Pochіdna funktsії - one of the folding topics in school programs. It’s not a skinny vipusknik to give power to the power supply, which is also stolen.

This statistic is simply and intelligently said about those that are also obsolete and for whom it is necessary... We will not at once be pragmatic about the mathematics of the Viclad. Naygolovnishhe - intelligence sense.

The memory of the meaning is:

Pochіdna - the price of change of function.

On a little - graphs of three functions. Yak vi vvazhate, what about them shvidshe growth?

The third is obvious. She has the greatest swiftness of serpent, so that the most obsolete.

Axle butt stock.

Kostya, Grisha, and Matviy immediately attacked the robot. We'll be surprised, yak zmіnyuvsya їхній dohіd stretching out to fate:

On the graph you can see everything at once, why is it wrong? Dohid Kostya for the food of good health. I have Grisha dohіd also zris, ale zovsіm trochas. And the arrival of Matvia changed to zero. Start with the same idea, but the speed of change of function, tobto lost, - Rizna. Well, before Matvia - his income is lost in a negative way.

Intuitively, it is easy to assess the flexibility of the function. Ale yak is pricey?

As a matter of fact, we marveled, it’s pretty steep to go up (or down) the graph of functions. In other words - naskіlki quickly zmіnyuєtsya in zіnіnoyu x. Obviously, the very function at different points can be different from the other, so it can change more or more.

The function is recognized.

It is shown how to know the graph.

The graph of the deyakoї function is painted. Look at the new speck with the abscissa. It is carried out at the same point exactly to the graph of the function. We want to assess how cool it is up to the top of the graph of functions. The value for the whole is equal - tangent kuta nahilu dots.

Similar to the function of the point to the tangent of the kuta nahil to the point, the graph of the function of the point is drawn.

To besmirch respect - yak kut nahil dotically we take kut mіzh dotically and positively directly on the axis.

It is necessary to power up some studies, but also, you can use the graph of functions. Tse is straightforward, I’m going to give a tribute to dilyantsi I’ll give one point to the graph, and so, it’s shown on the little one. It looks like a dots to the stake.

We know. Mi pam'yataєmo, which is the tangent of the gostny kut of the rectangular tricot, which is worn out of the prototype leg to the close one. 3 tricycle:

We know I will go for the help graph. I don’t know the formula of the function. This kind of work is often used in ADI from mathematics by number.

Є y іnshe more important spіvvіdnoshennya. Zgadaimo, it is straightforward to ask rivnyannyam

The value is called kutovim kofіtsієntom straight... Wonderful to the tangent of the kuta nahil straight up to the axis.

.

Mi otrimuєmo, scho

Zapam'yataєmo tsyu formula. I won’t catch geometric zm_st lonely.

The function is similar to the one for the point to the point of view, the graph of the function is carried out at the point of the point.

In other words, the tangent of the kuta is similar to the original.

We have already said that in one kind of function in different points it can be different. Surprisingly, yak is tied to the origin of the behavioral function.

Namaluєmo graph of deyakoї function. Do not go on some dilyankas for the growth function, on the others - to change, moreover, with a growth rate. The first step is the function of the point to the maximum and the minimum.

The point function of growth. Dotic to the graph, carried out at the point, setting the gostry kut with a positive straight axis. Also, the point is positive.

Our function will change. Stosovna at the tsy tochtsi set up a blunt kut with a positive straight axis. Oskіlki tangent of a blunt kuta is negative, at a point it is negative.

Axis to go:

As well as the function of growth, it is inherited and positive.

Iakshuyutsya, її obhіdna is negative.

And what about the points at the maximum and the minimum? Mi bachimo, at the points (point of maximum) and (point of minimum) is exactly horizontal. Also, the tangent of kuta nahil dotically at the same points to zero, and it is also expensive to zero.

The speck is the maximum point. At the same point of growth, the function is to change. Otzhe, the sign of an obscene change at the point from "plus" to "minus".

At point - point minimum - the same goes to zero, but the sign changes from minus to plus.

Visnovok: for the rest of the funniest, you can learn about the behavior of the function, how to trick us.

As long as it is positive, then the function of growth.

If it is negative, then the function changes.

At the point to the maximum, there is a zero and a change sign from "plus" to "minus".

At the point minimum, there is also a zero or a change sign from minus to plus.

We can write down the graphics at the table view:

growth point to the maximum change point minimum growth
+ 0 - 0 +

Two small clarifications. One of them will become familiar to you before the hour of the revision of the plant ЄDI. First of all - for the first course, for more serious education functions and older ones.

Mossy vipadok, if the function is lost from any point to zero, but not to the maximum, but not to the minimum, from the function of the point. Tse is so called :

At the point it is similar to the graph, horizontal and the same way to zero. However, up to the point of growth, the function grows - and from the point of prodding growth. The sign of obscenity does not change - you won’t be positive, so it’s gone.

Well, so, at the point of maximum and minimum, it is not lost. On the graph, it will lead to violent evil, although it would be unwise to carry it out at the point.

How can I know I’m going to, how is the function given not by a graph, but by a formula? Have a lot of vipadku stagnate

For task B9, a graph of the function is given, or even simple, which requires only one of the advancing values:

  1. The value of the negligible in the deyakiy point x 0
  2. Points to maximum or minimum (points to extremum),
  3. Intervals of growing and changing functions (intervals of monotony).

Functions that are old, presented in the whole task, are always uninterrupted, and will significantly forgive the solution. Do not marvel at those who love to share mathematical analysis, here at all the power to help the weakest scholars, fragments of some great theoretical knowledge are not needed here.

For the meaning of the obsolete, the points of extremum and the intervals of monotony, the simple and universal algorithms - all the stench will be seen lower.

Respectfully read the mind of the staff of B9, so as not to admit bad pardons: important minds, as they pour into the relocation, there is not a lot.

Estimated value of the obhidnoi. Two-point method

As in the problem, a graph of the function f (x) is given, but it is necessary to use the graph in the actual point x 0, and it is necessary to know the value of the old one in the whole point, the offensive algorithm will be stuck:

  1. Know on the graph the dotted two "adequate" points: їх coordinates can be used. Let the qi point A (x 1; y 1) і B (x 2; y 2). Correctly write the coordinates - the whole key moment of the decision, if it’s like making a mistake here, it’s wrong.
  2. Knowing the coordinates, it is easy to calculate the increase in the argument Δx = x 2 - x 1 and the increase in the function Δy = y 2 - y 1.
  3. Ambiguous, we know the value of the original D = Δy / Δx. In other words, the need to distribute the improved function for an improved argument - and it will be considered.

Once again, it’s audible: points A and B are required to shukati itself on the dotty, and not on the graph of the function f (x), as it is often done. One hundred percent obov'yazkovo revenge hocha b two such points - іnakse zavdannya folded incorrectly.

The point A (−3; 2) and B (−1; 6) can be seen and we know better:
Δx = x 2 - x 1 = −1 - (−3) = 2; Δy = y 2 - y 1 = 6 - 2 = 4.

We know the value of the same: D = y / Δx = 4/2 = 2.

Zavdannya. A graph of the function y = f (x) is depicted on the little screen, which is similar to the one at the point of the abscissa x0. Know the value of the common function f (x) at point x 0.

The point A (0; 3) and B (3; 0) are visible, we know the difference:
Δx = x 2 - x 1 = 3 - 0 = 3; Δy = y 2 - y 1 = 0 - 3 = −3.

Now we know the value of the same: D = Δy / Δx = −3/3 = −1.

Zavdannya. A graph of the function y = f (x) is depicted on the little screen, which is similar to the one at the point of the abscissa x0. Know the value of the common function f (x) at point x 0.

The points A (0; 2) and B (5; 2) are visible and we know better:
Δx = x 2 - x 1 = 5 - 0 = 5; Δy = y 2 - y 1 = 2 - 2 = 0.

It’s too late to know the value of the original: D = y / Δx = 0/5 = 0.

From the rest of the butt, you can formulate a rule: it is parallel to the axis OX, the function is similar to the point of origin to zero. For most people, there is no need for any rahuvati - just look at the graph.

Calculating points to a maximum and a minimum

One of the alternatives of the graph of the function of the plant B9 is given the graph of the old and it is necessary to know the point of the maximum of the minimum of the function. With such a layout, the two-point method is a more simple algorithm. A selection of visas is valid from the terminology:

  1. The point x 0 is called the point of the maximum of the function f (x), as in the vicinity of the point there is irregularity: f (x 0) ≥ f (x).
  2. The point x 0 is called the point of the minimum of the function f (x), as in the vicinity of the center of the point there is an inconsistency: f (x 0) ≤ f (x).

In order to know the points to the maximum and the minimum behind the graph of the obscene, there are enough viconati such crocs:

  1. Re-cross the graph of the obscene, taking away all the information you have received. I’ll show you the practice, give me a chance to make a decision. Tom is completely on the coordinate axis of the zero-obedient - and everything.
  2. Z'yasuvati old signs on the intervals between zeros. If for the actual point x 0 it seems that f '(x 0) ≠ 0, then there are only two options: f' (x 0) ≥ 0 or f '(x 0) ≤ 0. The bad sign is easily attributed to outgoing chairs: If the graph of the old one lies behind the OX axis, then f '(x) ≥ 0. In the first place, if the graph of the old one goes through OX, then f' (x) ≤ 0.
  3. I know the revision of zero and the signs of the obscene. There, de sign change from minus to plus, point minimum. I navpaki, as a sign of a bad change from plus to minus, tse point to the maximum. Seems to the right.

Qi scheme is correct only for uninterrupted functions - those tasks B9 cannot be used.

Zavdannya. On the little one there is a graph of the funky function f (x), assigned to the image [-5; 5]. Know the point to the minimum of the function f (x) in any given way.

We will have fun with the information you are interested in - it is too much for the border [-5; 5] and zeros common x = −3 and x = 2.5. Signs are also significant:

Obviously, at point x = −3, the sign of an abject change from minus to plus. Tse і є point to the minimum.

Zavdannya. The small picture shows the graph of the funky function f (x), assigned to the image [−3; 7]. Know the maximum point of the function f (x) at any given time.

The graph is overlaid, if it is overlapped on the coordinate axis from the line [−3; 7] and zeros of the old x = −1,7 and x = 5. Signs of the bad are significant for the rendered graph. Maєmo:

Obviously, at point x = 5, the sign of an obscure change from plus to minus - a point to the maximum.

Zavdannya. On the little one there is a graph of the funky function f (x), assigned to the image [-6; 4]. Know the number of points to the maximum of the function f (x), which should be similar to [−4; 3].

Would you like to know the slid, how to make it look like a part of the graph, surrounded by vidrizkom [-4; 3]. That will be a new graph, in which it is clear that there is no more between [−4; 3] that zeros of the old all-in-the-middle. And the point itself is x = −3.5 і x = 2. We can accept:

There is only one point to the maximum x = 2 on the whole graph. The very sign of an obnoxious change from plus to minus.

There is little respect for points with non-numeric coordinates. For example, in the last problem, the point x = −3.5 is looked at, but by the very success it is possible to take x = −3.4. As far as the factory is folded correctly, such a change is not to blame for the appearance, some points without a singing place of living do not take a middle-class part from the new member. Zrozumіlo, with tsіlim dots such a focus will not pass.

Knowing the intervals for growing and changing functions

In such tasks, it is possible to the maximum and minimum points, to appear behind the graph of the old ones in the field, in which the very function of growth or changes. For a cob, it is significant, as well as growing and falling:

  1. The function f (x) is called incrementally growing, for any two points x 1 and x 2, for the first time it is correct: x 1 ≤ x 2 ⇒ f (x 1) ≤ f (x 2). In other words, the more meaningful the argument, the more meaningful the function.
  2. The function f (x) is called a descending one, for any two points x 1 and x 2, for which one is correct: x 1 ≤ x 2 ⇒ f (x 1) ≥ f (x 2). Tobto. the greater the value of the argument is less important than the function.

I will formulate ample amount of growth and decline:

  1. The function f (x) has grown without interruption, it has grown up to be quite sufficient, but it’s going all the way down to the bottom of the ball is positive, so it’s. f '(x) ≥ 0.
  2. If the function f (x) is uninterrupted, it decreased by the end, enough, but the end of the line is negative, tobto. f '(x) ≤ 0.

Acceptable without proof. In such a rank, we can accept the scheme for the known intervals of growth and fall, as it is quite similar to the algorithm for calculating the points to the extremum:

  1. Take back all your information. On the graphical graph of the deprived one of us, to go to the front of the zero function, it will be deprived of it.
  2. Mark the old characters on the intervals with zeros. There, de f '(x) ≥ 0, the growth function, and de f' (x) ≤ 0, change. Yakshcho zavdannya was established interchangeably on the change x, which was previously recognized by the new graph.
  3. Now, since we see the behavior of the function and interconnection, we need to calculate the required value.

Zavdannya. The small picture shows the graph of the funky function f (x), assigned to the image [−3; 7.5]. Know the progress of the f (x) function change. At the top of the list there is a sum of whole numbers, which can be entered before the number of promises.

Yak depends, the graph is cross-referenced and clearly between [−3; 7.5], as well as zeros of the same x = −1.5 and x = 5.3. Possibly old signs. Maєmo:

Oscillations on the interval (- 1.5) are negative, while the interval of the function change. It has fallen asleep in all numbers when it is located in the middle interval:
−1 + 0 + 1 + 2 + 3 + 4 + 5 = 14.

Zavdannya. The small picture shows the graph of the funky function f (x), assigned to the example [−10; 4]. Know the progress of the f (x) function. At the top of the list, try the best dinner.

Let's have fun with the information. Overwhelmingly deprived of the line [−10; 4] and the zeros of the obscene, which developed the chotiri: x = −8, x = −6, x = −3 and x = 2. Significantly, the signs of the obscene that can be seen will come:

We are assigned during the development of functions, tobto. so, de f '(x) ≥ 0. There are two such prompts on the graph: (−8; −6) and (−3; 2). Numerous їх dozhini:
l 1 = - 6 - (−8) = 2;
l 2 = 2 - (−3) = 5.

Oscillations need to know at least the highest interval, in all cases the value l 2 = 5 is written down.

Tsey razdіl revenge zavdannya ЄDI from mathematics on those tied to pre-existing functions and old ones.

Demonstration options ЄDI 2020 rocky stench can be set up with a number 14 for basic level and number 7 for professional rivnya

Surprise respectfully at the three graphs of functions.
Have we remembered what kind of functions are in the singing sense "relatives"?
For example, quiet dilenks, de graph of the green function of the growth of zero, the red function of growth. On quiet houses, the graph of the green function is lower than zero, the red function changes.
You can create an analogous amount of respect for a red or blue graph.
You can also take note of the green function zeros (points x = −1 ta x = 3) are taken from the dots of the extremum of the red graph: at x = −1 per graph of hearts and bachimo local maximum, at NS = 3 on the red graph local minimum.
It’s not important to note, but local maxima and minimums of the blue graph can be reached in the quiet points themselves, but the chervony graph goes through the meaning y = 0.
It is possible to develop a few more pictures about the peculiarities of the behavior of these graphs, for that, the stench of fairness is linked with oneself. Surprise at the formulas of functions, retouching from the skin from the graphs, and counting on the way, but the skin is in front є common for the offensive and, apparently, the skin is offensive in one of the most different forward.

φ 1 (x ) = φ" 2 (x ) φ 2 (x ) = Φ 1 (x )
φ 2 (x ) = φ" 3 (x ) φ 3 (x ) = Φ 2 (x )

Zgadaimo, I know about going away:

Functions y = f(x) at point NS bends the speed of the function at the point x.

Physical zmist polyagaє in that, where the turn is lost, the speed is interrupted by the process, which is described by the fallowness y = f (x).

Geometric zmіst obhidnoї In the fact that it is meaningful in the point, how to look, it’s worthwhile to the top-of-the-line conference that was carried out before the graph of the differentiated function in the whole point.

And now, don't give me the heart of a little graph. It is permissible that we do not know the formulas of functions.

Chi can I feed you forgiven, tied with behavioral function φ 2 (x ), as vidomo, as vona є obedient function φ 3 (x ) that primary function φ 1 (x )?
I can. And on a rich diet, it is possible to give an exact view, even we know that it is similar to the characteristics of the quickness of the function, so we can judge about the features of the behavior of one of these functions, marveling at the graph of the function.

First of all, go to such a power supply, scroll the side up the hill, so that the top babies should be able to take revenge on the red graph. If you are going to need it, turn it back to change the result. I am deprived of the thought of wondering at the sea of ​​solution.

Uvaga: To make the most of the initial efect the answer to the decision zavanazhayutsya okremo for skin care after the last onslaught of the keyboard on the background. (If there is a lot of work, the buttons can appear from the lock. If the buttons are not visible, please change what is allowed in your browser JavaScript.)

1) Krystyuchitsya graph ryadnoї φ" 2 (x ) (we have a graph for greenery), but with two values ​​of the function more φ 2 (−3) or φ 2 (−2)?

Behind the graph of the obnoxious, it can be seen that in the case of [−3; −2] the meaning of suvoro is positive, the same function in the case of the development of x = −3 less, lower than the right value x = −2.

View: φ 2 (−3) φ 2 (−2)

2) Clown with the first graph Φ 2 (x ) (our vipad has a blue graph), if there are two more values ​​of the function φ 2 (−1) or φ 2 (4)?

Behind the graph of the first row you can see that point x = −1 to be on the stage of growth, but on the other hand, it is positively positive. Speck x = 4 to be on the date of decline that meaning of general objectionable negative. Oskilki are more positively meaningful more than a lot, robimo visnovok means an unprecedented function, like and є poor, at point 4 less, lower at point −1.

View: φ 2 (−1) > φ 2 (4)

You can put a lot of food on the daily graph, which makes the building more versatile because of a short explanation, prompted by the same scheme. Try them out.

Zavdannya viznachennya characteristics of the original function for the graph.


Malunok 1.


Malunok 2.

Zavdannya 1

y = f (x ), assigned to the interval (-10.5; 19). Visually, the number of points is positive.

The similar function is positive quietly, the de function of growth. On the little one you can see that there are intervals (−10.5; −7.6), (−1; 8.2) and (15.7; 19). It is possible to change the number of points in the middle of the intervals: "-10", "-9", "-8", "0", "1", "2", "3", "4", "5", "6", "7", "8", "16", "17", "18". Usogo 15 points.

View: 15

Respect.
1. If in problems about graphs of functions, you can name "points", as a rule, in respect of the lack of meaning to the argument x , Які є abscissae of the points of the raised on the graph. The ordinate points are the meaning of the function, they are fallow and can be easily calculated as needed.
2. When the points were overhauled, they did not break the edge of the intervals, so the function in these points does not grow and does not change, but "grows out". Pochіdna in such points is not positive and negative, won’t be expensive, so the stench is called stationary points. Besides, we are not able to distinguish between the areas of significance, and it is said for the reason that there is an interval.

Zavdannya 2

On baby 1 there is a graph of functions y = f (x ), assigned to the interval (-10.5; 19). Choose a number of points, which have lost functions f " (x ) Negative.

The similar function is negative quietly, the de-function is reduced. On the little one you can see that there are intervals (−7.6; −1) and (8.2; ​​15.7). Cycle points in all the middle cycle intervals: -7, -6, -5, -4, -3, -2, 9, 10, 11, 12 "13", "14", "15". Usogo 13 points.

View: 13

Div. zavazhennya to the peredniy zavdannya.

For the release of the onset of the buildings, they will have one designation.

The points to the maximum and the minimum of the function are to be recognized by the original name - points to extremum .

At these points, functions are lost, either to zero or not ( need to mind extremum).
However, it is necessary to have a mind - a sign, rather than a guarantee of the extremum of the function. Sufficient mind extremumє change of the sign of the obscene: which is lost at the point of change of the sign from "+" to "-", the point is to the maximum of the function; which is lost at the point of change the sign z "-" to "+", the point to the minimum of function; if at the point where the function is missing, it is zero, because it is not, but the sign of the poor one does not change when passing through the point, then the point is not designated as a point to the extremum of the function. Tse can be a point of a bend, a point of cutting or a point of evil graph of functions.

Zavdannya 3

On baby 1 there is a graph of functions y = f (x ), assigned to the interval (-10.5; 19). Know the number of points, for which the graph of the function is parallel to the straight line y = 6 hours to be with her.

Zgadaimo, ryvnyannya straight maє viglyad y = kx + b , de k- Coefficient nahila tsієї straight to the axis Ox... Our vipadku k= 0, tobto. straight y = 6 not healed, but parallel to the axis Ox... From the same shukany, but also to the same fault, parallel to the axis Ox and it is also the fault of the mother to the function. 0. Such power is also the exact same at the points of the extremum functions. In order to respond to nutrition, you just need to screw all the points to the extremum on the graph. Here їх 4 - two points to the maximum and two points to the minimum.

View: 4

Zavdannya 4

Functions y = f (x ), marked on the interval (-11; 23). Know the sum of points to the extremum of the function for a change.

There are 2 extremum points for the specified bachimo. The maximum function can be reached at the point x 1 = 4, minimum at points x 2 = 8.
x 1 + x 2 = 4 + 8 = 12.

View: 12

Zavdannya 5

On baby 1 there is a graph of functions y = f (x ), assigned to the interval (-10.5; 19). Know the number of points that have lost functions f " (x ) door 0.

The function goes to zero at the extremum points, which can be seen on the graph 4:
2 points to the maximum and 2 points to the minimum.

View: 4

Zavdannya viznachennya characteristics of the function for the graph її pohіdnoї.


Malunok 1.

Malunok 2.

Zavdannya 6

There is a graph for little 2 f " (x ) - funky function f (x ), marked on the interval (-11; 23). At the same point, the function [−6; 2] f (x ) nabuvaє of the most significant value.

On the designated end, it’s gone nowhere, but the function hasn’t matured. Vona ubuvala chi passed through stationary points. With such a rank, the most significant function reached on the left between: x = −6.

View: −6

Note: Behind the graph, it can be seen that there are zero trichi at the points [−6; 2]. x = −6, x = −2, x = 2. Ale at points x = −2 won for the sign, from the same point to the extremum of the function. Shvidshe for everything, there was a point beyond the graph of the vykhidnaya function.

Zavdannya 7

There is a graph for little 2 f " (x ) - funky function f (x ), marked on the interval (-11; 23). At the same point, the function of the pumping up is of the lowest meaning.

At the end of the day, it’s suvoro is positive, but the function of the whole business has grown. In such a rank, of the lowest value the function reached on the left of the line: x = 3.

View: 3

Zavdannya 8

There is a graph for little 2 f " (x ) - funky function f (x ), marked on the interval (-11; 23). Know the number of points to maximize the function f (x ), it should be like [-5; ten].

If necessary, provide the maximum functionality to the extremum maybe bootie at points, de її is lost to zero. At a given endpoint: x = −2, x = 2, x = 6, x = 10. Ale zgіdno with ample brain wіn surely you will only in quiet, de sign of a bad change from "+" to "-". On the graph of the ridiculous mi bachimo, from the transition points such a point x = 6.

View: 1

Zavdannya 9

There is a graph for little 2 f " (x ) - funky function f (x ), marked on the interval (-11; 23). Know the number of points to the extremum of the function f (x ), which should be overlaid.

Extremum functions can be found in quiet points, de її looses the road 0. On a given graph of looser items, there are 5 such points: x = 2, x = 6, x = 10, x = 14, x = 18. Ale at point x = 14 lost did not change the symbol, from the її requirement to view. This rank is filled with 4 points.

View: 4

Zavdannya 10

There is a graph for little 1 f " (x ) - funky function f (x ), assigned to the interval (-10.5; 19). Know the progress of the growing function f (x ). At the top of the list, try the best dinner.

The advances in the growth of the function are made up of the advances of obnoxious positivity. On the graphs mi bachimo їkh three - (-9; -7), (4; 12), (18; 19). The one who found another from them. Yogo dojina l = 12 − 4 = 8.

View: 8

Zavdannya 11

There is a graph for little 2 f " (x ) - funky function f (x ), marked on the interval (-11; 23). Know the number of points that have a graph of the function f (x ) is parallel to the straight line y = −2x − 11 for being with her.

Kutoviy kofіtsієnt (win tangent kuta nakhilu) given straight k = −2. We are parallel to each other and to be similar, tobto. straight with such a naughty one. Going out of the geometric sense of the funniest - the cool functionality of the dots in the graph of the function, where you can see the points that are in the home −2. There are 2 such points per little 9. Ouch.

View: 9

Yak bachyte, according to the same graph, you can put the best nutritional behavior and behavioral functions. Also, the same food can be brought up to the graphs of other functions. Be respectful when you visit your child, and you will see it very easy. Іnshі vidan tsih zdannya - to the geometrical zmіst of the primordial - will be seen in the dividing line.

Sergiy Nikiforov

As the function of sign-posting on the interval is lost, but the function itself is uninterrupted from the first boundary, then the boundary points come as far as the progress of growth, and until the beginning of the decline, when

Farit Yamaev 26.10.2016 18:50

Good afternoon. Yak (at what point) can be built, but at the point, deceased to zero, the function of growth. Provide arguments. Inakshe, it’s just a whim. For which theorem? And also proved. Thank you.

Pidtrim service

The meaning of the obsolete at the point is not directly related to the development of the function for the advance. Take a look, for example, the functions - all the stench grows on

Vladlen Pisarєv 02.11.2016 22:21

Also, the function of growth on the interval (a; b) is assigned and uninterrupted at points a and b, it grows on an alternating basis. Tobto. point x = 2 to enter at the end of the line.

If you want, let the growth and decline look at the wrong direction, but in the interval.

Ale at point x = 2, the function is small local minimum. I will explain to children, if the stench is whispering points of growth (change), then points local extremum not vvazhaєmo, but promises of growth (change) - to enter.

Vrahoyuchi, scho persha chastina ЄDI for the "middle group" baby cage", then melodiously such nuance-overburning.

Okremo, I am glad for "Virisha ЄDI" to all sportsmen and teachers.

Sergiy Nikiforov

In a simple way, it is possible to delineate how it is displayed as a function of growth / change. I guess it should sound like this: the function is called growth / change to progress, if the larger argument is the function of the larger / less significant function. This is not a vikorist understanding of the obscene, the food about the point, which is degraded to turn to zero, cannot be victorious.

Iryna Ishmakova 20.11.2017 11:46

Good day. Here at the comments I bach perekonannya, shho cordoni treba turn on. Perhaps I will wait a little longer. Ale wonder, be we caress, your solution is up to the task 7089. There, for the order of prominence, the cordon does not turn on. First and foremost injected. Tobto. revision of the factory 6429 and 7089 supervise one to one. Clarify, be weasel, ciu situation.

Oleksandr Ivanov

The staff have 6429 and 7089 meals.

In one progression there is growth, and in the other progression with a positive obscenity.

Superhealth is dumb.

Extremes to enter at the midpoint of growth and decline, albeit a point, for those that are lost, they are zero, do not enter for those that are old, positive.

A Z 28.01.2019 19:09

Colleagues є understanding of growth in point

(Div. Fikhtengolts for the first time)

and your growth rate at point x = 2 is superb to compare with the classic value.

Growth and decline is a process, although it is necessary to meet the limitation principle.

Whether it is an interval, such as avenging the point x = 2, the function is not simple. The inclusion of a given point x = 2 is a process of singularities.

Zvychay, sob to find out the rogues about the inclusion of intervals, to say ok.

Oleksandr Ivanov

The function y = f (x) is called a function that is capable of doing something more meaningful, an argument for a more meaningful function.

At point x = 2, the function is differentiated, but in the interval (2; 6), it is old positive, then, for some time)