the tangent lies under the graph, and attached above the graph (or vice versa)

Conditions of existence

A necessary condition for the existence of an inflection point: if a function f(x), which is twice differentiable in some neighborhood of the point , has an inflection point, then.

A sufficient condition for the existence of an inflection point: if a function is continuously differentiable in some neighborhood of the point times, and odd and, u, and a, then the function has an inflection point.