Lesson and presentation in algebra in grade 10 on the topic: "Research of a function for monotonicity. Research algorithm"

Additional materials
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Manuals and simulators in the Integral online store for grade 10 from 1C
Algebraic problems with parameters, grades 9–11
Software environment "1C: Mathematical Constructor 6.1"

What we will study:
1. Decreasing and increasing functions.
2. Relationship between derivative and monotonicity of a function.
3. Two important theorems on monotonicity.
4. Examples.

Guys, earlier we looked at a lot of various functions and built their graphs. Now let's introduce new rules that work for all the functions that we have considered and will continue to consider.

Decreasing and increasing functions

A function is a correspondence y= f(x), in which each value of x is associated with a single value of y.

Let's look at the graph of some function:

Our graph shows: the larger x, the smaller y. So let's define a decreasing function. A function is said to be decreasing if higher value argument matches lower value functions.

If x2 > x1, then f(x2) Now let's look at the graph of this function:
This graph shows that the larger x, the larger y. So let's define an increasing function. A function is called increasing if a larger value of the argument corresponds to a larger value of the function.
If x2 > x1, then f(x2 > f(x1) or: the greater x, the greater y.

If a function increases or decreases over a certain interval, then it is said that it is monotonic on this interval.

Relationship between derivative and monotonicity of a function

If you look at our tangents or visually draw any other tangent, you will notice that the angle between the tangent and the positive direction of the x-axis will be acute. This means the tangent has a positive slope. Tangent slope equal to the value derivative in the abscissa of the point of tangency. Thus, the value of the derivative is positive at all points in our graph. For an increasing function performs the following inequality: f"(x) ≥ 0, for any point x.

Guys, now let's look at the graph of some decreasing function and construct tangents to the graph of the function.

Let's look at the tangents and visually draw any other tangent. We will notice that the angle between the tangent and the positive direction of the x-axis is obtuse, which means the tangent has a negative slope. Thus, the value of the derivative is negative at all points in our graph. For a decreasing function, the following inequality holds: f"(x) ≤ 0, for any point x.

So, the monotonicity of a function depends on the sign of the derivative:

If a function increases on an interval and has a derivative on this interval, then this derivative will not be negative.

If a function decreases on an interval and has a derivative on this interval, then this derivative will not be positive.

Important, so that the intervals on which we consider the function are open!

Two important theorems on monotonicity

Theorem 1. If at all points of the open interval X the inequality f’(x) ≥ 0 holds (and the equality of the derivative to zero either does not hold or holds, but only at finite set points), then the function y= f(x) increases on the interval X.

Theorem 2. If the inequality f'(x) ≤ 0 holds at all points of an open interval X (and the equality of the derivative to zero either does not hold or holds, but only at a finite set of points), then the function y= f(x) decreases on the interval X.

Theorem 3. If at all points of the open interval X the equality
f’(x)= 0, then the function y= f(x) is constant on this interval.

Examples of studying a function for monotonicity

1) Prove that the function y= x 7 + 3x 5 + 2x - 1 is increasing on the entire number line.

Solution: Let's find the derivative of our function: y"= 7 6 + 15x 4 + 2. Since the degree at x is even, then power function accepts only positive values. Then y" > 0 for any x, which means by Theorem 1, our function increases on the entire number line.

2) Prove that the function is decreasing: y= sin(2x) - 3x.

Let's find the derivative of our function: y"= 2cos(2x) - 3.
Let's solve the inequality:
2cos(2x) - 3 ≤ 0,
2cos(2x) ≤ 3,
cos(2x) ≤ 3/2.
Because -1 ≤ cos(x) ≤ 1, which means our inequality is satisfied for any x, then by Theorem 2 the function y= sin(2x) - 3x decreases.

3) Examine the monotonicity of the function: y= x 2 + 3x - 1.

Solution: Let's find the derivative of our function: y"= 2x + 3.
Let's solve the inequality:
2x + 3 ≥ 0,
x ≥ -3/2.
Then our function increases for x ≥ -3/2, and decreases for x ≤ -3/2.
Answer: For x ≥ -3/2, the function increases, for x ≤ -3/2, the function decreases.

4) Examine the monotonicity of the function: y= $\sqrt(3x - 1)$.

Solution: Let's find the derivative of our function: y"= $\frac(3)(2\sqrt(3x - 1))$.
Let's solve the inequality: $\frac(3)(2\sqrt(3x - 1))$ ≥ 0.

Our inequality is greater than or equal to zero:
$\sqrt(3x - 1)$ ≥ 0,
3x - 1 ≥ 0,
x ≥ 1/3.
Let's solve the inequality:
$\frac(3)(2\sqrt(3x-1))$ ≤ 0,

$\sqrt(3x-1)$ ≤ 0,
3x - 1 ≤ 0.
But this is impossible, because square root is defined only for positive expressions, which means our function has no decreasing intervals.
Answer: for x ≥ 1/3 the function increases.

Problems to solve independently

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 not zero, then the function is called strictly monotonous. A monotonic function is a function that changes in the same direction.

A function is incremented if a larger argument value corresponds to a larger function value. A function decreases if a larger value of the argument corresponds to a smaller value of the function.

Let the function be given. Then

A (strictly) increasing or decreasing function is called (strictly) monotonic.

Definition of extremum

A function y = f(x) is said to be increasing (decreasing) in a certain interval if, for x1< x2 выполняется неравенство (f(x1) < f(x2) (f(x1) >f(x2)).

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

(f" (x)< 0).

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

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

Extremum points

Necessary conditions for an extremum. If the point xо is the extremum point of the function f(x), then either f "(xо) = 0, or f (xо) does not exist. Such points are called critical, and the function itself is in critical point determined. The extrema of a function should be sought among its critical points.

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

Second sufficient condition. Let the function f(x) have a derivative f " (x) in the vicinity of the point xо and a second derivative at the point xо itself. If f " (xо) = 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 functions .Inflection points.

Inflection point of a function

This term has other meanings, see Inflection point.

Inflection point of a function internal point domain of definition, such that is continuous at this point, there is a finite or a certain sign infinite derivative at this point, is simultaneously the end of the interval of strict convexity upward and the beginning of the interval of strict convexity downward, or vice versa.

Unofficial

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

Increasing, decreasing and extrema of a function

Finding intervals of increasing, decreasing and extrema of a function is as follows: an independent task, and the most important part of other tasks, in particular, full function study. Initial information about the increase, decrease and extrema of the function are given in theoretical chapter on derivative, which I highly recommend for preliminary study (or repetition)– also for the reason that the following material is based on the very essentially derivative, being a harmonious continuation of this article. Although, if time is short, then a purely formal practice of examples from today’s lesson is also possible.

And today there is a spirit of rare unanimity in the air, and I can directly feel that everyone present is burning with desire learn to explore a function using its derivative. Therefore, reasonable, good, eternal terminology immediately appears on your monitor screens.

For what? One of the reasons is the most practical: so that it is clear what is generally required of you in a particular task!

Monotonicity of the function. Extremum points and extrema of a function

Let's consider some function. To put it simply, we assume that she continuous on the entire number line:

Just in case, let’s immediately get rid of possible illusions, especially for those readers who have recently become acquainted with intervals of constant sign of the function. Now we NOT INTERESTED, how the graph of the function is located relative to the axis (above, below, where the axis intersects). To be convincing, mentally erase the axes and leave one graph. Because that’s where the interest lies.

Function increases on an interval, if for any two points of this interval, connected by relationship, the inequality is true. That is, a larger value of the argument corresponds to a larger value of the function, and its graph goes “from bottom to top”. The demonstration function grows over the interval.

Likewise, the function decreases on an interval if for any two points of a given interval such that , the inequality is true. That is, a larger value of the argument corresponds to a smaller value of the function, and its graph goes “from top to bottom”. Our function decreases on intervals .

If a function increases or decreases over an interval, then it is called strictly monotonous at this interval. What is monotony? Take it literally – monotony.

You can also define non-decreasing function (relaxed condition in the first definition) and non-increasing function (softened condition in the 2nd definition). A non-decreasing or non-increasing function on an interval is called a monotonic function on a given interval (strict monotony - special case“just” monotony).

The theory also considers other approaches to determining the increase/decrease of a function, including on half-intervals, segments, but in order not to pour oil-oil-oil on your head, we will agree to operate with open intervals with categorical definitions - this is clearer, and for solving many practical problems quite enough.

Thus, in my articles the wording “monotonicity of a function” will almost always be hidden intervals strict monotony(strictly increasing or strictly decreasing function).

Neighborhood of a point. Words after which students run away wherever they can and hide in horror in the corners. ...Although after the post Cauchy limits They’re probably not hiding anymore, but just shuddering slightly =) Don’t worry, there won’t be any proofs of theorems now mathematical analysis– I needed the surroundings to formulate definitions more strictly extremum points. Let's remember:

Neighborhood of a point called the interval that contains this point, while for convenience the interval is often assumed to be symmetrical. For example, a point and its standard neighborhood:

Actually, the definitions:

The point is called strict maximum point, If exists her neighborhood, for everyone values ​​of which, except for the point itself, the inequality . In our specific example this is the point.

The point is called strict minimum point, If exists her neighborhood, for everyone values ​​of which, except for the point itself, the inequality . In the drawing there is point “a”.

Note : the requirement of neighborhood symmetry is not at all necessary. In addition, it is important the very fact of existence surroundings (even tiny, even microscopic), satisfying specified conditions

The points are called strictly extremum points or just extremum points functions. That is, it is a generalized term for maximum points and minimum points.

How do we understand the word “extreme”? Yes, just as directly as monotony. Extreme points of roller coasters.

As in the case of monotonicity, loose postulates exist and are even more common in theory (which, of course, the strict cases considered fall under!):

The point is called maximum point, If exists its surroundings are such that for everyone
The point is called minimum point, If exists its surroundings are such that for everyone values ​​of this neighborhood, the inequality holds.

Note that according to the last two definitions, any point of a constant function (or a “flat section” of a function) is considered both a maximum and a minimum point! The function, by the way, is both non-increasing and non-decreasing, that is, monotonic. However, we will leave these considerations to theorists, since in practice we almost always contemplate traditional “hills” and “hollows” (see drawing) with a unique “king of the hill” or “princess of the swamp”. As a variety, it occurs tip, directed up or down, for example, the minimum of the function at the point.

Oh, and speaking of royalty:
– the meaning is called maximum functions;
– the meaning is called minimum functions.

Common name – extremes functions.

Please be careful with your words!

Extremum points– these are “X” values.
Extremes– “game” meanings.

! Note : sometimes the listed terms refer to the “X-Y” points that lie directly on the GRAPH OF the function ITSELF.

How many extrema can a function have?

None, 1, 2, 3, ... etc. ad infinitum. For example, sine has infinitely many minima and maxima.

IMPORTANT! The term "maximum of function" not identical the term " maximum value functions." It is easy to notice that the value is maximum only in a local neighborhood, and there are “cooler comrades” at the top left. Likewise, “minimum function” is not the same as “ minimum value functions”, and in the drawing we see that the value is minimal only in a certain area. In this regard, extremum points are also called local extremum points, and the extrema – local extremes . They walk and wander nearby and global brethren. So, any parabola has at its vertex global minimum or global maximum. Further, I will not distinguish between the types of extremes, and the explanation is voiced more for general educational purposes - the additional adjectives “local”/“global” should not take you by surprise.

Let’s summarize our short excursion into the theory with a test shot: what does the task “find the monotonicity intervals and extremum points of the function” mean?

The wording encourages you to find:

– intervals of increasing/decreasing function (non-decreasing, non-increasing appears much less often);

– maximum and/or minimum points (if any exist). Well, to avoid failure, it’s better to find the minimums/maximums themselves ;-)

How to determine all this? Using the derivative function!

How to find intervals of increasing, decreasing,
extremum points and extrema of the function?

Many rules, in fact, are already known and understood from lesson about the meaning of a derivative.

Tangent derivative brings the cheerful news that function is increasing throughout domain of definition.

With cotangent and its derivative the situation is exactly the opposite.

The arcsine increases over the interval - the derivative here is positive: .
When the function is defined, but not differentiable. However, at the critical point there is a right-handed derivative and a right-handed tangent, and at the other edge there are their left-handed counterparts.

I think it won’t be too difficult for you to carry out similar reasoning for the arc cosine and its derivative.

All of the above cases, many of which are tabular derivatives, I remind you, follow directly from derivative definitions.

Why explore a function using its derivative?

To better understand what the graph of this function looks like: where it goes “bottom up”, where “top down”, where it reaches minimums and maximums (if it reaches at all). Not all functions are so simple - in most cases we have no idea at all about the graph of a particular function.

It's time to move on to more meaningful examples and consider algorithm for finding intervals of monotonicity and extrema of a function:

Example 1

Find intervals of increase/decrease and extrema of the function

Solution:

1) The first step is to find domain of a function, and also take note of the break points (if they exist). IN in this case the function is continuous on the entire number line, and this action V to a certain extent formally. But in a number of cases, serious passions flare up here, so let’s treat the paragraph without disdain.

2) The second point of the algorithm is due to

a necessary condition for an extremum:

If there is an extremum at a point, then either the value does not exist.

Confused by the ending? Extremum of the “modulus x” function .

The condition is necessary, but not enough, and the converse is not always true. So, it does not yet follow from the equality that the function reaches a maximum or minimum at point . Classic example already highlighted above - this is a cubic parabola and its critical point.

But be that as it may, necessary condition extremum dictates the need to find suspicious points. To do this, find the derivative and solve the equation:

At the beginning of the first article about function graphs I told you how to quickly build a parabola using an example : “...we take the first derivative and equate it to zero: ...So, the solution to our equation: - it is at this point that the vertex of the parabola is located...”. Now, I think, everyone understands why the vertex of the parabola is located exactly at this point =) In general, we should start with a similar example here, but it is too simple (even for a dummies). In addition, there is an analogue at the very end of the lesson about derivative of the function. Therefore, let's increase the degree:

Example 2

Find intervals of monotonicity and extrema of the function

This is an example for independent decision. Complete solution and an approximate final sample of the task at the end of the lesson.

The long-awaited moment of meeting with fractional-rational functions has arrived:

Example 3

Explore a function using the first derivative

Pay attention to how variably one and the same task can be reformulated.

Solution:

1) The function suffers infinite discontinuities at points.

2) We detect critical points. Let's find the first derivative and equate it to zero:

Let's solve the equation. A fraction is equal to zero when its numerator equal to zero:

Thus, we get three critical points:

3) We plot ALL detected points on the number line and interval method we define the signs of the DERIVATIVE:

I remind you that you need to take some point in the interval and calculate the value of the derivative at it and determine its sign. It’s more profitable not to even count, but to “estimate” verbally. Let's take, for example, a point belonging to the interval and perform the substitution: .

Two “pluses” and one “minus” give a “minus”, therefore, which means the derivative is negative over the entire interval.

The action, as you understand, needs to be carried out for each of the six intervals. By the way, note that the numerator factor and denominator are strictly positive for any point in any interval, which greatly simplifies the task.

So, the derivative told us that the FUNCTION ITSELF increases by and decreases by . It is convenient to connect intervals of the same type with the join icon.

At the point the function reaches its maximum:
At the point the function reaches a minimum:

Think about why you don't have to recalculate the second value ;-)

When passing through a point, the derivative does not change sign, so the function has NO EXTREMUM there - it both decreased and remained decreasing.

! Let's repeat important point : points are not considered critical - they contain a function not defined. Accordingly, here In principle there can be no extremes(even if the derivative changes sign).

Answer: function increases by and decreases by At the point the maximum of the function is reached: , and at the point – the minimum: .

Knowledge of monotonicity intervals and extrema, coupled with established asymptotes already gives a lot good show O appearance function graphics. A person of average training level is able to verbally determine that the graph of a function has two vertical asymptotes and oblique asymptote. Here is our hero:

Try once again to correlate the results of the study with the graph of this function.
There is no extremum at the critical point, but there is graph inflection(which, as a rule, happens in similar cases).

Example 4

Find the extrema of the function

Example 5

Find monotonicity intervals, maxima and minima of the function

…it’s almost like some kind of “X in a cube” holiday today....
Soooo, who in the gallery offered to drink for this? =)

Each task has its own substantive nuances and technical subtleties, which are commented on at the end of the lesson.

Numerical 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).

U even function the graph is symmetrical about the Oy axis.

Function y = f(X), which is defined on the set X, counts odd, if fulfilled following conditions: a) many X symmetrical about zero; b) for anyone xЄ X, f(X) = -f(-X).

U odd function the graph is symmetrical about the origin.

Function at = f(x), xЄ X, called periodic 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 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. Smallest of positive periods of a given function (if it exists) is called its main period.

In case T is the main period of the function, then to construct its graph, you can plot part of the graph on any of the intervals of the domain of determination of the length T and then do parallel transfer this section of the graph along the O axis X by ± T, ±2 T, ....

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

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

Function considered limited on a set X(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 that for anyone XЄ X the inequalities are satisfied 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(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, called increasing (non-decreasing) on a subset M WITH X when for everyone 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 a set TO, if a larger value of the argument from this set corresponds to a larger value of the function.

Function at = f(X), XЄX, called decreasing (non-increasing) on a subset M WITH X when for everyone 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, called monotonous on a subset M WITH X, if it is decreasing (non-increasing) or increasing (non-decreasing) by 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 a set M.

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

Number m called lowest value functions y on the set TO, if this number is the value of the function at a certain value X 0 arguments from the set TO, and for other values ​​of the argument x from the set TO function values less number m.

Basic properties of a function , from which it is better to begin its study and research, this is the area of ​​​​its definition and significance. You should remember how graphs are depicted elementary functions. Only then can you move on to constructing more complex graphs. The topic "Functions" has wide applications in economics and other fields of knowledge. Functions are studied throughout the entire mathematics course and continue to be studied in higher educational institutions . There, functions are studied using the first and second derivatives.