Twenty-five graduates of one of the eleventh grades of school No. 4 in city N passed the profile level of the Unified State Examination in mathematics. The lowest score obtained by exactly two of these graduates is 18, and the highest is 82. The threshold is 27 points. Select statements that follow from this information.

1) Among these graduates there is at least one who received 82 points for the Unified State Examination in mathematics.
2) Among these graduates, there are exactly two who did not score the threshold score.
3) Among these graduates there are at least two people with equal scores for the Unified State Examination in mathematics.
4) Points for the Unified State Examination in mathematics of any of these graduates are not higher than 82.

In 1312, in the city of Blaviken, the price of amulets against dark forces increased by 12% compared to 1311, and in 1314 - by 38% compared to 1312. Which of the following statements follow from these data?

1) In 1315, the price of amulets against dark forces will increase, but not much compared to 1314.
2) For three years, the price has increased by one and a half times compared to 1311.
3) There are many dark forces in the city.
4) None of the proposed.

In your answer, indicate the numbers of the selected statements without spaces, commas and other additional characters.

There are 36 subscribers in the public Mythology of the ancient Kyrgyz, of which 25 know English, 14 know German, and only four know French. Choose the statements that follow from the given data.

In public:
1) there is not a single person who knows all three of these languages
2) at least two subscribers know both English and German
3) each subscriber knows at least one foreign language
4) at least one subscriber knows both German and French

In your answer, indicate the numbers of the selected statements without spaces, commas and other additional characters.

Among the four tallest boys in the class, Petya is taller than Sasha, Misha is taller than Andrey, Andrey is shorter than Petya, and Sasha is fatter than Andrey. Choose the statements that follow from the given data.

1) Petya is the tallest in the class.
2) Andrei is the shortest of these four boys.
3) Andrei is not the tallest in the class.
4) If you add up the heights of Petya and Sasha, then the result will be greater than the sum of the heights of Misha and Andrey.

In your answer, indicate the numbers of the selected statements without spaces, commas and other additional characters.

Graduate Barankin passed the exam in four subjects. He showed the lowest result in mathematics - 33 points (for other exams, the points are higher). The average score of Barankin for four passed exams is 45 points. Choose the statements that follow from the given data.

1) The average score for three exams, except for mathematics, is 49.
2) All subjects, except for mathematics, Barankin passed 45 points or better.
3) Barankin did not even get 80 points in any of these four subjects.
4) In some subject, Barankin received more than 48 points.

In your answer, indicate the numbers of the selected statements without spaces, commas and other additional characters.

14 cats live in Antonina Petrovna's apartment. Each cat is over a year old but under 17 years old. Select the statements that follow from the given information.

1) 7 cats in this apartment are under 9 years old.
2) There is a cat in this apartment that is over 11 years old.
3) The oldest cat in this apartment is less than 22 years older than the youngest.
4) There are no 6-month-old kittens in this apartment.

In your answer, indicate the numbers of the selected statements without spaces, commas and other additional characters.

At the Winter Olympics in Sochi, the Zimbabwean team won fewer medals than the Kazakh team, the Cameroon team - less than the Danish team, and the Russian team - more than the teams of all these four countries combined. Choose the statements that are true under the given conditions.

1) The Russian team won five times more medals than the teams of Cameroon and Zimbabwe together.
2) The Danish team won more medals than the Kazakhstan team.
3) National teams of Cameroon and Zimbabwe won the same number of medals.
4) The Russian team won more medals than each of the other four teams.

In your answer, indicate the numbers of the selected statements without spaces, commas and other additional characters.

When Ivan Valeryevich is fishing, he always switches his phone to silent mode. Select the statements that are true under the given condition.

1) If Ivan Valeryevich's phone is on silent mode, then he is fishing.
2) If Ivan Valeryevich is on a catfish fishing, then his phone is on silent mode.
3) If Ivan Valeryevich's phone is not on silent mode, then he is not fishing.
4) If Ivan Valeryevich's phone is not on silent mode, then his wife did not let him go fishing.

In your answer, indicate the numbers of the selected statements without spaces, commas and other additional characters.

Among the residents of house number 23 there are those who work, and there are those who study. And also there are those who do not work and do not study. Some residents of house number 23, who study, also work. Choose the statements that are true under the given conditions.

1) At least one of the working residents of house No. 23 is studying.
2) All residents of house number 23 work.
3) Among the residents of house No. 23 there are no those who do not work and do not study.
4) At least one of the residents of house No. 23 works.

Before the volleyball tournament, the height of the players of the volleyball team of the city N was measured. It turned out that the height of each of the volleyball players of this team is more than 190 cm and less than 210 cm. Choose statements that are true under the specified conditions.

1) The volleyball team of the city N must have a player whose height is 220 cm.
2) The volleyball team of city N does not have players with a height of 189 cm.
3) The height of any volleyball player of this team is less than 210 cm.
4) The difference in height of any two players of the city N volleyball team is more than 20 cm.

In your answer, write down the numbers of the selected statements without spaces, commas, or other additional characters.

Some employees of the company in the summer of 2014 rested in the country, and some - on the sea. All employees who did not rest on the sea, rested in the country. Choose the statements that are true under the given conditions.

1) Each employee of this company rested in the summer of 2014, either in the country, or at sea, or both.
2) An employee of this company, who did not rest at sea in the summer of 2014, did not rest in the country either.
3) If Faina did not rest in the summer of 2014 either at the dacha or at sea, then she is an employee of this company.
4) If an employee of this company did not rest at sea in the summer of 2014, then he rested in the country.
In your answer, write down the numbers of the selected statements without spaces, commas, or other additional characters.

In the country of Dotalandia, there are more men than women. The most common male name is Ivan, the female name is Maria. Select the statements that follow from the given data.
In the country "Dotaland":

1) there are more women with the name Maria than with the name Avdotya
2) there are more men with the name Evsikaky than with the name Eustathius
3) at least one woman has the name Maria
4) there are more men with the name Anton than women with the name Dulcinea

In your answer, indicate the numbers of the selected statements without spaces, commas and other additional characters.

The school purchased a table, blackboard, tape recorder and printer. It is known that a printer is more expensive than a tape recorder, and a board is cheaper than a tape recorder and cheaper than a table. Select the statements that are true under the given conditions.

1) A tape recorder is cheaper than a board.
2) The printer is more expensive than the board.
3) The board is the cheapest of purchases.
4) Printer and board cost the same.

In your answer, write down the numbers of the selected statements without spaces, commas, or other additional characters.

There are 30 students in the class, 20 of them attend the biology circle, and 16 attend the geography circle. Select the statements that are true under the given conditions.

1) There are at least two of this class who attend both circles.
2) Each student from this class attends both circles.
3) There are 11 people who do not attend any circle.
4) There will not be 17 people from this class who attend both circles.

In your answer, write down the numbers of the selected statements without spaces, commas, or other additional characters.

The hostess bought a cake, pineapple, juice and cold cuts for the holiday. Cake cost more than pineapple, but cheaper than cold cuts, juice cost less than cake. Select the statements that are true under the given conditions.

1) Pineapple was cheaper than cold cuts.
2) They paid more for juice than cold cuts.
3) Cold cuts are the most expensive purchase.
4) Cake is the cheapest purchase.

In your answer, write down the numbers of the selected statements without spaces, commas, or other additional characters.

1) The table is cheaper than a copier.
2) The rack is more expensive than a copier.

In your answer, write down the numbers of the selected statements without spaces, commas, or other additional characters.

Vitya is taller than Kolya, but shorter than Masha. Anya is no taller than Vitya. Select the statements that are true under the given conditions.

1) Masha is the tallest of these four people.

2) Anya and Masha are the same height.

3) Vitya and Kolya are the same height.

4) Kolya is lower than Masha.

In your answer, write down the numbers of the selected statements without spaces, commas, or other additional characters.

Twenty graduates of one of the eleventh grades passed the exam in social science. The lowest score obtained was 36 and the highest was 75. Select the statements that are true under the given conditions.

1) Among these graduates there are twenty people with equal scores for the Unified State Examination in social studies.
2) Among these graduates there is a person who received 75 points for the Unified State Examination
in social science.
3) Points for the Unified State Examination in social studies of any of these twenty people
not less than 35.
4) Among these graduates there is a person who received 20 points for the Unified State Examination in social studies.

In your answer, write down the numbers of the selected statements without spaces, commas, or other additional characters.

1) Each student of this class attends both circles.
2) There are at least two from this class who attend both circles.
3) If a student from this class goes to a circle in history, then he must go to a circle in mathematics.
4) There will not be 11 people from this class who attend both circles.

In your answer, write down the numbers of the selected statements without spaces, commas, or other additional characters.

In a pet store, 30 fish were launched into one of the aquariums. The length of each fish is more than 2 cm, but does not exceed 8 cm. Choose the statements that are true under the given conditions.

1) Seven fish in this aquarium are shorter than 2 cm.
2) There is no fish 9 cm long in this aquarium.
3) The difference in the length of any two fish is not more than 6 cm.
4) The length of each fish is more than 8 cm.

In your answer, write down the numbers of the selected statements without spaces, commas, or other additional characters.

The company purchased a rack, a table, a projector and a photocopier. It is known that a rack is more expensive than a table, and a copier is cheaper than a table and cheaper than a projector. Select the statements that are true under the given conditions.

1) The table is cheaper than a copier.
2) The rack is more expensive than a copier.
3) Xerox is the cheapest purchase.
4) Rack and copier cost the same.

Olya is younger than Alice, but older than Ira. Lena is not younger than Ira. Select the statements that are true under the given conditions.

1) Alice and Ira are the same age.
2) Among these four people there is no one younger than Ira.
3) Alice is older than Ira.
4) Alice and Olya are the same age.

If an athlete participating in the Olympic Games sets a world record, then his result is also an Olympic record.

Choose the statement that is true under the given condition.

1) If the result of an athlete participating in the Olympic Games is not an Olympic record, then it is not a world record either.

2) If the result of an athlete participating in the Olympic Games is not an Olympic record, then it is a world record.

3) If the result of an athlete participating in the Olympic Games is a world record, then it is not an Olympic record.

4) If an athlete participating in the Olympic Games sets a world record in the 100m, then his result is also an Olympic record.

In your answer, indicate the numbers of the selected statements without spaces,
commas and other additional characters.

Among summer residents in the village there are those who grow grapes, and there are those who grow pears. And also there are those who do not grow either grapes or pears. Some summer residents in this village who grow grapes also grow pears. Select the statements that are true under the given conditions.

1) If a summer resident from this village does not grow grapes, then he grows pears.
2) Among those who grow grapes, there are summer residents from this village.
3) There is at least one summer resident in this village who grows both pears and grapes.
4) If a summer resident in this village grows grapes, then he does not grow pears.

In your answer, write down the numbers of the selected statements without spaces, commas, or other additional characters.

Among those who are registered on VKontakte, there are schoolchildren from Tver. Among schoolchildren from Tver there are those who are registered in Odnoklassniki. Select the statements that are true under the given conditions.

1) All schoolchildren from Tver are not registered on either VKontakte or Odnoklassniki.
2) Among schoolchildren from Tver, there are no those who are registered on VKontakte.
3) Among schoolchildren from Tver there are those who are registered in VKontakte.
4) At least one of the Odnoklassniki users is a student from Tver.

In your answer, write down the numbers of the selected statements without spaces, commas, or other additional characters.

Company N has 50 employees, 40 of whom know
English and 20 German. Select the statements that are true under the given conditions.
1) In firm N, at least three employees know both English and German.
2) There is not a single employee in this firm who knows both English and German.
3) If an employee of this company knows English, then he also knows German.
4) No more than 20 employees of this firm know both English and German.
In your answer, write down the numbers of the selected statements without spaces, commas, or other additional characters.

When physics teacher Nikolai Dmitrievich teaches a lesson, he always turns off his phone. Select the statements that are true under the given condition.
1. If Nikolai Dmitrievich's phone is turned on, he does not teach a lesson.
2. If Nikolai Dmitrievich's phone is turned on, then he is teaching a lesson.
3. If Nikolai Dmitrievich conducts laboratory work in physics at the lesson, then his phone is turned off.
4. If Nikolai Dmitrievich is teaching a physics lesson, then his phone is on.

2) If the house has gas stoves, then this house has less than 13 floors.
3) If the house has more than 17 floors, then gas stoves are installed in it.
4) If the house has gas stoves, then it has no more than 12 floors.
In your answer, write down the numbers of the selected statements without spaces, commas, or other additional characters.

1) There are 10 people in this company who do not use either the Odnoklassniki network or the VKontakte network.

2) There are at least 5 people in this company using both networks.

3) There is not a single person from this company who uses only the Odnoklassniki network.

4) No more than 10 people from this company use both networks.

In your answer, write down the numbers of the selected statements without spaces, commas, or other additional characters.

2) If Ivan Petrovich's phone is turned on, it means that he is teaching a lesson.

3) If Ivan Petrovich is doing a math test, then his phone is turned off.

4) If Ivan Petrovich is teaching a math lesson, then his phone is on.

In your answer, indicate the numbers of the selected statements without spaces, commas and other additional characters.

There are 20 students in the class, 13 of them attend the history circle, and 10 attend the mathematics circle. Select the statements that are true under the given conditions.

1) Each student of this class attends both circles.
2) If a student from this class goes to a circle on history, then he definitely goes to a circle on mathematics.
3) There are at least two of this class who attend both circles.
4) There will not be 11 people from this class who attend both circles.
1) Vitya is taller than Sasha.
2) Sasha is shorter than Anya.
3) Kolya and Masha are the same height.
4) Vitya is the tallest of all.
In your answer, indicate the numbers of the selected statements without spaces, commas and other additional characters.

USE 2017. Mathematics. Task 18. Tasks with a parameter. Sadovnichiy Yu.V.

M.: 2017. - 128 p.

This book is devoted to tasks similar to task 18 of the Unified State Examination in mathematics (task with a parameter). Various methods for solving such problems are considered, and much attention is paid to graphic illustrations. The book will be useful for high school students, mathematics teachers, tutors.

Format: pdf

The size: 1.6 MB

Watch, download:drive.google

CONTENT
Introduction 4
§one. Linear equations and systems of linear equations 5
Tasks for independent solution 11
§2. Investigation of the square trinomial using the discriminant 12
Tasks for independent solution 19
§3. Vieta's theorem 20
Tasks for independent solution 26
§four. Location of the roots of the square trinomial 28
Tasks for independent solution 43
§5. Application of graphic illustrations
to the study of the square trinomial 45
Tasks for independent solution 55
§6. Function limitation. Finding the range 56
Tasks for independent solution 67
§7. Other properties of functions 69
Tasks for independent solution 80
§eight. Logic tasks with parameter 82
Tasks for independent solution 93
Illustrations on the coordinate plane 95
Tasks for independent solution 108
Okha method 110
Tasks for independent solution 119
Answers 120

This book is devoted to tasks similar to task 18 of the Unified State Examination in mathematics (task with a parameter). Along with problem 19 (a problem that uses the properties of integers), problem 18 is the most difficult in the variant. Nevertheless, the book attempts to systematize problems of this type according to various methods for their solution.
Several paragraphs are devoted to what seems to be such a popular topic as the study of the square trinomial. However, sometimes such tasks require different, sometimes the most unexpected approaches to their solution. One such non-standard approach is demonstrated in example 7 of paragraph 2.
Often, when solving a problem with a parameter, it is necessary to investigate the function given in the condition. The book formulates some statements concerning such properties of functions as boundedness, parity, continuity; after that, examples demonstrate the application of these properties to solving problems.

The wording of the task limits the material only to cases of setting commas. This is a significant narrowing of the topic.

Commas are used in the following cases:

The subordinate clause is separated from the main comma if it comes before or after the main:

When she entered the room, I stood up.

(When…), .

I stood up when she entered the room.

, (when…).

The subordinate clause is separated from the main one by commas on both sides, if it is inside the main one:

Yesterday, when Ivan called, I was busy.

[ , (when…), ].

Homogeneous subordinate clauses connected without a union are separated by a comma:

He knew that the teacher would call his mother, his mother would be extremely unhappy, he would get hit.

, (what …), (), ().

Homogeneous clauses are connected by repeated unions, commas are placed in the same way as with homogeneous members:

He knew that the teacher would call his mother, and that his mother would be extremely unhappy, and that he would fly into it.

, (what...), and (what...), and (what...).

Relative clauses with complex subordinating conjunctions because, due to the fact that, in view of the fact that, instead of, in order to, after, while and others similar are separated from the main by one comma, which is placed on the border of the main and subordinate clauses:

As he spoke, I became more and more perplexed.

(As…),.

I became more and more perplexed as he spoke.

, (as...).

As he spoke, I became more and more perplexed.

[ (as...) ].

Compound unions can fall into two parts if:

1) there is a negative particle in front of them not:

She is not I answered because I was afraid.

2) there are particles in front of them only, just, just etc., expressing a restrictive meaning:

She answered only because she was scared.

Attention:

Unions while, as if, even if, only when do not break.

If there are two subordinating unions nearby, then a comma is placed between them in all cases, except for those when these are complex unions with then.

Need a comma: They decided that if the weather was good in the morning, they would go out of town.
No comma: They decided that if the weather was fine in the morning, then they go out of town.

Definitive clauses with an allied word which the. A comma after an allied word that is not put. This rule works even if the word which the included in the adverbial turnover:

I do not know how to react to a situation from which I see no way out.

We settled down on the shore of the lake, the shores of which were overgrown with lingonberries.

(Comma after adverbial phrase knowing which not set).

In contact with

Classmates

Handbook for preparing for the exam

• Task 16. Punctuation marks in sentences with separate members (definitions, circumstances, applications, additions)
• Task 17. Punctuation marks in sentences with words and structures that are not grammatically related to the members of the sentence

USE in mathematics profile level

The work consists of 19 tasks.
Part 1:
8 tasks with a short answer of the basic level of complexity.
Part 2:
4 tasks with a short answer
7 tasks with a detailed answer of a high level of complexity.

Run time - 3 hours 55 minutes.

Examples of USE assignments

Solving USE tasks in mathematics.

For a standalone solution:

1 kilowatt-hour of electricity costs 1 ruble 80 kopecks.
The electricity meter on November 1 showed 12625 kilowatt-hours, and on December 1 it showed 12802 kilowatt-hours.
How much do you need to pay for electricity in November?
Give your answer in rubles.

Problem with solution:

In a regular triangular pyramid ABCS with a base ABC, the edges are known: AB \u003d 5 roots out of 3, SC \u003d 13.
Find the angle formed by the plane of the base and the straight line passing through the midpoint of the edges AS and BC.

Solution:

1. Since SABC is a regular pyramid, then ABC is an equilateral triangle, and the remaining faces are equal isosceles triangles.
That is, all sides of the base are 5 sqrt(3), and all side edges are 13.

2. Let D be the midpoint of BC, E the midpoint of AS, SH the height from point S to the base of the pyramid, EP the height from point E to the base of the pyramid.

3. Find AD from the right triangle CAD using the Pythagorean theorem. You get 15/2 = 7.5.

4. Since the pyramid is regular, point H is the intersection point of heights / medians / bisectors of triangle ABC, which means it divides AD in a ratio of 2: 1 (AH = 2 AD).

5. Find SH from right triangle ASH. AH = AD 2/3 = 5, AS = 13, by the Pythagorean theorem SH = sqrt(13 2 -5 2) = 12.

6. Triangles AEP and ASH are both right-angled and have a common angle A, hence similar. By assumption, AE = AS/2, hence both AP = AH/2 and EP = SH/2.

7. It remains to consider the right triangle EDP (we are just interested in the angle EDP).
EP = SH/2 = 6;
DP = AD 2/3 = 5;

Angle tangent EDP = EP/DP = 6/5,
Angle EDP = arctg(6/5)

Answer:

At the exchange office 1 hryvnia costs 3 rubles 70 kopecks.
Vacationers exchanged rubles for hryvnia and bought 3 kg of tomatoes at a price of 4 hryvnia per 1 kg.
How much did this purchase cost them? Round your answer to the nearest whole number.

Masha sent SMS messages with New Year greetings to her 16 friends.
The cost of one SMS-message is 1 ruble 30 kopecks. Before sending the message, Masha had 30 rubles in her account.
How many rubles will Masha have after sending all the messages?

The school has triple tourist tents.
What is the smallest number of tents to take on a hike with 20 people?

The Novosibirsk-Krasnoyarsk train leaves at 15:20 and arrives at 4:20 the next day (Moscow time).
How many hours does the train travel?

Do you know what?

Among all figures with the same perimeter, the circle will have the largest area. Conversely, among all figures with the same area, the circle will have the smallest perimeter.

Leonardo da Vinci derived the rule that the square of the diameter of a tree trunk is equal to the sum of the squares of the diameters of the branches, taken at a common fixed height. Later studies confirmed it with only one difference - the degree in the formula is not necessarily equal to 2, but lies in the range from 1.8 to 2.3. Traditionally it was believed that this pattern is due to the fact that a tree with such a structure has an optimal mechanism for supplying branches with nutrients. However, in 2010, the American physicist Christoph Elloy found a simpler mechanical explanation for the phenomenon: if we consider a tree as a fractal, then Leonardo's law minimizes the likelihood of branches breaking under the influence of wind.

Laboratory studies have shown that bees are able to choose the best route. After localizing the flowers placed in different places, the bee makes a flight and returns in such a way that the final path is the shortest. Thus, these insects effectively cope with the classic “traveling salesman problem” from computer science, which modern computers, depending on the number of points, can spend more than one day to solve.

If you multiply your age by 7, then multiply by 1443, the result is your age written three times in a row.

We consider negative numbers to be something natural, but this was far from always the case. For the first time negative numbers were legalized in China in the III century, but were used only for exceptional cases, as they were considered, in general, meaningless. A little later, negative numbers began to be used in India to denote debts, but they did not take root to the west - the famous Diophantus of Alexandria argued that the equation 4x + 20 = 0 is absurd.

American mathematician George Dantzig, being a graduate student at the university, one day was late for class and took the equations written on the blackboard for homework. It seemed to him more complicated than usual, but after a few days he was able to complete it. It turned out that he solved two "unsolvable" problems in statistics that many scientists struggled with.

In Russian mathematical literature, zero is not a natural number, but in Western literature, on the contrary, it belongs to the set of natural numbers.

The decimal number system we use arose due to the fact that a person has 10 fingers on his hands. The ability for abstract counting did not appear in people immediately, and it turned out to be most convenient to use fingers for counting. The Mayan civilization and, independently, the Chukchi historically used the decimal number system, using not only the fingers, but also the toes. The basis of the duodecimal and sexagesimal systems common in ancient Sumer and Babylon was also the use of hands: the phalanges of other fingers of the palm, the number of which is 12, were counted with the thumb.

One familiar lady asked Einstein to call her, but warned that her phone number is very difficult to remember: - 24-361. Remember? Repeat! Surprised Einstein answered: - Of course, I remember! Two dozen and 19 squared.

Stephen Hawking is one of the greatest theoretical physicists and popularizer of science. In a story about himself, Hawking mentioned that he became a professor of mathematics, having not received any mathematical education since high school. When Hawking began teaching mathematics at Oxford, he read his textbook two weeks ahead of his own students.

The maximum number that can be written in Roman numerals without violating Schwartzman's rules (rules for writing Roman numerals) is 3999 (MMMCMXCIX) - you cannot write more than three digits in a row.

There are many parables about how one person offers another to pay him for some service as follows: he will put one grain of rice on the first cell of the chessboard, two on the second, and so on: each next cell is twice as much as the previous one. As a result, he who pays in this way is bound to be ruined. This is not surprising: it is estimated that the total weight of rice will be more than 460 billion tons.

In many sources there is a statement that Einstein flunked mathematics at school or, moreover, generally studied badly in all subjects. In fact, this was not the case: Albert at an early age began to show talent in mathematics and knew it far beyond the school curriculum.

USE 2020 in mathematics task 18 with a solution

Demo version of the Unified State Examination 2020 in mathematics

Unified State Examination in Mathematics 2020 in pdf format Basic level | Profile level

Tasks for preparing for the exam in mathematics: basic and profile level with answers and solutions.

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USE 2020 in mathematics task 18

USE 2020 in mathematics profile level task 18 with a solution

USE in mathematics

Find all positive values ​​of the parameter a,
for each of which the equation and x = x has a unique solution.

Let f(x) = a x , g(x) = x.

The function g(x) is continuous, strictly increasing over the entire domain of definition, and can take any value from minus infinity to plus infinity.

At 0< a < 1 функция f(x) - непрерывная, строго убывающая на всей области определения и может принимать значения в интервале (0;+бесконечность). Поэтому при любых таких a уравнение f(x) = g(x) имеет ровно одно решение.

For a = 1, the function f(x) is identically equal to one, and the equation f(x) = g(x) also has a unique solution x = 1.

For a > 1:
The derivative of the function h(x) = (a x - x) is
(a x - x) = a x ln(a) - 1
Let's equate it to zero:
a x ln(a) = 1
a x = 1/ln(a)
x = -log_a(ln(a)).

The derivative has a single zero. To the left of this value, the function h(x) decreases, to the right it increases.

Therefore, it either has no zeros at all, or has two zeros. And it has one root only if it coincides with the found extremum.

That is, we need to find a value a for which the function
h(x) = a x - x reaches an extremum and vanishes at the same point. In other words, when the line y = x is tangent to the graph of the function a x .

A x = x
a x ln(a) = 1

Substitute a x = x into the second equation:
x ln(a) = 1, whence ln(a) = 1/x, a = e (1/x) .

Substitute again into the second equation:
(e (1/x)) x (1/x) = 1
e 1 = x
x = e.

And we substitute this into the first equation:
a e = e
a = e (1/e)

Answer:

(0;1](e (1/e) )

USE in mathematics

Find all values ​​of the parameter a for which the function
f(x) = x 2 - |x-a 2 | - 9x
has at least one maximum point.

Solution:

Let's expand the module:

For x<= a 2: f(x) = x 2 - 8x - a 2 ,
for x > a 2: f(x) = x 2 - 10x + a 2 .

Derivative of the left side: f "(x) \u003d 2x - 8
Derivative of the right side: f "(x) \u003d 2x - 10

Both the left and right sides can only have a minimum. This means that the function f(x) can have a unique maximum if and only if at the point x=a 2 the left side increases (that is, 2x-8 > 0), and the right side decreases (that is, 2x-10< 0).

That is, we get the system:
2x-8 > 0
2x-10< 0
x = a2

Where
4 < a 2 < 5

a ~ (-sqrt(5); -2) ~ (2; sqrt(5))

Answer:(-sqrt(5); -2) ~ (2; sqrt(5))