Find function gaps. "increase and decrease function"

Function called increasing in the interval
if for any points

inequality holds
(a larger value of the argument corresponds to a larger value of the function).

Similarly, the function
called decreasing in the interval
if for any points
from this interval under the condition
inequality holds
(a larger value of the argument corresponds to a lower value of the function).

Increasing on interval
and decreasing in the interval
functions are called monotonic in the interval
.

Knowing the derivative of a differentiable function allows you to find the intervals of its monotonicity.

Theorem (sufficient condition for increasing function).
the functions
positive on the interval
then the function
monotonically increases in this interval.

Theorem (sufficient condition for a decrease of a function). If the derivative is differentiable on the interval
the functions
negative on the interval
then the function
monotonously decreases in this interval.

Geometric meaning of these theorems consists in the fact that on the intervals of decreasing functions the tangents to the graph of the function form with the axis
obtuse angles, and at intervals of increase - sharp (see fig. 1).

Theorem (a necessary condition for the monotonicity of a function).If the function
differentiable and
(
) on the interval
, then it does not decrease (does not increase) in this interval.

Algorithm for finding intervals of monotonicity of a function
:

Example. Find the intervals of monotonicity of the function
.

Point called function maximum point

such that for all satisfying the condition
, the inequality
.

Maximum function Is the value of the function at the maximum point.

Figure 2 shows an example of a graph of a function with maxima at points
.

Point called function minimum point
if some number exists
such that for all satisfying the condition
, the inequality
. Naris. 2 function has a minimum at a point .

For highs and lows there is a common name - extreme points . Accordingly, the maximum and minimum points are called extremum points .

A function defined on a segment can have a maximum and a minimum only at points inside this segment. It is also impossible to confuse the maximum and minimum of a function with its largest and smallest value on a segment - these are fundamentally different concepts.

At the points of extremum, the derivative has special properties.

Theorem (a necessary condition for an extremum). Let at a point function
has an extremum. Then either
does not exist either
.

Those points from the domain of the function at which
does not exist or in which
are called critical points of function .

Thus, the extremum points lie among the critical points. In general, a critical point does not have to be an extremum point. If the derivative of the function at some point is equal to zero, then this does not mean that the function has an extremum at this point.

Example. Consider
. We have
but the point
is not an extremum point (see fig. 3).

Theorem (first sufficient condition for an extremum). Let at a point function
continuous and derivative
when crossing a point changes sign. Then - point of extremum: maximum, if the sign changes from "+" to "-", and minimum, if from "-" to "+".

If when crossing a point derivative does not change sign then at there is no extremum.

Theorem (second sufficient condition for an extremum). Let at a point derivative of a twice differentiable function
equal to zero (
), and its second derivative at this point is nonzero (
) and is continuous in some neighborhood of the point . Then - point of extremum
; at
this is the minimum point, and when
this is the maximum point.

The algorithm for finding extrema of a function using the first sufficient condition for an extremum:

Find the derivative.

Find critical points of a function.

Examine the sign of the derivative to the left and right of each critical point and conclude that there are extrema.

Find the extreme values \u200b\u200bof the function.

The algorithm for finding the extrema of a function using the second sufficient condition for an extremum:

Example. Find function extrema
.

1. Find the scope of the function

2. Find the derivative of the function

3. Set the derivative to zero and find the critical points of the function

4. Mark critical points on the definition area

5. Calculate the sign of the derivative in each of the obtained intervals

6. Find out the behavior of the function in each interval.

Example: Find the intervals of increasing and decreasing functionsf(x) = and the number of zeros of this function in the interval.

Decision:

1. D ( f) \u003d R

2. f"(x) =

D ( f") \u003d D ( f) \u003d R

3. Find the critical points of the function by solving the equation f"(x) = 0.

x(x – 10) = 0

critical points of function x \u003d 0 and x = 10.

4. Define the sign of the derivative.

f"(x) + – +

f(x) 0 10 x

in the intervals (-∞; 0) and (10; + ∞), the derivative of the function is positive at the points x \u003d 0 and x \u003d 10 function f(x) is continuous, therefore, this function increases in the intervals: (-∞; 0];.

We define the sign of the values \u200b\u200bof the function at the ends of the segment.

f(0) = 3, f(0) > 0

f(10) = , f(10) < 0.

Since the function decreases on the interval and the sign of the values \u200b\u200bof the function changes, then on this interval one zero of the function.

Answer: the function f (x) increases in the intervals: (-∞; 0];;

in the interval, the function has one function zero.

2. Function extremum points: maximum and minimum points. Necessary and sufficient conditions for the existence of an extremum of a function. Rule of function research on extremum .

Definition 1: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 the function if the value of the function at this point is less than (greater than) the nearest function.

It should be borne in mind that the maximum and minimum in this case are local.

In fig. 1. local maxima and minima are shown.

The maximum and minimum functions are united by a common name: extremum of a 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 a minimum at this point, then its derivative vanishes at,.

Theorem 2 (a sufficient sign of the existence of an extremum of a function). If a continuous function has a derivative at all points of a certain interval containing a critical point (with the exception of this point itself), and if the derivative changes sign from plus to minus when passing the argument from left to right through the critical point, then the function has a maximum at this point, and when passing the sign from minus to plus, it has a minimum.

Very important information about the behavior of the function is provided by the intervals of increase and decrease. Their finding is part of the process of researching a function and plotting. In addition, extreme points at which a change occurs from increasing to decreasing or from decreasing to increasing are given special attention when finding the largest and smallest values \u200b\u200bof the function in a certain interval.

In this article, we give the necessary definitions, formulate a sufficient sign of the increase and decrease of the function on the interval and sufficient conditions for the existence of an extremum, and apply this whole theory to solving examples and problems.

Page navigation.

The increase and decrease of the function on the interval.

Definition of increasing function.

The function y \u003d f (x) increases on the interval X if, for any and inequality holds. In other words, a larger value of the argument corresponds to a larger value of the function.

Definition of a decreasing function.

The function y \u003d 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 lower value of the function.

NOTE: if the function is defined and continuous at the ends of the interval of increasing or decreasing (a; b), that is, with x \u003d a and x \u003d b, then these points are included in the interval of increasing or decreasing. This does not contradict the definitions of increasing and decreasing functions on the interval X.

For example, from the properties of the basic elementary functions, we know that y \u003d sinx is defined and continuous for all real values \u200b\u200bof the argument. Therefore, from an increase in the function of the sine in the interval, we can state about the increase in the interval.

Points of extremum, extrema of the function.

Point called maximum point function y \u003d f (x), if for all x from its neighborhood the inequality holds. The value of the function at the maximum point is called maximum function and denote.

Point called minimum point function y \u003d f (x), if for all x from its neighborhood the inequality holds. The value of the function at the minimum point is called minimum function and denote.

By a neighborhood, points mean the interval where is a sufficiently small positive number.

The minimum and maximum points are called extremum points, and the values \u200b\u200bof the function corresponding to the points of the extremum are called extrema functions.

Do not confuse the extrema of the function with the largest and smallest value of the function.

In the first figure, the greatest value of the function in the interval is reached 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 reached at the point x \u003d b, which is not the maximum point.

Sufficient conditions for increasing and decreasing functions.

Based on sufficient conditions (signs) of the increase and decrease of the function, the intervals of increase and decrease of the function are found.

Here are the wordings of the signs of increasing and decreasing functions on the interval:

if the derivative of the function y \u003d f (x) is positive for any x from the interval X, then the function increases on X;
if the derivative of the function y \u003d 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 the function, it is necessary:

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

Example.

Find the intervals of increasing and decreasing functions.

Decision.

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

We proceed to find the derivative function:

To determine the intervals of increasing and decreasing functions by a sufficient criterion, we solve inequalities in the domain of definition. We use a generalization of the interval method. The only valid root of the numerator is x \u003d 2, and the denominator vanishes at x \u003d 0. These points break down the domain into intervals in which the derivative of the function retains its sign. We mark these points on the number line. The pluses and minuses will arbitrarily denote the intervals at which the derivative is positive or negative. The arrows below schematically show the increase or decrease of the function in the corresponding interval.

Thus, and .

At the point x \u003d 2, the function is defined and continuous; therefore, it should be added to both the increase and the decrease. At the point x \u003d 0, the function is not defined; therefore, this point is not included in the desired intervals.

We give a graph of the function for comparing the results obtained with it.

Answer:

The function increases when , decreases on the interval (0; 2].

Sufficient conditions for the extremum of a function.

To find the maxima and minima of the function, you can use any of the three signs of the 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 \u003d f (x) be differentiable in the-neighborhood of a point, and be continuous at the point itself.

In other words:

Algorithm for finding extremum points by the first sign of the 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 in which the derivative does not exist (all the points listed are called points of possible extremepassing through these points, the derivative can just change its sign).
These points divide the domain of the function into the intervals in which the derivative retains the sign. We determine the signs of the derivative on each of the intervals (for example, calculating the value of the derivative of a function at any point in a given interval).
We choose the points at which the function is continuous and, passing through which, the derivative changes sign - they are the points of extremum.

Too many words, we’ll better consider a few examples of finding points of the extremum and extrema of a function using the first sufficient condition for the extremum of the function.

Example.

Find the extrema of the function.

Decision.

The domain of the function is the entire set of real numbers, except x \u003d 2.

We find the derivative:

The zeros of the numerator are the points x \u003d -1 and x \u003d 5, the denominator vanishes at x \u003d 2. Mark these points on the numerical axis.

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 \u003d -2, x \u003d 0, x \u003d 3 and x \u003d 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 extreme points.

At the point x \u003d -1 the function is continuous and the derivative changes sign from plus to minus, therefore, according to the first sign of extremum, x \u003d -1 is the maximum point, the maximum of the function corresponds to it .

At the point x \u003d 5 the function is continuous and the derivative changes sign from minus to plus, therefore, x \u003d -1 is the minimum point, the minimum of the function corresponds to it .

Graphic illustration.

Answer:

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

Example.

Find the extremum points and extrema of the function .

Decision.

The domain of a function is the whole set of real numbers. The function itself can be written as:

Find the derivative of the function:

At the point x \u003d 0, the derivative does not exist, since the values \u200b\u200bof the one-sided limits do not coincide when the argument tends to zero:

At the same time, the original function is continuous at x \u003d 0 (see the section on examining a function for continuity):

Find the value of the argument at which the derivative vanishes:

We mark all the points obtained on the number line and determine the sign of the derivative on each of the intervals. For this, we calculate the values \u200b\u200bof the derivative at arbitrary points of each interval, for example, for x \u003d -6, x \u003d -4, x \u003d -1, x \u003d 1, x \u003d 4, x \u003d 6.

I.e,

Thus, according to the first sign of extremum, the minimum points are , the maximum points are .

We calculate the corresponding function minima

We calculate the corresponding function maxima

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 a point.

Function increase and decrease

function y = f(x) is called increasing on the segment [ a, b] if for any pair of points x and x ", and ≤ x the inequality f(x) ≤ f (x "), and strictly increasing - if the inequality f (x) f(x ") Decreasing and strict decreasing of a function are defined in a similar way. For example, the function at = x 2 (fig. , a) strictly increases on the segment, and

(fig. , b) strictly decreases on this segment. Increasing functions are denoted by f (x), and decreasing f (x) ↓. In order to 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 the function on the segment, the increase and decrease of the function at the point are considered. Function at = f (x) is called increasing at the point x 0 if there is such an interval (α, β) containing the point x 0 that for any point x from (α, β), x\u003e x 0, the inequality f (x 0) ≤ f (x), and for any point x from (α, β), x 0, the inequality f (x) ≤ f (x 0). Similarly, a strict increase in the function at the point 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 in this interval.

S. B. Stechkin.

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

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Extreme functions

Definition 2

The point $ x_0 $ is called the maximum point 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) $ holds.

Definition 3

The point $ x_0 $ is called the maximum point of the function $ f (x) $ if there is a neighborhood of this point such that for all $ x $ from 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. We introduce its definition.

Definition 4

$ x_0 $ is called the critical point of the function $ f (x) $ if:

1) $ x_0 $ is the internal point of the definition domain;

2) $ f "\\ left (x_0 \\ right) \u003d 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

A sufficient condition for extremum

Let the point $ x_0 $ be critical for the function $ y \u003d 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 is $ f "\\ left (x \\ right)\u003e 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 is $ f "\\ left (x \\ right) 0 $, then the point $ x_0 $ is the minimum point for this function.

3) If both the interval $ (a, x_0) $ and the interval $ (x_0, b) $ the derivative $ f "\\ left (x \\ right)\u003e 0 $ or the derivative $ f" \\ left (x \\ right)

This theorem is illustrated in Figure 1.

Figure 1. A sufficient condition for the existence of extrema

Examples of extremes (Fig. 2).

Figure 2. Examples of extreme points

Rule of function research on extremum

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

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

Function increase and decrease

We introduce, for starters, the definition of increasing and decreasing functions.

Definition 5

The function $ y \u003d f (x) $ defined on the interval $ X $ is called increasing if for any points $ x_1, x_2 \\ in X $ for $ x_1

Definition 6

The function $ y \u003d f (x) $ defined on the interval $ X $ is called decreasing if for any points $ x_1, x_2 \\ in X $ for $ x_1f (x_2) $.

Investigation of the function of increasing and decreasing

You can explore the functions of increasing and decreasing using the derivative.

In order to investigate the function for intervals of increase and decrease, it is necessary to do the following:

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

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

3) Find the points at which the equality $ f "\\ left (x \\ right) \u003d 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 this function;

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

7) To conclude: at intervals where $ f "\\ left (x \\ right) 0 $ the function increases.

Examples of tasks for studying functions of increasing, decreasing, and the presence of extreme points

Example 1

Investigate the function of increasing and decreasing, and the presence of maximum and minimum points: $ f (x) \u003d (2x) ^ 3-15x ^ 2 + 36x + 1 $

Since the first 6 points are the same, let’s start with them.

1) Scope - all real numbers;

2) $ f "\\ left (x \\ right) \u003d 6x ^ 2-30x + 36 $;

3) $ f "\\ left (x \\ right) \u003d 0 $;

\ \ \

4) $ f "(x) $ exists at all points in the domain of definition;

5) Coordinate line:

Figure 3

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

\ \}