What does it mean for a function to decrease or increase? Function Properties

Monotone

A very important property of a 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.

The following types of monotony of functions are distinguished:

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 value of the argument corresponds to a smaller value of the function;

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 a function does not belong to any of the specified types, then it is called neither even nor odd or general function. Such functions have no symmetry.

Such a function, for example, is the linear function we recently considered with a graph:

3. A special property of functions is periodicity.

The fact is that the periodic functions that 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, trigonometric functions with complex arguments may have a non-standard period. We are talking 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 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. Guys, try to draw a graph of a limited function yourself.

Topic: Properties of functions: intervals of increasing and decreasing; highest and lowest values; 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;

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. 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 characteristic features: intervals of decrease and increase, 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.

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 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:

\ \, if for any pair of points X And X", a ≤ x the inequality holds f(x) ≤ f (x"), and strictly increasing - if the inequality f (x) f(x"). Decreasing and strictly decreasing functions are defined similarly. For example, the function at = X 2 (rice. , a) strictly increases on the segment , and

(rice. , b) strictly decreases on this segment. Increasing functions are designated f (x), and decreasing f (x)↓. In order for a differentiable function f (x) was increasing on the segment [ A, b], it is necessary and sufficient that its derivative f"(x) was non-negative on [ A, b].

Along with the increase and decrease of a function on a segment, we consider the increase and decrease of a function at a point. Function at = f (x) is called increasing at the point x 0 if there is an interval (α, β) containing the point x 0, which for any point X from (α, β), x> x 0 , the inequality holds f (x 0) ≤ f (x), and for any point X from (α, β), x 0 , the inequality holds f (x) ≤ f (x 0). The strict increase of a function at the point is defined similarly x 0 . If f"(x 0) > 0, then the function f(x) strictly increases at the point x 0 . If f (x) increases at each point of the interval ( a, b), then it increases over this interval.

S. B. Stechkin.

Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .

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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, a larger argument value corresponds to a larger function value.

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.