Find the intervals of monotonic increase of the function online. Topic "Increasing and decreasing quadratic function" Find by
The function is called increasing on the interval
, if for any points 
inequality holds 
(a larger argument value corresponds to a larger function value).
Likewise, the function 
called decreasing on the interval
, if for any points 
from this interval if the condition is met 
inequality holds 
(a larger argument value corresponds to a smaller function value).
Increasing on the interval 
and decreasing on the interval 
functions are called monotonic on the interval
.
Knowing the derivative of a differentiable function allows one to find intervals of its monotonicity.
Theorem (sufficient condition for an increase in a function).
functions 
positive on the interval 
, then the function 
increases monotonically over this interval.
Theorem (sufficient condition for a function to decrease). If the derivative is differentiable on the interval 
functions 
negative on the interval 
, then the function 
decreases monotonically over this interval.
Geometric meaning
of these theorems is that on intervals of decreasing functions, tangents to the graph of the function form with the axis 
obtuse angles, and at increasing intervals – acute (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 (increase) on this interval.
Algorithm for finding intervals of monotonicity of a function
:
Example. Find intervals of monotonicity of a function 
.
Dot 
called maximum point of the function
such that for everyone 
, satisfying the condition 
, the inequality holds 
.
Maximum function is the value of the function at the maximum point.
Figure 2 shows an example of a graph of a function that has maxima at the points 
.
Dot 
called minimum point of the function
, if there is some number 
such that for everyone 
, satisfying the condition 
, the inequality holds 
. Fig. 2 function has a minimum at point 
.
There is a common name for highs and lows - extremes . 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 located inside this segment. One should also not confuse the maximum and minimum of a function with its greatest and lowest value on a segment – these are fundamentally different concepts.
At extremum points, the derivative has special properties.
Theorem (necessary condition for extremum). Let at the point 
function 
has an extremum. Then either 
does not exist, or 
.
Those points from the domain of definition of the function at which 
does not exist or in which 
, are called critical points of the function
.
Thus, the extremum points lie among the critical points. In general, the critical point does not have to be an extremum point. If the derivative of a function at a certain point is equal to zero, this does not mean that the function has an extremum at this point.
Example. Let's consider 
. We have 
, but point 
is not an extremum point (see Figure 3).
Theorem (the first sufficient condition for an extremum). Let at the point 
function 
is continuous, and the derivative 
when passing through a point 
changes sign. Then 
– extremum point: maximum if the sign changes from “+” to “–”, and minimum if from “–” to “+”.
If, when passing through a point 
the derivative does not change sign, then at the point 
there is no extreme.
Theorem (second sufficient condition for extremum). Let at the 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 
– extremum point 
; at 
this is the minimum point, and at 
this is the maximum point.
Algorithm for finding extrema of a function using the first sufficient condition for an extremum:
Find the derivative.
Find the critical points of the function.
Examine the sign of the derivative to the left and right of each critical point and draw a conclusion about the presence of extrema.
Find extreme values of the function.
Algorithm for finding extrema of a function using the second sufficient condition for an extremum:
Example. Find the extrema of the function 
.
Let a rectangular coordinate system be specified on a certain plane. The graph of some function , (X-domain of definition) is the set of points of this plane with coordinates, where .
To construct a graph, you need to depict on a plane a set of points whose coordinates (x;y) are related by the relation.
Most often, the graph of a function is some kind of curve.
The simplest 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 resulting points are marked on the plane, and a curve is drawn through them.
An example of constructing a function graph using points:
Let's build a table.
Now let's build a graph.
But in this way it is not always possible to construct a sufficiently accurate graph - for accuracy you need to take a lot of points. Therefore they use various methods function studies.
The complete research scheme of the function is familiarized with in higher education. educational institutions. One of the points of studying a function is to find the intervals of increase (decrease) of the function.
A function is said to be increasing (decreasing) on a certain 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, on intervals 
increases, and decreases in the interval (-5;3). That is, in the intervals The schedule is going uphill. And in the interval (-5;3) “downhill”.
Another point in the study of function is the study of function for periodicity.
A function is called periodic if there is 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 Example even function y=x2.
A function is called odd if Example odd function y=x 3 .
The graph of an even function is symmetrical about the op-amp axis (axial symmetry).
The graph of an odd function is symmetrical about the origin (central symmetry).
Examples of graphs of an even (left) and odd (right) function.
"Increasing and decreasing functions"
Lesson objectives:
1. Learn to find intervals of monotony.
2. Development of thinking abilities that ensure analysis of the situation and development of adequate methods of action (analysis, synthesis, comparison).
3. Forming interest in the subject.
Lesson progressToday we continue to study the application of the derivative and consider the question of its application to the study of functions. Front work
Now let’s give some definitions to the properties of the “Brainstorming” function.
1. What is a function called?
2. What is the name of the variable X?
3. What is the name of the variable Y?
4. What is the domain of a function?
5. What is the value set of a function?
6. Which function is called even?
7. Which function is called odd?
8. What can you say about the graph of an even function?
9. What can you say about the graph of an odd function?
10. What function is called increasing?
11. Which function is called decreasing?
12. Which function is called periodic?
Mathematics is the study of mathematical models. One of the most important mathematical models is the function. There are different ways descriptions of functions. Which one is the most obvious?
– Graphic.
– How to build a graph?
- Point by point.
This method is suitable if you know in advance what the graph approximately looks like. For example, what is a graph quadratic function, linear function, inverse proportionality, function y = sinx? (The corresponding formulas are demonstrated, students name the curves that are graphs.)
But what if you need to plot a graph of a function or even more complex one? You can find multiple points, but how does the function behave between these points?
Place two dots on the board and ask students to show what the graph “between them” might look like:
Its derivative helps you figure out how a function behaves.
Open your notebooks, write down the number, great job.
Objective of the lesson: learn how the graph of a function is related to the graph of its derivative, and learn to solve two types of problems:
1. Using the derivative graph, find the intervals of increase and decrease of the function itself, as well as the extremum points of the function;
2. Using the scheme of derivative signs on intervals, find the intervals of increase and decrease of the function itself, as well as the extremum points of the function.
Similar tasks are not in our textbooks, but are found in tests of the same state exam(part A and B).
Today in class we will look at small element work of the second stage of studying the process, studying one of the properties of the function - determining the intervals of monotonicity
To solve this problem, we need to recall some issues discussed earlier.
So, let's write down the topic of today's lesson: Signs of increasing and decreasing functions.
Signs of increasing and decreasing function:
If the derivative of a given function is positive for all values of x in the interval (a; b), i.e. f"(x) > 0, then the function increases in this interval.
If the derivative of a given function is negative for all values of x in the interval (a; b), i.e. f"(x)< 0, то функция в этом интервале убывает
The order of finding intervals of monotonicity:
Find the domain of definition of the function.
1. Find the first derivative of the function.
2. decide for yourself on the board
Find critical points, investigate the sign of the first derivative in the intervals into which the found critical points divide the domain of definition of the function. Find intervals of monotonicity of functions:
a) domain of definition,
b) find the first derivative:
c) find the critical points: ; , And
3. Let us examine the sign of the derivative in the resulting intervals and present the solution in the form of a table.
point to extremum points
Let's look at several examples of studying functions for increasing and decreasing.
A sufficient condition for the existence of a maximum is to change the sign of the derivative when passing through the critical point from “+” to “-”, and for the minimum from “-” to “+”. If, when passing through the critical point, the sign of the derivative does not change, then there is no extremum at this point
1. Find D(f).
2. Find f"(x).
3. Find stationary points, i.e. points where f"(x) = 0 or f"(x) does not exist.
(The derivative is 0 at the zeros of the numerator, the derivative does not exist at the zeros of the denominator)
4. Place D(f) and these points on the coordinate line.
5. Determine the signs of the derivative on each of the intervals
6. Apply signs.
7. Write down the answer.
Consolidation of new material.
Students work in pairs and write down the solution in their notebooks.
a) y = x³ - 6 x² + 9 x - 9;
b) y = 3 x² - 5x + 4.
Two people are working at the board.
a) y = 2 x³ – 3 x² – 36 x + 40
b) y = x4-2 x³
3. Lesson summary
Homework: test (differentiated)
Increasing and decreasing function function y = f(x) is called increasing on the interval [ a, b], 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.
Big Soviet encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .
See what “Increasing and decreasing functions” are in other dictionaries:
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A branch of mathematics that studies derivatives and differentials of functions and their applications to the study of functions. Design of D. and. into an independent mathematical discipline is associated with the names of I. Newton and G. Leibniz (second half of 17 ... Great Soviet Encyclopedia
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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 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 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. 
