Unified State Exam 2017. Mathematics. Task 18. Problems with a parameter. Sadovnichy Yu.V.

M.: 2017. - 128 p.

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

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CONTENT
Introduction 4
§1. Linear equations and systems linear equations 5
Problems for independent solution 11
§2. Studying the quadratic trinomial using discriminant 12
Problems for independent solution 19
§3. Vieta's theorem 20
Problems for independent solution 26
§4. Location of roots of quadratic trinomial 28
Problems for independent solution 43
§5. Use of graphic illustrations
to the study of the quadratic trinomial 45
Problems for independent solution 55
§6. Limited function. Finding the range of values 56
Problems for independent solution 67
§7. Other properties of functions 69
Problems for independent solution 80
§8. Logic problems with parameter 82
Problems for independent solution 93
Illustrations on coordinate plane 95
Problems for independent solution 108
Method "Okha" 110
Problems for independent solution 119
Answers 120

This book is devoted to problems similar to problem 18 of the Unified State Exam in mathematics (problem with a parameter). Along with problem 19 (a problem whose solution uses the properties of integers), problem 18 is the most difficult in the variant. However, the book makes an attempt to systematize problems of this type according to various methods for solving them.
Several paragraphs are devoted to a seemingly popular topic such as the study of the quadratic trinomial. However, sometimes such problems require different, sometimes the most unexpected, approaches to solving them. One of these non-standard approaches is demonstrated in example 7 of paragraph 2.
Often, when solving a problem with a parameter, it is necessary to examine the function given in the condition. The book formulates some statements concerning such properties of functions as boundedness, parity, continuity; Then examples demonstrate the application of these properties to problem solving.

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

Commas are used in the following cases:

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

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 I received a call from Ivan, I was busy.

[ , (When…), ].

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

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

, (What …), (), ().

Homogeneous subordinate clauses are connected by repeating conjunctions, commas are placed in the same way as with homogeneous clauses:

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

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

Subordinate clauses with complex subordinating conjunctions because, thanks to the fact that, in view of the fact that, instead of, in order that, after as, while and other similar ones are separated from the main one by one comma, which is placed at the border of the main and subordinate clause:

As he talked, I became more and more perplexed.

(As…),.

I became more and more perplexed as he talked.

, (as...).

As he talked, I became more and more perplexed.

[ (as...) ].

Complex unions can fall into two parts if:

1) there is in front of them negative particle Not:

She Not I answered because I was scared.

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

She answered only because I was scared.

Attention:

Unions then, as if, even if, only when don't break.

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

A comma is needed: They decided that if the weather was good the next morning, they would go out of town.
There is no comma: They decided that if the weather was good the next morning, That they will go out of town.

Subordinate clauses with a conjunction word which. A comma after a conjunction word that is not placed. This rule works even if the word which included in the participial phrase:

I don’t know how to react to a situation from which I don’t see a way out.

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

(Comma after participial phrase having learned which not placed).

In contact with

Classmates

Handbook for preparing for the Unified State Exam

Task 16. Punctuation marks in sentences with isolated members (definitions, circumstances, applications, additions)
Task 17. Punctuation marks in sentences with words and constructions that are grammatically unrelated to the members of the sentence

Twenty-five graduates of one of the eleventh grades of school No. 4 in city N took the specialized level of the Unified State Exam 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 the statements that follow from this information.

1) Among these graduates there is at least one who received 82 points for the Unified State Exam in mathematics.
2) Among these graduates there are exactly two who did not achieve the threshold score.
3) Among these graduates there are at least two people with equal scores for the Unified State Exam in mathematics.
4) The Unified State Exam scores in mathematics of any of these graduates are no 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) Over three years, the price has increased one and a half times compared to 1311.
3) There are many dark forces in the city.
4) None of the proposed ones.

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

There are 36 subscribers in the public Mythology of the Ancient Kyrgyz people, of which 25 know English language, 14 - German and only four speak French. Select the statements that follow from the given data.

In the 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 or 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. Select the statements that follow from the given data.

1) Petya is the tallest in the class.
2) Andrey is the shortest of these four boys.
3) Andrey is not the tallest in the class.
4) If you add up the heights of Petya and Sasha, 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 or other additional characters.

Graduate Barankin passed the Unified State Exam in four subjects. He showed the lowest result in mathematics - 33 points (in other exams the scores were higher). Average score Barankin’s score on four passed Unified State Examinations is 45 points. Select the statements that follow from the given data.

1) The average score for three exams, except mathematics, is 49.
2) Barankin passed all subjects except mathematics with 45 points or better.
3) Barankin did not even receive 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 or other additional characters.

There are 14 cats living in Antonina Petrovna’s apartment. Each cat is more than a year old but less than 17 years old. Select the statements that follow from this information.

1) 7 cats in this apartment are under 9 years old.
2) There is a cat in this apartment who 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 or other additional characters.

At the Winter Olympics in Sochi, the Zimbabwe team won fewer medals than the Kazakhstan team, the Cameroon team - less than the Danish team, and the Russian team - more than the teams of all four countries together. 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) The 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 or other additional characters.

When Ivan Valerievich fishes, he always switches his phone to silent mode. Choose the statements that are true under the given conditions.

1) If Ivan Valerievich’s phone is on silent mode, it means he is fishing.
2) If Ivan Valerievich is on a catfish fishing trip, then his phone is on silent mode.
3) If Ivan Valerievich’s phone is not on silent mode, it means he is not fishing.
4) If Ivan Valerievich’s phone is not on silent mode, it means that his wife did not let him go fishing.

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

Among the residents of house No. 23 there are those who work, and there are those who study. And there are also those who do not work and do not study. Some residents of house No. 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 No. 23 work.
3) Among the residents of house No. 23 there are no those who do not work or 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 the statements that are true under the specified conditions.

1) The volleyball team in city N must have a player whose height is 220 cm.
2) In the volleyball team of city N there are no 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 volleyball team of city N is more than 20 cm.

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

In the summer of 2014, some of the company's employees vacationed at the dacha, and some at the seaside. All employees who did not vacation at sea vacationed at the dacha. Choose the statements that are true under the given conditions.

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

In the country "Dotalandia" there are more men than women. Most common male name- Ivan, female - Maria. Select the statements that follow from the given data.
In the country "Dotalandia":

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

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

The school purchased a table, a blackboard, a tape recorder and a 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 purchase.
4) The printer and the 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 people in the class, of which 20 people attend a biology club, and 16 attend a geography club. Select the statements that are true under the given conditions.

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

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. The cake was more expensive than pineapple, but cheaper than cold cuts, and the juice was cheaper than cake. Select the statements that are true under the given conditions.

1) Pineapple was cheaper than cold cuts.
2) They paid more for the juice than for the 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) A table is cheaper than a photocopier.
2) A rack is more expensive than a photocopier.

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 not 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 shorter 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 took the Unified State Exam in social studies. 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 Exam in social studies.
2) Among these graduates there is a person who received 75 points for the Unified State Exam
in social studies.
3) Scores for the Unified State Exam in social studies of any of these twenty people
not lower than 35.
4) Among these graduates there is a person who received 20 points for the Unified State Exam in social studies.

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

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

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 put 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 specified conditions.

1) Seven fish in this aquarium are shorter than 2 cm.
2) There are no fish 9 cm long in this aquarium.
3) The difference in the length of any two fish is no 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) A table is cheaper than a photocopier.
2) A rack is more expensive than a photocopier.
3) Copier is the cheapest purchase.
4) The rack and the photocopier cost the same.

Olya is younger than Alisa, 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 statements that are true under the given conditions.

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

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 100 m race, 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 the summer residents in the village, there are those who grow grapes, and there are those who grow pears. And there are also those who grow neither grapes nor 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 registered on VKontakte there are schoolchildren from Tver. Among the 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 VKontakte or Odnoklassniki.
2) There are no schoolchildren from Tver who are registered on VKontakte.
3) Among the schoolchildren from Tver there are those who are registered on VKontakte.
4) At least one of the Odnoklassniki users is a school 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, of which 40 people know
English, and 20 - German. Choose the statements that are true under the specified conditions.
1) In company N, at least three employees speak both English and German.
2) There is not a single employee in this company 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 company speak 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. Choose the statements that are true under the given conditions.
1. If Nikolai Dmitrievich’s phone is turned on, he is not teaching a lesson.
2.If Nikolai Dmitrievich’s phone is turned on, then he is teaching a lesson.
3.If Nikolai Dmitrievich conducts a lesson laboratory work according to physics, that means his phone is turned off.
4.If Nikolai Dmitrievich is teaching a physics lesson, then his phone is on.

2) If gas stoves are installed in the house, 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) In this company there are 10 people who do not use either the Odnoklassniki network or the VKontakte network.

2) There are at least 5 people in this company who use 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 on, it means he is teaching a lesson.

3) If Ivan Petrovich conducts test according to math, that means 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 or other additional characters.

There are 20 people in the class, of which 13 people attend a history club, and 10 attend a mathematics club. Select the statements that are true under the given conditions.

1) Each student in this class attends both clubs.
2) If a student from this class goes to a history club, then he must go to a math club.
3) There will be at least two from this class who attend both clubs.
4) There are not 11 people from this class who attend both clubs.
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 or other additional characters.

Unified State Examination in mathematics profile level

The work consists of 19 tasks.
Part 1:
8 short answer tasks of basic difficulty level.
Part 2:
4 short answer tasks
7 tasks with detailed answers high level difficulties.

Running time - 3 hours 55 minutes.

Examples of Unified State Examination tasks

Solving Unified State Examination tasks in mathematics.

To solve it yourself:

1 kilowatt-hour of electricity costs 1 ruble 80 kopecks.
The electricity meter showed 12,625 kilowatt-hours on November 1, and 12,802 kilowatt-hours on December 1.
How much should I pay for electricity for November?
Give your answer in rubles.

Problem with solution:

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

Solution:

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

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

3. Find AD from the right triangle CAD using the Pythagorean theorem. It turns out 15/2 = 7.5.

4. Since the pyramid is regular, point H is the point of intersection of the altitudes/medians/bisectors of triangle ABC, and therefore divides AD in the ratio 2:1 (AH = 2 AD).

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

6. Triangles AEP and ASH are both right angles and have a common angle A, hence similar. By condition, AE = AS/2, which means AP = AH/2 and EP = SH/2.

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

Angle tangent EDP = EP/DP = 6/5,
Angle EDP = arctan(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 many rubles did this purchase cost them? Round your answer to a whole number.

Masha sent SMS messages with New Year's 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 left after sending all the messages?

The school has three-person camping tents.
What is the smallest number of tents you need to take on a camping trip involving 20 people?

The Novosibirsk-Krasnoyarsk train departs 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 the figures with the same perimeter, the circle will have the largest area. Conversely, among all shapes with the same area, the circle will have the smallest perimeter.

Leonardo da Vinci derived a rule according to which 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 explained by the fact that a tree with such a structure has an optimal mechanism for supplying its branches with nutrients. However, in 2010, American physicist Christophe Alloy 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 optimal route. After localizing the flowers placed in different places, the bee makes a flight and returns back in such a way that the final path turns out to be 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 solving.

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

We believe negative numbers something natural, but this was not always the case. Negative numbers were first legalized in China in the 3rd 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 in the west they did not take root - the famous Diophantus of Alexandria argued that the equation 4x+20=0 was absurd.

The American mathematician George Danzig, while a graduate student at the university, was once late for class and mistook the equations written on the blackboard for homework. It seemed more difficult to him 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 had struggled with.

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

The decimal number system we use arose because humans have 10 fingers. The ability for abstract counting did not appear in people right away, and it turned out to be most convenient to use fingers for counting. The Mayan civilization and, independently of them, the Chukchi historically used the twenty-digit number system, using fingers not only on the hands, but also on the toes. The duodecimal and sexagesimal systems common in ancient Sumer and Babylon were also based on the use of hands: the phalanges of the other fingers of the palm, the number of which is 12, were counted with the thumb.

One lady friend asked Einstein to call her, but warned that her phone number was very difficult to remember: - 24-361. Do you remember? Repeat! Surprised, Einstein replied: “Of course I remember!” Two dozen and 19 squared.

Stephen Hawking is one of the leading theoretical physicists and popularizer of science. In his story about himself, Hawking mentioned that he became a professor of mathematics without receiving any mathematical education since high school. When Hawking began teaching mathematics at Oxford, he read the textbook two weeks ahead of his own students.

The maximum number that can be written in Roman numerals without violating Shvartsman'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 invites another to pay him for some service in the following way: on the first square of the chessboard he will put one grain of rice, on the second - two, and so on: on each subsequent square twice as much as on the previous one. As a result, the one who pays in this way will certainly go bankrupt. 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 failed mathematics at school or, moreover, generally studied very poorly in all subjects. In fact, everything was not like that: Albert was still in early age began to show talent in mathematics and knew it far beyond the school curriculum.

Unified State Exam 2020 in mathematics task 18 with solution

Demo Unified State Exam option 2020 in mathematics

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

Assignments for preparing for the Unified State Exam in mathematics: basic and specialized level with answers and solutions.

Unified State Exam 2020 in mathematics task 18

Unified State Exam 2020 in mathematics profile level task 18 with solution

Unified State Exam in Mathematics

Find all positive values of 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 equal to
(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 of 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) )

Unified State Exam 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:

At 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) = 2x - 8
Derivative of the right side: f"(x) = 2x - 10

Both the left and right parts can only have a minimum. This means that the function f(x) can have a single 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 = a 2

Where
4 < a 2 < 5