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1/333
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v 04
LATEX
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2011-
2012
(MEd ).
.
1 - . -
.
4.2.25 . -
.
.
.
( -
)
.
pattern
. -
1 (2011-2012)
. -
,
.
2012
.. ,
(PhD, MSc)
i
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i
1 1
1.1 . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 . . . . . . . . . . . . . . . . . . . 4
1.2.2 . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 . . . . . . . . . . . . . . . . . . . . . . 6
1.2.4 . . . . . . . . . . . . . . . . . . 6
1.2.5 . . . . . . . . . . . . . . . . . . . . . . 7
1.2.6 , . . . . . . . . . . . . 13
1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.1 . . . . . . . . . . . . . . . . . . . 16
1.3.2 . . . . . . . . . . . . . . . . . . 17
1.3.3 . . . . . . . . . . . . . . . . . . 18
1.3.4 . . . . . . . . . . . . . . . . . . . . . 19
1.4 . . . . . . . . . . . . . . . . . . . . . . 20
1.4.1 . . . . . . . . . 20
1.4.2 . . . . . . . . . . 21
1.4.3 . . . . . . . . . . . . . 22
1.4.4 . . . . . . . . . . . . . . . . 24
1.4.5 . . . . . . . . . . . . . . 27
1.4.6 . . . . . . . . . . . . . . . . . 271.4.7 . . . . . . . . . . 29
1.4.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.4.9 . . . . . . . . . . . . . . . 31
1.4.10 . . . . . . . . . . . . 32
iii
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1.4.11 . . . . . . . . . . . . . . . . . . . . 33
1.4.12 . . . . . . . . . . . 37
1.4.13 . . . . . . . . . . . . . . . . . . . . . 43
1.5 . . . . . . . . . . . . . . . . . . . . . . . . 44
1.5.1 Pons Asinorum . . . . . . . . . . . . . . . . . . . . . . . . 44
1.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 471.5.3 . . . . . . . . . . . . . . . . . . . . . . 48
1.5.4 . . . . . . . . . . . . . . . . . . . . . . . . 48
1.5.5 . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.5.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.5.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
1.5.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2 55
2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.1.1 . . . . . . . . . . . . . . . . . . . . 55
2.1.2 . . . . . . . . . . . . . . . . . . . . . 55
2.1.3 , . . . 56
2.1.4 . . . . . . . . . . . . . . . . . . . . 57
2.1.5 . . . . . . . . . . . . . . 58
2.1.6 . . . . . . . . . . . . . . 58
2.1.7 Euler-Venn . . . . . . . . . . . . . . . . 58
2.1.8 R . . . . . . . . . . . . . . . . . . . . . . . 59
2.1.9 . . . . . . . . . . . . . . . . . . . 63
2.1.10 . . . . . . . . . . . . . . . . . . . . 64
2.1.11 . . . . . . . . . . . . . . . . . . . 65
2.1.12 . . . . . . . . . . . . . . . . . . 66
2.1.13 . . . . . . . . . . . . . . . . . . . 67
2.1.14 . . . . . . . . . . . . . . 67
2.1.15 . . . . . . . . . . . . . . . 69
2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.2.1 . . . . . . . . . . . . . . . . . . 71
2.2.2 . . . . . . . . . . . 73
2.2.3 . . . . . . . . . . . . . . . . . . . . . 75
2.2.4 . . . . . . . . . . 79
2.2.5 : A B . . . . . . . . . . 79
2.2.6 : A . . . . . . . . . . . . . . . . . . . 792.2.7 : A B . . . . . . . . . . . . . . . . . . . . . . 80
2.2.8 : A B . . . . . . . . . . . . . . . . . . . . . . . 80
2.2.9 : A B . . . . . . . . . . . . . . . . . . 81
2.2.10 . . . . . . . 82
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3 , , , 85
3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.1.1 2 = 2 2+ 2 . . . . . . . . . . 853.1.2
3
=3
32
+32
3 . . . . . 87
3.1.3 2
2
=
+
. . . . . . . . . . . . . 88
3.1.4 . . . . . . . . . . . . . . . . 903.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.1.6 . . . . . . . . . . . . . 99
3.1.7 . 100
3.1.8 - . . 100
3.1.9 . . . . . . . . . . . . . 101
3.1.10 102
3.1.11 - . 103
3.1.12 . . . . . . . . . . . . 103
3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.2.1 . . . . . . . . . . . . . . . . . . . . 105
3.2.2 . . . . . . . . . . . . 1073.2.3 . . . . . . . . . . . . . . . 109
3.2.4 . . . . . . . . . . . . . . . . . . 112
3.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.2.6 . . . . . . . . 115
3.2.7 . . . . . . . . . . . . . . 116
3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.3.1 x + = 0 . . . . . . . . . . . . . . . . . . . . . 1213.3.2 x + = 0 . . . . 1223.3.3 . . . . . . . . . . . . . . . . . . . . . . . 124
3.3.4 x2
+x
+
=0 . . . . . . . . . . . . . . . . . 126
3.3.5 x2 + x + = 0 . 1293.3.6 Vieta . . . . . . . . . . . . . . . . . . . . . 1313.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.4.1 x + > 0 . . . . . . . . . . . . . . . . . . . . 1333.4.2 () . . . . . . . . . 136
3.4.3 x + , 0. . . . . . . . . . . . . . . . . . 1373.4.4 -. . . . . . . . . . . . . . . . 139
3.4.5 x2 + x + > 0, 0 . . . . . . . . . . . . 1423.4.6 x2 + x + > 0, 0 . . . . . 1423.4.7 x2 + x + , 0 . . . . . . 1443.4.8 x2
+x
+
>0,
0 . . . . . 147
4 1514.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.1.1 . . . . . . . . . . . . 151
4.1.2 154
4.1.3 . . . . . . . . . 156
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4.1.4 . . . . . . . . . 159
4.1.5 . . . . . . . . . . . . . . . . . . 162
4.1.6 . . . . . . . . . . . 168
4.1.7 . . . . . . . . . . . . . . . . . . . 169
4.1.8 . . . . . . . . . . . 170
4.1.9 . . . . . . . . . . . 1734.1.10 . . . . . . . . . . . . . . . 175
4.1.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
4.1.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
4.2.1 . . . . . . . . . . . . . . . . . . 181
4.2.2 . . . . 184
4.2.3 . . . . . . . . . . . . . . . . . . . . . 190
4.2.4 x . . . . . . . . . . . . . . . . . . . . . . . 191
4.2.5 - . . . . . . . . . . . . . . . . . 193
4.2.6 - . . . . . . . . . . . . . . . . . . 198
4.2.7 - . . . . . . . . . . . . . . . . . . . 2024.2.8 x = . . . . . . . . . . . . . . . . . . . . . . . . 2034.2.9 . . . . . . . . . . . . . . . . . . 204
4.2.10 fx = x + . . . . . . . . . . . . . . . . . 2084.2.11 , . . . . . . . . . . . . . . 210
4.2.12 y = 1x + 1 , y = 2x + 2. 2124.2.13 x + = 0 x + > 0 . . . . . 2164.2.14 fx = x . . . . . . . . . . . . . . . . . . 2194.2.15 fx = x2 + x + . . . . . . . . . . . . . 2224.2.16 fx = x2 + x + > 0. . . . . . . 2234.2.17 f
x
=x2
+x
+
>0 ; . . . . . 226
4.2.18 f
x
=x2
+x
+
>0. . . . . . . . . 227
4.2.19 fx = x2 + x + . . . . . . . 2294.2.20 fx = x2 + x + > 0. . . . . 2304.2.21 fx = x2 + x + < 0 . . . . . . . . . . . . 2334.2.22 fx = x2 + x + . . . . . . . . . . . . . 235
5 237
5.1 1 . . . . . . . . . . . . . . . . . . . . . . 237
5.2 2 . . . . . . . . . . . . . . . . . . . . . . 248
5.3 3 . . . . . . . . . . . . . . . . . . . . . . 256
5.4 4 . . . . . . . . . . . . . . . . . . . . . . 279
6 289
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1
. .
1.1
0, 1, 2, 3, 4, ...
0, 1 0, 2
1 .. .
. 0 .
1. 2011; 20112011;
a;
:
2 + 3 = 5 5 + 11 = 16 32 + 39 = 71 32 39
61. ,
+ . .
.
2 3 = 6 5 11 = 55 32 39 = 1248 ,
. .
1
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2 1.1.
2. , 6.
;
:
7 2 = 5 = 2 7 = 5 ,
.
. : ,
.
3. ,
. , ;
:
8 4 = 84= 2 =
4 8 = 48= 1
2= 0, 5
8
5
=8
5
=1, 6
5 8 = 58= 0, 625
, .
, .
: 1
1 .
;
. = = .
= 0 1 (1)
0 (1) . -
: .
http://lyk-evsch-n-smyrn.att.sch.gr
..
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1. 3
-
0. 0 :
0
=0
0
1.2
2 7 = 5 5 . - 5.
:
0, , 1, ,
1, , 2, ,
2, , 3, ,
3, , 4, ,
4, ,...
0, ,1, ,2, ,3, ,4, ,... (...) .
.
:
... 4, 3, 2, 1, 0, 1, 2, 3, 4,... -
. a a
+1 a
1.
4. .. 2.
5.
.
5
.
2.
2. ;
;
= 0 (1) . 0 = 0
. . !
.
. .
.
.
..
www.nsmavrogiannis.gr
http://lyk-evsch-n-smyrn.att.sch.gr
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4 1.2.
,
.
2
+7
=5
8
+7
= 15
2
7
= 14
8
7
=56
6 12 = 6 12 6 = 18 , , -
.
.
.
1.2.1
-
.
:
0
1 .
.
1.2.2 .
2
0 0, 2, 4, 6, ... 0 2, 4 6 .... :
... 6, 4, 2, 0, 2, 4, 6 ... .
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..
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1. 5
:
2
2. 2
.
= 2 = 2
2.
( -
) 2
.
( : 2= )
2=
=2
= 2 . ( ) =2 .
. :
= 2 .
= 2 .
= 2. 5.
- : = 2 (2)
- = 2
..
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6 1.2.
- = 2.- .
- .
;
1.2.3
1
0 - 2
... 5, 3, 1, 1, 3, 5, 7, ... .
. :
.
.
:
+ 1. = 2
+ 1 = 2+ 1. :
= 2+ 1 .1.2.4
4
(, , )
. -
:
http://lyk-evsch-n-smyrn.att.sch.gr
..
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1. 7
- : -
.
:
.
. .
180 -
180 .
2
.
. -
. :
.
-
.
. 3 6 .
.
.
1.2.5
.
.
- .
.
.
. :
2 1.5 44.
..
www.nsmavrogiannis.gr
http://lyk-evsch-n-smyrn.att.sch.gr
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8 1.2.
1. .
.
2 2 .
3 :
3 4 = 2 2 . 4 :
4 8
=2
2
2 .
5 ; ; 16
. :
5 16 = 2 2 2 2 .
http://lyk-evsch-n-smyrn.att.sch.gr
..
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1. 9
2 2 2 ... 21
;
. 6 32 .
:
:
6 2 2 2 2 2 = 32 31 .3
Leo Moser
1921-1970 . .
, .
2. .
2 + 8 = 10 , 12 + 24 = 36, 18 + 24 = 42, 100 + 98 = 198 .
.
. -
. .
3
Moser.
463+23218+24
24
..
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10 1.2.
.
. 2
8, 12 24 ...
+.
+ .
. . :
.
, = 2 = 2. + . 2 + 2 2 + . . + , . + = 2 . , ,
:
+
= 2
+ 2
= 2 +
= 2 =
4. :
1.2.1. .
;
. -
:
1.2.2. .
, ,
:
+
= 2
+ 1 + 2
+ 1 = 2 + 2+ 2 = 2 + + 1
= 2 = :
1.2.3. .
2)
6. 1.2.3.
4, ( ), ( ), (
( )
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..
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1. 11
1.2.4. 1) .
2) .
. 1) = 2 + 1, = 2+ 1 , . = 2 + 1 2+ 1
= 4+ 2 + 2+ 1 2 4, 2, 2 -
=2
2
+
+
+1
= 2+ +
2 (
), .
.
= 2 + 1 .
2) = 2 .
.
= 2 2 2 :
= 2
=2
. ;
.
:
..
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12 1.2.
1.2.5.
.
.
1 .
=2
= 2+ 1 . + = 2 + 2+ 1 = 2 + + 1 = 2 + 1
= + . + .2
. , + , . x = + :
1 x
2 x
.
. ;
+ = x x .
= x : .
1.2.3 . (
) ( ). ,
.
. ...
1. ...
(
).
. Q.E.D.
quod erat demonstrandum
( ). 2. 1.2.5 -
( ).
. .
. ( -
)
.
http://lyk-evsch-n-smyrn.att.sch.gr
..
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1. 13
( 1.2.5
).
( ) -
.
:
-
-
7. , . ( ) :
1. + + 20112.
+
1
3. + 4. 2 + 1
8. + 1 . .
9. + 1 .1.2.6 ,
-
2. 0 -
2 - 2 .
2+ 1 . 0 2
3 3
..
www.nsmavrogiannis.gr
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1. 15
y = xz. y x y x x y.
3. 3 3
1 3
= 1
3
.
12 36 3
3 12 = 36. y = xz y xz x.
z x
:
...5x, 4x, 3x, 2x, x, 0 x, x, 2x, 3x, 4x, 5x ... 12. 12
=1
a,
= 1
a
:
1 a. :
1. 0.
2. 1.3. 1,.
1. 0
2. 13. 1,
2 x 3 x 4 x x 1, 0 2 3 4...
276 .. - 195 .. 0,
1, 1 .
1
.
-
0, 1, 1 2, 4,.... . - 0, 1, 1 2, 4,... .
.
..
www.nsmavrogiannis.gr
http://lyk-evsch-n-smyrn.att.sch.gr
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16 1.3.
1.1: 1 400
2 4, 6 ... -
3.
6, 9 ..
1 400.
-
.
.
1.3
1.3.1
( )
1
2,
5
3,
34
,6
3
,
( 0). 5 -
-
.
36
72= 50
100= 14
28= 816 = 12
5 . 1.4.12
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..
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1. 17
() . -
(
). (
)
.
4. :12560
= 2512
144
64= 9
4
77121 = 711 13. 122
56, 7080
, 7181
1.3.2
:
=
= =
=
1.2.2
=
= .
=
, , .
.
:
0 0
=0.
14. ( ,, ) =
15. 2358987812345912345923589878
= 7812345923589858987812345923
;
16. x, y x3y
.
x;
,
, >
<
.
>
>.
00 = 0a+ + ... + < 0
=
a ...
> 01 a
0,
=0
1
a
1
a
...
1
a
=1
a
0
1.4.10
, ,
.
: < < < (1.24)
< < (1.25) < < (1.26) (1.27)
(1.24) (1.25), (1.26), (1.27)
.
2 < x x < y 2 < y. a < b, b < c, c < d a < c c < d
a
0
+
>0. :
> 0, > 0 + > 0 (1.33)
.
:
0, > 0 + > 0 (1.34)
0, 0 + 0 (1.35)
.
: > 0 - > 0 > . < 0 > 0. :
> 0 < 0 (1.36) (1.29)
.
. .
< > 0. > 0 ( (1.30) ) > 0. > 0 > 0 < (1.37) < < 0 > 0 < < > . :
< < 0 > (1.38) (1.30)
+ + = +.
.
13.
13
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1. 35
Teubner, 1893.
> 0 < 0 > 0 > 0 (1.30) > 0. = > 0 < 0.
> 0 < 0 < 0 (1.39) + = . < 0 < 0 =
+
+> 0 :
< 0 < 0 > 0 (1.40) = +
:
.
< < 0 > 0 < 0 > . :
< < 0 > (1.41) -
. . 1 = 1. 1 > 0 > 0 1 > 0 . < 0 1
< 0
.
> 0 1> 0 (1.42)
0, < 0
1.
+
2.
3. + 1 14.
+
5.+1
6.
+1+1
1.4.12
x
.
x. x
x.
x
O x
x
x
x
x
.
2, 3
=2,
7
= 7,
[1
2] =0,
[2
2] = 2.
:
x = x =
x
(1.48)
..
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38 1.4.
x x < x + 1 (1.49)
. x ,
, , .
x.
x = x + x (1.50)
x = x x (1.51) 3, 2 3, 2 3 = 0, 2 4, 78 0, 22.
,
.
10:
...
104 = 11000
,
103 = 11000
,
102 = 1100
,
101 = 110
,
100 = 1,101 = 10,102 = 100
...
(1.52)
(1.52) 10 .
10 .
.
x. x 10
: x x
(1.52) x
10 10
.
x = 8243
. x 102 = 100 103
=1000. :
102 < 8243 < 103
102 103
102 100.
100 = 1 100, 2 100, 3 100, 4 100, 5 100, 6 100, 7 100, 8 100, 9 100 = 1000
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40 1.4.
100 = 102 . 101.
2
102
+1
10, 2
102
+2
10, 2
102
+2
10, ..., 2
102
+10
10
x x
. x
2 102 + 7 10 < 8243< 2 102 + 8 10
10
100
=1
2 1002 + 7 10 + 4 100 < 8243 < 2 1002 + 7 10 + 5 100
101 = 110
. :
2 1002 + 7 10 + 4 100 + 6 101 < 8243< 2 1002 + 7 10 + 4 100 + 7 101
: 102, 103 .
2 102, 7 101, 4 100, 6 101, 6 102, 6 103, 6 104, .. (;;) (
10 ) -
x. (bold)
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1. 41
:
2 102 7 101 4 10 6 101 6 102 6 103, 6 104 ...
2 7 4 6 6 6 6 ...
2 7 4 6 6 6 6 ...
.
10 .
.
,:
2 7 4 , 6 6 6 6 ...
:
8 2 4 3
- 6 2 7 4, 6 6 6
2 2
- 2 1
1 4
- 1 1 2
2 0
- 1 8
2 0 0
- 1 8
2 0
x.
x = 8243
.
10 -
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
( 0 )
.
.
..
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42 1.4.
-
( )
( ).
-
.
.
.
:
: .
: .
0.
-
. , ,
.
9. -
. 0, 999...
1. 0, 9999
a
=0, 9999 10a
=9, 9999... 10a
=9
+a 9a
=9 a
=1.
9 -
1 0, 9999.
mm1...0
, 122...
64. 197
. .
65. 3 0, 23
66. a2a1a0 b2b1b0
a2 + b2 = a1 + b1 = a0 + b0 = 11. a2a1a0 + b2b1b0. 67. 0, 3 ( )
68. ( ):
0, 1 01
1 0 02
1 0 0 03
1 0 0 0 01
. . .
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1. 45
.
. ( ) . 60 .
1. , ,
.16
2.
.17
3. , ,
, .18
4. .
5. AB ,
.
6. .
7. ....
8. .
16, , . . .
Legendre. 19 .17
: , 20 .18
: , 20 .
..
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46 1.5.
9.
10. : ...
-
.
.
-
.
. -
.
: .
384 .. - 322 ..
; 1,
2, 4, 6, 7, 9. 5 9
. .
. 8
.
. , , -
. .
, , .
19
.
.
.Augustus De Morgan
1806 - 1871
George Boole
1815 - 1864
. ( )
. - De Morgan Boole.
(
) .
- -
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1. 47
.
1.5.2
.
7.
. -
.
11. 1. 3 > 5+ > .
2. 11
.
3. 3 > 5 4. 33 > 5 .5. x + 2 = 3
x .
. x 3
3 + 2 = 3 . x 1 1
+2
=3.
7 .
. :
.
, ,
:
.19.
.
.
19 ( ) . -
. .
Goldbach:
2 .
428. 1742!
..
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48 1.5.
1.5.3
x + 2 = 3 - (
x) .
0 x = 2, x + 2 = 3 0 x = 0. .
.
-
.
-
20.
1.5.4
:
P
Q
.
P Q ()
().
.
.
. .
.
:
= + = +
. ,
=
+
=
+.
:
= + = + 20
:
.
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50 1.5.
1.5.5
( - ):
.
AB :
= B =
B
=
=
.
:
= B = .
P Q
P Q -
.
.
76. ;
1. > 5 2 > 102. 2 < 0 4 < 03. > 0 > 4. > 0 + > 05. > 2 > 7
77. ;
1. x
2
x
+2
4
2. x 2 x > 23. x y x2 y24. x = y + 1 x y5. x = 1 x2 1 = 0
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1. 51
1.5.6
1 < 2 < 3 : 1 < 2 2 < 3. 1 < 2 < 3 1
32. 2 33. x < x + 14. 2 75.
a
+
=
1.5.9
. -
-
.
.
P1
P2 ... P
P1 P2, P2 P3,P1 P . : P1
P1 P2 P2 ( ). P2 P2 P3 P3 .
P .
12. x < y 2x3
4< 2y3
4. :
x < y 2x < 2y 2x 3 < 2y 32x
3
4 0
=0 .
#
## # . + < + + = + . :
+ +
##
:
+ + + + + + + + . :
+ + (3.37)
i, ii,
3.37
. (3.36)
1 12 2...
1 + 2 + ... + 1 + 2 + ... + (3.38)
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3. , , , 107
:
.
201. x
0
.
3.2.6
< 1 > . > < 1. :
< > (3.50) 3.50 : -
0
:
.
. :
< 1
, 1
; :
2 < 3. 12> 1
3
2
1
.
: 1> 1
1
1
> 0
> 0.
> 0, > 0 . ,
< . > 0 <
1
< 1
1
> 1
.
:
, < 1> 1
(3.51)
227. (3.51)
.
; 228. , < 1
> 2
+ > 1 . 229. A = ,1, B = 1, 0, C =0, 1 D = 1,+. x (4 ) A,B,C,D 1
x;
230. p,q,r p < qr< 1
p.
1. p < 1.2. p < r
q< 1
p.
3.2.7
. .
+ + +4 9 13 2 3
13 5
16 9 25 4 3 5 7
0 16 16 0 4 4 4
144 25 169 12 5 13 17
+ + , .
+ = +.
+ = + , :
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3. , , , 117
231. , + = + = 0 = 0
232. ,
=
;
. - :
4 9 36 2 3 6 6
16 9 144 4 3 12 12
0 16 0 0 4 0 0
4 25 100 2 5 10 10
-
=. 3.2.5: =
2 = 2 2 = ()2 2
2 = 22
=
. , :
= (3.52) (3.52)
. ..: = =
(3.52)
= (3.52)
:
4 9 49 2 3 23 23
16 9 169
4 3 43
4
3
0 16 0 0 4 0 0
4 25 425
2 5 25
2
5
..
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118 3.2.
=
. -
(3.52):
=
2 =
2
2 = ()2()2
2
=
2
2
=
, 0 > 0
=
(3.53)
(3.53)
(3.52):
=
= (3.52)
=
=
. 2 = . - .
2
2.
2.
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3. , , , 119
0. < 0 ( 2 = 2) . :
0
2
=
2 = { 0 < 0(3.54)
233.
:
1.
22 34 58 = 2 32 542.
311 519 114 = 35 59 112153.
35 73 117 = 32 7 113231 234.
,, . :
1. A = 242. B = 824
3. =2484. = 577
235. ,, . :
1. A = 446
2. B = 3283.
=
6
4
84
4. = 444926 236.
2 3 = 1
2
2
3 12
2
237. t1 = +, t2 = :1. t1 + t2 = 22. t1t2 = 2 3. t
2
1+ t2
9= 22 + 2
4. t31+ t3
2= 23 + 6
238. :
1. 322.
323. 32
4. 325.
(3 3)26.
(1 3)2..
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120 3.3.
3.3
() . .
.
.
.
x + 12 = x2 + 2x + 1 x .
x .
x -
.
2 3
x
+1
2
=x2
+3x
+1
x = 0. 0.
1 2
x + 12 = x2 + 2x + 2 x. .
x 3 = 0 x y = 4 . x 3 = 0 x = 3 . x y = 4 : x = 4, y = 0 x
=5, y
=1 ... 4 0
x 4 y 0 . y 4
x 0 0 4 = 4. .
:
1 , 2 , ..
. -
, ..
.
x
y
=4
x, y
-
x, y, 5, 1, 11, 7, 11,15 ... x 2y + 4z = 7 - x, y, z. 1, 1, 8 x = 1, y = 1, z = 8 1, 8, 1 x = 1, y = 8, z = 1 .
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3. , , , 121
239. x = 3 3x = 9; x3= 9;
27
x= 9; x3 = 27; xx = 27;
240. 2, 3, 6 x + y + z = 11. .
3.3.1 x + = 0 x + = 0 ( -
) . .
.
-
:
23. : 2x + 8 = 0.: 2x
+8
=0
2x
= 8
2x2
=82
x
=82
x
= 4
24. : 0x + 8 = 0.: 0x + 8 = 0 0x = 8.
x 0 = 8. .
25. : 4 x + 5 = 2.: : 4 x + 5 = 2
4x + 20 = 2 4x = 2 20 4x = 18 x = 184 x = 9
2
26. : 4
x
+5
=4x
+20.
: 4 x + 5 = 4x+20 4x+20 = 4x+20 4x4x = 2020 0x = 0. 0x = 0 ( ) x .
. -
: .
x + = 0 x = x =
0.
=0,
0.
= = 0. 241. ( x):
..
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122 3.3.
1. 2x + 8 = 02. 3x 8 = 133. x + t = 0, ( 0)
4. 1
x + t = 0, ( 0)5. + 1x = 0, ( 1)6. x + 4 = 1, ( 0)
242. :
1. 6 x + 3 5x = 262. 6 x + 3 5x = 26 + x3. 4 4 + 2x = 60 3x
4. 60x + 1 = 3 3 + 4x5. 6 x x + 4 = 8 x26. x 32 5 10 + x = x2 8
243. :
1. 3x + 100 = x3+ x
2 4
2. 5x114 x1
10= 11x1
12
3. 4x+2
+7
x+3
=27
x+2x+3
4. 30x+5 153 5+4xx+5 = 05. 3 x 1 x 2 x 3 x 4 = 2x + 7 x+ 46. 2x 2x
9= 1
9(16x 3
2)
3.3.2 x + = 0 243
x+= 0 . :
27. :
x3 + x + 14 = 8
. :
. x3+ x+1
4= 8
12( x3+ x+1
4) = 12 8 4x + 3 x + 1 = 96 4x + 3x + 3 = 96 x = 93
7
28. :
2
x
+1
=3
4
. : x + 1.
x + 1 0 x 1. .
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3. , , , 123
x 1. : 2x+1 = 34
2 4 = 3 x + 1 8 = 3x + 3 x = 53
1. 29. :
3x + 2 + 6 + xx + 2 x + 1 = 5x + 1
. x + 2, x + 1 x 2,1.
x + 2 x + 1. :3
x + 2 + 6 + xx + 2 x + 1 = 5x + 1
x + 2 x + 1 3
x + 2 +6
+x
x + 2 x + 1 = x + 2 x + 15
x + 1 3 x + 1 + 6 + x = 5 x + 2 3 x + 1 + 6 + x = 5 x + 2 ... x = 1
.
.
30. :
4
3
x
2
2x
1
x
1
=0
.
:
4 3 x 2 2x 1 x 1 = 0
4 3 x 2 = 0
2x 1 = 0
x
1
=0
x =10
3 x =1
2 x = 1 -
x = 103
, x = 22
, x = 1. 244. :
..
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124 3.3.
1. 2x 6 x+ 35x 6 = 02. 3 2x 4x 20 x + 2 4 + 2x = 0
245. :
1. x2 4 = 02. 4x2 9 = 0
3. x2 2x + x 2 = 03.3.3
31.
v0 = 3msec a = 2msec2 . v = 6m
sec;
a
=1m
sec2; a
=5m
sec2;
. .
:
a = 2msec2
. v = v0 + at 6 = 3 + 2t 2t = 3 t = 3
2
t = 32
sec
a = 1msec2
. 6 = 3 + 1 t t = 3 t = 3sec
a = 5msec2
. 6 = 3 + 5t 5t =3
t
=3
5 t
=3
5sec
: .
a
. :
6 = 3 + at at = 3 t = 3a
-
a.
t = 3a
a
.
6 = 3 + at t a . . -
.
.
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3. , , , 125
32. x 2x + 2 = + 4x.. .
:
2x + 2 = + 4x 2x 4x = 2 (2 4)x = 2 2 4 . .
2 4 = 0 2 + 2 = 0 = 2 = 2 :
=2 0x
=0 .
= 2 0x = 4 . 2 4
2 x x = 2
2 4 x = 2 2 + 2 x = 1 + 2 246. :
x + 3 = 5 x + 1 247.
1x 2 + 1 = 0 x ;
248. t
t
x
+1
3t
=x
t
x = 5; 249.
x, y xy= .
.
..
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126 3.3.
3.3.4 x2 + x + = 0 x2 + x + = 0.
= 0 x + = 0 ;;.
0 x2
+ x + = 0 x2 ( 0x2 +x + = 0 ). .
.
.
-
.
,
x2 + x 2 = 0
, -
: ,
. -
AB (A )
, 2
.
A= 2
. E, Z
B. AE. ZAE . , ZA .
BAE = 90 A = A =ZA AEB, ABZ B AEB =BZA.
AB
BZ= BE
AB= AE
AZ
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3. , , , 127
x + 2+
2
= x
2 = x x + x x2 + x 2 = 0.
19.
x2 + x + = x + 22 2 4
42
:
x + 22 2 4
42 =
x2 + x+ 2
42 2 4
42 =
x2 + x+ 2
42 2
42 4
42 =
x2 + x+
=
x
2
+ x+ 0 x2+x+= 0 x +
22 2 4
42 = 0 x +
222 4
42= 0
2 4 - . x +
22 24
42= 0
x +
22
42 = 0 < 0
42 .
x + 22
.
..
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128 3.3.
. 0 =
2
x
+
2
2
2
2
=0
x + 2
2x +
2+
2 = 0 (3.55)
x + 2
2= 0 x +
2+
2= 0
x = 2+
2 x =
2
2
= 0 x
=
2. :
x2 + x + = 0 0 = 2 4: < 0 0. > 0
() x1,2=
2. = 0
x = 2
250.
1. 4x2
+92x
+333
=0
2. x2 + 23x + 76 = 03. x2 + 23x + 102 = 0
4.
2x2
+3x
+20
=0
5. 2x2 + 3x + 77 = 06. 64x2 64x 273 = 0
251. - :
1. x2 +x = 02. 2x2 + 4x = 03. x2 + 2x 2= 0
4. 2x2 + 5x 122 = 05. x2 + x + = 06. x2 + x + = 0
252. :
x2 + 2xy y2 = 0 ():
1. x ( x).
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3. , , , 129
2. y.
253. :
1. x2
+3x
+2
=0 x
= 1
2.
2
x
2
3x + = 0 x = 13.3.5 x2 + x + = 0
x2 + x + = 0 . -
:
33. :
2
2 x +1
2 =4
2x x
2
. 2 x 2x x2 = x 2 x .
x 0 x 2. :
2
2 x + 12 = 42x x2 2
2 x + 12 = 4x 2 x 2x2x2x 2 x
2
2 x +1
2 = 2x 2 x4
x 2 x 4x + x 2 x = 8
6x x2 = 8x2 + 6x 8 = 0
x2 6x + 8 = 0 .
x2
6x
+8
=0 x
=2
x
=4. x
0 x
2
2 4.
34. :
2x 12 3 2x 1 + 2 = 0..
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130 3.3.
.
.
x -
2x
1.
2x
1
=u
u.
x.
2x1 u u23u+2 = 0 u1 = 1 u2 = 2. 2x 1 = u 2x 1 = 1 2x 1 = 2 x1 = 1 x2 = 32 . 35. :
x4 5x2 36 = 0. x4 = (x2)2
(x2)2 5x2 36 = 0 (x2)2 . x2 = u u x.
u2 5u 36 = 0 u1 = 4 u2 = 9. u x2 = u :
x2 = 4 x2 = 9 x1 = 3 x2 = 3. 254.
1. 2x + 21x+4 = 9
2. 23x1
+3
+70
x24x+3
=0
3. 3x + 5x2 = 2
4. 1x+ 3
2= 1
x+35.
3x3+4+x33x2+4+x2
=7
6. x2+x+1
x2x+1 = 32+22+32 255.
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3. , , , 131
1. x4 5x2 + 4 = 02. 2x+ 12 3 2x + 1 1 = 03. (2x+3
x1 )2 3 (2x+3x1 ) + 2 = 04. 4
x2+4 + 5x2+5 = 25. (x2 + x + 1)(x2 + x + 2) 12 = 06. 7 (x + 1
x) 2(x2 + 1
x2) = 9
3.3.6 Vieta
x2 + x + = 0 . Francois Viete
(Franciscus Vieta)
1540 - 1603
.
x1 = 2 x2 = +2 . :x1 + x2 =
2+ +
2= () + (+)
2= 2
2=
x1 x2 = 2 +2 = () (+)42 =2 242
= 2 42
= 2 (2 4)42
= 442
=
3 S P.
S= x1 + x2 =
P = x1 x2 =
Vieta (3.56)
36. -
2x2
4x
1
=0 .
. (;). S = 42 = 2 P = 1
2= 1
2.
.
37. x2+x+= 0 x1 x2. x21 + x22
. x21 + x22 : x1 x2 x1 x2
. x1 x2
x2 x1 :
x21 + x22 x22 + x21
3 summa, productum.
..
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132 3.3.
x1, x2 (
4 ) ,
, S = x1 + x2 P
=x1
x2. :
x2
1 + x22 = x1 + x22 2x1x2 = S2 2P ,,
:
S2 2P = 2 2
= 2
2 2
= 2 2
2
x21 + x22 = 2 22
Vieta
S P.
= S
= P = S = P
x2 Sx + P = 0 (x2 Sx + P) = 0 x2 Sx +P = 0. :
S P
x2 Sx +P = 0 (3.57) 38. 4 2.
. S
=4
2. x2 4x + 2 = 0 . . :
x1 = 2 +2 x2 = 2 2. 256. .
1. 2x2 58x+ 12 = 02. t2 + 1500t 2 = 0 3. 2x
2 2x + 6 = 04. tx2 2tx+ t2 = 0
257. x2
+x
+
=0 1, 2.
,, :
1. 1 2 + 1 + 2 1 + 1 2. 11+1 + 12+1 258.
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3. , , , 133
1. 1 42. 5
8 1
3
3. 16+ 1
6
13 1
6 1
6
13
4.
3 13
259. 1. .
1. 2x2 + 928x 930 = 0 2. x2 8193x+ 8192 = 0 260. S P . .
1. S= 15, P = 142. S= 19, P = 84 3. S= 12, P = 34. S= 2, P = 2 2
261. .
1. 7x2 + 14x 1 = 02. 3x2 9x + 2 = 0 3. 6x
2 14x + 1 = 04. 3x2 + 5x + 1 = 03.4
()
. -
(. 3.3.1). -
. x
2
>0
2
2,+. x + 1 (y2 + 1) > 0 x, y x, y x > 1, y R.3.4.1 x + > 0
2x + 4 > 0 -:
2x + 4 > 0 2x > 4 2x2> 4
2 x > 2
x
x
> 2
2,
+.
:
:
..
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134 3.4.
2x + 4 > 0 :2x + 4 > 0 2x > 4 2x
2 < 42 x < 2 2 .
:
:
- x
+
>0
x + > 0 x >
x
>
>0
x <
< 0
0 > = 0
> 0 < 0 : -
,+
, . = 0
. x + > 0 0 > . x
+
>0 x 0
>
>0.
>0 x
+
>0
x R. > 0 0 x + > 0 x .
:
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3. , , , 135
x + > 0:
,+ > 0.
,
< 0. R = 0 > 0. = 0 0.
4. x + 0, x + < 0, x + 0 x + > 0. 39. 3x + 2 0..
3x + 2 03x 2 $$$3x3 23 x 23
40. x
+1
x.
. : x + 1 x x x 1 1x 1. :
1 > 0 > 1. x x 11 .
1 > 0 < 1. x x 11 .
1 = 0 = 1. 0 1 x .
262. :
1. 2x > 32. 2x > 33. 2x < 3
4. 2x < 35. 4x + 2 < 16. 8 x + 2 > 4
263. :
..
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136 3.4.
1. x > 4 > 0.2. 2x + 2 0 > 2
3. 2 1x < + 1 > 12
4. (2 + 2 + 1)x < 3 + 2 264. :
1. x+13 x14 > 1 2. 2 x 1 + 2x14 < 3
265.
3x + 4 < 5 t 2t 1 ; 266. 2x 3y > 1.
1. x.
2. y.
267. m 2x2 3x+m+ 1 = 0 x ;
3.4.2 ()
-
.
.
.
41. 4 3x > 0 5 6x > 0.
(, 43) (, 5
6).
: (, 43) (, 5
6) = (, 5
6).
42. 4 < 3x 1 < 5.
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3. , , , 137
. x
: 4 < 3x 1 3x 1 < 5. (5
3,+)
, 2
.
:
(5
3,
+), 2
=(53 , 2).
43. m -
m
2 5 6, 6.. 6 < m2 5 < 6
.
.
6 < m2 5, m2 5 < 6 12
2 0
269. 3 5 6x < 58
.
270. x2 4x + 5 1 = 0 .
3.4.3 x + , 0. ,
0
x x+= 0 ( ) x x + > 0 x + < 0 ( ). 4 x +
4
..
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3. , , , 139
x
3x + 4 4
3+
+ 0 45. x (3 1)x + (3 + 1).. = 3 1 > 0 (3 1)x + (3 + 1)
x = 3+131 = (3+1)(3+1)(31)(3+1) = 4+232 = 2 3. x
x
.
x
(3
1
)x
+ (3
+1
)
2 3 +
0
+3.4.4 -.
x + , 0 .
46. x 2x 1 4 3x.. -
.
:
x
2x 1 1
2+
0 +x
4 3x 4
3+
+ 0 -
-. (, 12) ,( 1
2, 4
3) , (4
3,+).
x
:
..
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3. , , , 141
x2x 1
5x + 10
2x15x+10
2 22
+ 0 + + ++ 0 +
271. 46 x = 2012 .
272. 47 12012
.
273. 3 x 5 + 4x 3x 1.
x
3 x5 + 4x3x
1
3 x 5 + 4x 3x 1
54
1
33 +0
0
0
0 0 0
274. x :
1. 2 4x x + 52. (x 1
2) 2x 1
3.
1
x
2
x
3
x
275. x :2x 1 x + 35x + 7
.
..
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142 3.4.
3.4.5 x2 + x + > 0, 0 x2+x +> 0, 0
<
>,
,
. x2
+x
+
x + - x2 +x + -
. x2 + x + = 0 .
3.4.6 x2 + x + > 0, 0 0 :
x2
+x
+
=
x
+
2
2
2 4
42
(3.58)
0 (. 128, (3.55) )
x2 + x + = x + 2
2x +
2+
2
2
2,
2+
2
x1 = 2 + 2 x2 = 2 2 x2 + x + = 0 () x2
+x
+.
x2 + x + 0 x2 + x + = x x1 x x2 x1,2 = 2 (3.59)
-
. -
px + q ( 278). = 0
(3.58).
x2
+x
+
=
x
+
2
2
(3.55) ): -
x1 = x2
x2 + x + = x x1 x x1 = ax x12
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3. , , , 143
x1 . x1 = 2
x2+x+= ax 22 = ax +
22.
48. 3x2
+2x
1.
.
x1 = 1 x2 = 13 3x2 + 2x 1 = 3 x 1 (x 13) 3x2 + 2x 1 = 3 x + 1 x 1
3
3 (x 13)
3x2 + 2x 1 = x + 1 3x 1. 49. 3x2+x
+1.
. x1 = 1+136 x2 = 1136 . 3x2 + x + 1 = 3x 1 +136
x 1 136
50. 9x2 12x + 4.
. 23
9x2
12x
+4
=9
x
2
3
x
2
3
=9
x
2
3
2
9x2 12x + 4 = 3x 22. 9 (x 2
3)2 = 32 (x 2
3)2 = (3(x 2
3))2 = 3x 22.
51. 22+11+ 152.
.
.
, , :
2
2
+11
+152
112 4 2 152 = 2 1122 5
2 3.
22 + 11+ 152 = 2 52
3 = 2 + 52
+ 3..
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144 3.4.
2 + 52
22 + 11+ 152 = 2 + 5 + 3 276. :
1. 2x2 5x + 22. 4x2 + 4x + 13. x2 8x 33
277.
1. x2+6x91
x2+8x105
2. 2a2+8a90
3a236a+105
3. b2
9bx+14x2
b2bx2x2
278. x2 +x + - x2 + x + = px + q px + q p, q, p, q .
3.4.7 x2 + x + , 0 -
x2 + x + . :
x2 + x + = x + 22 42 x2 + x +
x + 22
42
-
x
x + 22 42 x -
,
.
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..
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3. , , , 145
< 0 x + 22
42 :
x
+
2
2
42
= x
+
2
2
0 +42
>0>0 >0
x2 +x + x.
= 0 x + 22
42= x +
22.
.
x 2
x = 2
.
x
2 .
> 0 x1 < x2 x + 22
42= x x1 x x2.
:
x
x x1x
x2
x x1 x x2
x1 x2 + 0 + +
0
++ 0 0 + x
, x (
)
.
:
x2 + x + x < 0. x = 0. x > 0. (3.60)
..
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146 3.4.
:
+
+
0
52. 2x2 + 4x + 1. 1 1
2
6
1+ 12
6. = 2
. x
.
53. (7 3) x 2 x + 3.. ( )
x2
7 3 3 2. 3 2 .
279. :
1. 3x2 + 18x 212. 9x2
+3x
2
3. 5x2
35x + 404. x2 + x + 15. 3x
1
4x2
6. 1 6x+ 9x2
280. ,
1. 2 + + 2 02. 2 + 2 0
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3. , , , 147
281. x, y :
x2 2x + 6 + y2 4y > 0 282. x
=1 x
=2
x
=3. x
=4;
3.4.8 x2 + x + > 0, 0 x2 + x + > 0
x
.
54. 2x2 2x 24 > 0.. 2x2 2x 24 = 2 > 0 3 4.
x < 3 4 < x.
,
3
4,
+.
55. 3x2 2x 1 < 0.. 3x2 2x 1 = 3 > 0 1
3 1.
.
x 13
1. x 13< x < 1.
(13
, 1). 56. 3x2 11x 4 0.
.
1
3 4.
. (,13] 4,+. 57. 2x2 x + 1.
. -
= 2 . x .
.
58. x22x+43 = 0 .
.
2
2
4
1
4
3
42 16 + 12 > 0. , 1 3,+.
,
:
..
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148 3.4.
x
2x2
7x
4
3x2 16x + 5(2x2 7x 4) (3x2 16x + 5)
12
1
34 5 +
+0
0
+ ++ + 0 0 ++ 0 0 + 0 0 +
(2x2 7x 4) (3x2 16x + 5) - 0 1
2 x 1
3 4 x 5.
[1
2, 1
3
] 4, 5
.
59. x2x+2 2x34x1
. x 2, 14
.
.
. x 2, 14
:
x
2
x
+2
2x
3
4x
1
x 2x + 2 2x 34x 1 0
x 2 4x 1 x + 2 2x 3x + 2 4x 1 02x2 10x + 8x + 2 4x 1 0
(2x2 10x + 8) x + 2 4x 1 0 :
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3. , , , 149
x
2x2 10x + 8x + 2
4x 1(2x2 10x + 8) x + 2 4x 1
2 14
1 4 ++ + + 0 0 + + + + 0 + + 0 + 0 ++ + 0 0 +
,2 (14
, 1]
4,
+ 283. , , 1. x + 2 x 2 < 02. x + 3 x+ 7 03. (x + 1
2) (x 3
4) < 0
4. 2x + 3 3x 4 05. x + 5 2x + 5 > 06. 3 2x 3x + 2 > 0
284. 0 < 0 285.
1. x2 x 12 > 02. 3x2 + 2x + 1 > 03. 4x2 4x+ 1 > 04. 3x2 4x+ 1 < 0
5. 4x2 + x 1 > 06. x2 + 4x + 6 < 07. 4x2 + 4x 1 08. 4x2 + 4x + 9 > 0
286.
1. 1 < x2 x < 22. 1 < x
x2+1 23. 1
x< x
x+1 < 4..
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4
4.1
4.1.1 .
-
.
( )
.
O B 0 1 -
12
A 2 .
1.
x
x.
,
.
1, 2, 3, 1,2, 3 1, 2, 3.
1, 2, 3 1
-
151
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4. 153
... - -
... .... 2 3 =
2
+3
2
3
= 2
+3.
, . .
x 0 x x = 0. (4.6)
( ):
x = x x. (4.7) x
x.
x
= x
.
x
x
x
:
x x x x. (4.8) 291. :
1. 7 =2. 1
3 =
3. 32 11 =4. 5 32000 =5. 1
4111 1
8111 =
6. x2 + 3 = 292. :
1. A = 1 2 32. B = 1
3 4 2
3 1
2
3. = (5 252) (3
4 1)
4. = 3567
5. E= 4 8 7 9 116. Z= 1 334 7 334
293. > :
1. A = + 12. B
=33
3. = 3 3 + 4.
= 22+
+
294.
+ + = + + 295. x > 1. x31
x2+x+1
..
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154 4.1.
296. x > 5. x 2 3 x 5
297. 1
0 2x 1 = 2x 1.
:
x
2x 12x 1
1
2 + 0 +1 2x 0 2x 1
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4. 155
2x 1 =
1 2x x < 12
0 x = 12
2x
1 x
>1
2
. x = 1
2
. . :
2x 1 = { 1 2x x < 122x 1 x 1
2
x = 12
-
: 2x 1 x = 12
0 .
61. x 2x 1 + x 1.. .
.
:
x
2x
1
2x 1x + 1x + 1
2x 1 + x 1
1 12
+
0
+1 2x 1 2x 2x 1 0 + +
x 1 x + 1 x + 13x 2 x 3x
. :2x 1 + x + 1 =
3x x < 12 x 1 x < 1
2
3x x 12
..
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156 4.1.
62. x
2x 1 x + 1 x.. 2x 1 x + 1.
x
2x 1x + 1
2x 1x + 12x 1x + 12x 1x + 1 x
1 12 + 0 + 0 + ++ 0 0 +
2x 1 x + 1 2x 1x + 1 2x 1 x + 12x
2 1 2x2 2x + 1 2x2 1:
2x 1 x + 1 x = 2x2 1 x < 12x2 2x + 1 1 x < 1
2
2x2 1 x 12
299. x
2 x + 4 + x. 300. x
x 2 x + 2 x.4.1.3
.
.
x = . x .
0
.
x = < 0. 0 = 0. > 0. (4.9)
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160 4.1.
. :
x <
x < x 0
x
< x
2 x < .. ... < x
306. :
1. x < 7 ... < x < ...2. x 12 ... x ...3. x > 11 x < ... x > ...
4. x 3 x ... x ...5. 1
2< x < 1
2 x < ...
6. x < 8 x > 8 ... > .....
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164 4.1.
2 = 2
2
=2
2
2 = 2
2
2
2= 2
2()
:
. 4.1.2. -
:
=
=
=
=
71. = 3, = 5 = = 3 5 = 15
= = 35
4.1.2 :
4.1.4. 3 1
2
...
1
= 1
2
...
1
(4.13)
.
1 2 ... 1 = 1 2 ... 1 =1 2 ... 1 = 1 2 ... 1 == 1 2 ... 1 = ... =1 2 ... 1
4.1.5.
= (4.14)
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4. 165
. = ...
= ...
=
, , :
72. xyz3 = xyz3 = x yz3
309. ,, = 2, = 4, = 12
.
:
1.
2. 3. 2
4. 2
5. + 6. 3 + 3 2
310. xx (x 0) 1. 311. :
x1+x2 12 , x R.
.
-
.
:
x y x + y x y x + y x + y2 3 5 2 3 5 5
2 3 1 2 3 1 52 3 1 2 3 1 5
2
3
5 2 3 5 5
2 0 2 2 0 2 2
0 3 3 0 3 3 32 0 2 2 0 2 20 3 3 0 3 3 3
..
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166 4.1.
x + y x + y. x, y
.
x
+y
x
+ y
.
4.1.6. ,
+ + (4.15). : (4.8)
:
+ + + A = + X= + . A 0.
A X A A > 0 X A. A = 0 X= 0 X A.
X
A
X, A X, A :
+ + : + + + 2 + 2 + 2 2 + 2 + 2
+
2
2
+2
+2
2 + 2+ 2 2 + 2 + 2 2 2 (4.8)
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4. 167
: + + ; 0.
=
+ = + =
= 0. : + = + 0 (4.16)
: = a + a + = + :
+
(4.17)
73. x 1 y 9 x + y 10
. x + y x + y 1 + 9 = 10. 74.
+ + .
=
+
+
=
+
+
+
.
75. x = y = 2 2x 3y 10. 2x 3y 2x + 3y = 2 x + 3 y = 2 2 + 3 2 = 10. 312. ; 313. x, y x < 2, y < 3. :
1. x + y < 52.
x
+2y
1, y > 2 x1y2 > 0
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180 4.1.
KM =KT M, T -
. M x 2y+ 3 = 0 .
(
340) :
Kx1, y1 x2, y2 :x2 x1 y y1 = y2 y1 x x1 (4.25) (4.25) xx y1 = y2 x1 x2 (4.25) y = y1
yy x1 = x2 y1
y2 (4.25) x
=x1.
337. A2, 4, B 1, 3. 338. :
339. 2x 3y = 1,x + y = 2. 340. (*) -
Kx1, y1 x2, y2 x2 x1 y y1 = y2 y1 x x1.
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4. 183
f,g,h,, ..
f
A B f A B ( f A B). f x
A y
B
f
x
y y
=f
x
. x
A
y = fx f x. x y
.
x y x
y.
x1 = x2 fx1 = fx2 (4.26) f
f
2
=f
(4
) f
(x2
+2x
+1
) =f
(x
+1
2
) 2
= 4 x2
+2x
+1
=x + 12
. fx1 = fx2 x1 = x2. fx = x2 f2 = f2 2 = 2!
A B -
. -
.
x fx ().
83. A = R, B = R f A B f
x
= x
+1
x
+2
+1.
f1 = 1 + 1 1 + 2 + 1 = 7f3 = 3 + 1 3 + 2 + 1 = 3
f2 = 2 + 12 + 2 + 1 = 32 + 5 84. A = 0,+, B = R q A B g x = x + x2.
g 0 = 0+ 02 = 0g 4 = 4 + 42 = 18
85. A = N A = 1, 2, 3, 4,... B = R A R = 2 + 3. 1 = 2 + 3 1 = 5 2 = 2 + 3 2 = 8
..
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4. 185
:
-
.
.
.
: :
-
.
87. f R R, g R R fx =x2, g x = 2x 1 :
88. -
:
..
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186 4.2.
x g 2x1 f. f . 2x 1 2x 12.
.
-
( x) ( y).
.
f
x
=x3.
x -3 -2 -1 0 1 2 3 4 5
fx -27 -8 -1 0 1 8 27 64 125 . -
. f A R A ( ) R f
x A y = fx Mx, fx.
x Mx, fx . A
.
:
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188 4.2.
f.
y f.
1. x fx > 0: xx.
2. x f
x
x. : x > x
x>0 1 > xx 1 > x 1 >
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4. 193
x 1 > x . : 0 < x < 1
,
< x
>x
(4.28)
x 1 :
:
x > 1 , < x < x (4.29)
: x < x x>0 1 < xx 1 < x
1
x ()
351. :
1. 23...24
2. (23)5 ...( 2
3)9
3. (2 1)33 ...(2 1)444. 2+1
3+14 ...2+13+175. 1 + x2012 ...1 + x2000 x 06. 1 + (2
3)11 ...1 + (2
3)15
352. 0 < x < 1.1. 0 < x < 1 x.
;
2. (4.28)
4.2.5 - .
x
g
x
=x2.
0 yy
x
0 g
x
=.
: yy ,
, xx: x gx = x2 = . x
.
..
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194 4.2.
hx = x3. 0 x 0 hx = x3 = . x
3
.
x (=
) .
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196 4.2.
< 1 :
.
. :
x = 0x 0x =
(4.31)
= (4.32) :
1
= , 1 = 1, 0 = 0 (4.33)
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4. 197
, ,
:
95. ) 4
81 = 3 34 = 81. ) 4625 4
4
625. (
) 4
625 = 5. 96. m.
3
m3
=m
m3
m. 3m6 = m2 m6 = (m2)3.
5
m15 = m3 (m3)5 =m15. 353. :
1.4
1296 =2.
5
a10 = a > 03. 4
16
81=
4.r
art
=, a
>0
5.33a36 = , a > 0
6.4
83
8 = 354. 4
= 3 4= 5 + .
355. :
1. (1 + 32)3 = 3 + 3 32 + 3 3222. ( 33 32)3 = 1 3 332 32 + 3 33 3223.
3
3
3
2
=1
3
32
+3
33
2+3
22
356. :
1. ( 3x + 3y)3 = x + 3 3x2 3y + 3 3x 3y2 + y2. ( 3x +y)6 = x2 + 2xy + y
..
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198 4.2.
3. ( 3x +y)6 ( 3xy)6 = 4xy 357. :
1.
(3
x
+3
y
)3
=x
+3 3
x23
y
+3 3
x 3
y2
+y
2. ( 3x +y)6
= x2
+ 2xy + y3. ( 3x +y)6 ( 3xy)6 = 4xy
358. :3
5 + 6 32 + 3 322 = 1 + 34
359. 3
= x 3 = y. x+ y3 3xy x + y = + 360. 3x + 13x3 = x + 3 3x + 33x + 1x .
4.2.6 - .
(3.52), (3.53) . , :
= (4.34)
=
(4.35)
:
=
=
=
=
=
=
=
=
= =
.
(4.34) .
1, 2,..., :
12 ... = 1 2... (4.36) 1, 2,...,
:
=
(4.37)
= = (4.32) 0 :
= = (4.38)
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4. 199
4.38 -
:
- .
(
).
.
. (
a). x 0 . x ( x
a). y 0 x. :
y
y x x
. y = y = . y . y = y = . y = a. :
a
=
(4.39)
(4.39) :
a = a = ( ) a = ( a) = a =
.
.
- : = .
= . :
= (4.40)..
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202 4.2.
4.2.7 - .
x
x 0. y
> > . - .
. ,
0 x = < 0 :x = x = x = x =
x = 0 :
x
=
x
=
x
=
371. :
1. x3 = 82. x3 = 8 3. x
3 = 1 24. x3 = 8
27
372. :
1. x12 = 2122. x13 = 213
4.2.9
. . ,
,
:
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4. 205
1 = 1 = 1
=1
1
=1
0
=0
0
=0
= = =
= = =
= + =
= + =;
= =;
.
!
. = + .
.
.
= . > 0 . ; .
= (
)
= (
)
( ) = ( )
=
( )( ) = ( )( )( ) = ( )
= +
=
+ = + = =
+. (.. = 3, = 3 = 12
7).
..
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206 4.2.
= . :
+ = + =
= +
+ = +
.
:
>0 t, s
t
s = st = st(4.44)
75
3 = 375 = 316807 25, 6 3 53 = 335 = 3243 = 6, 2. : 5x6 = x 54 3x4 = x 43 . :
1
s = s (4.45) :
= 12 , 3 = 13 , 4 = 14 , 4 = 14 , ... > 0 t, s
ts
ts = 1
ts= 1s
t(4.46)
. (4.44)
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4. 207
(4.46) 0 = 1. x
53 = x 53 = 1
x53
13
x5 x
54 = x 54 = 1x54
= 14x5
.
:
pq = p+q (4.47)p
q= pq (4.48)
pq = pq (4.49)p = pp (4.50)
p
=p
p(4.51)
. (4.47)
p =
q=
:
pq = = = = = + =
+ = + = p+q
( ) :
> 1 :p
0, < 0 = 0)
y = x+ xx y x 0.
= 0. 0 xx xx. x y .
.
y = x + : yy 0, xx 0
, 0 (4.54) 105. y = x + 3 = 1. 1 - 45 xx 45. 106. y = x 2 = 1. 1 = 135 xx 135. 107. y
=2x
+1.
=2
=1.
=
1
2. x
x y
y
(12 , 0) 0, 3 . 382. y = x+ A1, 2 xx. 60 383. y = 3x+2 . 384. P1,2 . ;
4.2.12 y
=1x
+1 , y
=2x
+2.
y = 1x + 1 y = 2x+2 . .
. m
m, 1m + 1
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4. 213
.
.
1m
+1
=2m
+2
1 2m + 1 2 = 0 m
Am +B = 0 A = 1 2 B = 1 2.
. m
R m
. A = B = 0 1 2 = 0 1 2 = 0.
y = 1x + 1 y = 2x + 2 1 = 2 1 = 2 (4.55)
.
. m
R
. A
=0 B
0 1
2
=0
1 2 0. : y = 1x + 1 y = 2x + 2 1 = 2 1 2 (4.56)
. A 0. :
y
=1x
+1 y
=2x
+2
1
2
(4.57)
-
: y = 1x + 1. y = 2x + 2
xx:..
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214 4.2.
xx 1
=1.
1
=1 .
.
yy 1 = 2. 1 2 :
. y = 1x+1, y = 2x+2 . . y =1x+, y = 2x+ .
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4. 215
y
=1x, y
=2x 1
A1
1, 1
, A2
1, 2
y
=1x, y
=2x.
A1A2O A1A2 :1 02 + 1 022 +1 02 + 2 022 = 1 12 + 1 222
1 02 + 1 02 + 1 02 + 2 02 = 1 12 + 1 22 1 + 21 + 1 + 22 = 1 22
1 + 21 + 1 +22 = 21 212 + 22 2
= 212
12 = 1 12 = 1 ( 389) .
y = 1x + 1 y = 2x + 2 12 = 1 (4.58)
108. y = 2x + 3 y
= 2
2
x
+
+2
.
2 2 = 2 + 2 3. = 2 2 + 2 3. 109. y = (1 2)x + 1 y =(2 1)x + 1 ...
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216 4.2.
1. (1 2)(2 1) = (2 + 1) (2 1) = 22 1 = 1. 385. t y
= 4
2t
x
+1
y
=3x
2.
386. t y = 2t + 4x + 4, y = 3t 1 .
387. y = x + : P1 3, y = 3x + 4
388. y = x + : P
2, 4
,
y = 8x + 3 389. y = 1x+1 y = 2x+2 12 < 0. .
4.2.13 x + = 0 x + > 0 fx = x+
x + = 0 x + > 0 . 2x
1
=0. x
f
x
=2x
1
xx. 2x 1 > 0 x fx = 2x 1 xx.
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4. 217
2x 1 = 0 12
2x 1 = 0 > 0 0,+. x + = 0
x
+
>0.
=0.
= 0 fx = x + xx.
xx = = 0 xx > 0 = .
0 fx = x+ -.
.
:
f
.
f x
fx. x1 x2 f
x1
fx2. fx = x + > 0 :
x1 < x2 >0 x1 < x2 x1 + < x2 + fx1 < fx2
..
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218 4.2.
fx = x + < 0 :x1 < x2
x2 x1 + > x2 + fx1 > fx2 - .
x = x > fx > 0 x < fx < 0. :
x > fx > f fx > 0x < fx < f fx < 0
: f = 0 x > fx < f fx < 0x < fx > f fx > 0
110.
fx = x+. 2. x > 2 fx > 0 x < 2 fx < 0.
111.
fx = x+ . fx < 0 fx > 1.
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4. 219
. f(52) = 0 f3 = 1.
fx < 0 x > 52
fx > 1 x < 3. 390. f
x
=x
+.
1. fx = 32. fx = 33. fx > 14. 5 < fx < 3
4.2.14 fx = x f
x
= x
x
:
fx = { x , x < 0x , x 0
y = x y = x. ..
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220 4.2.
0 -
. Geogebra
: f(x)=abs(x).
fx =x x = x <
x >
0 () x = (), () , , ,+,+.
112.
2 < x < 3. yy 2 3 xx . 2 3
3
2 2 3.
3,2 2, 3.
391. Geogebra
g x = x 1 x 1 < 3...
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4. 223
113. , ,
fx = x2 + x + A 0,1, B 1, 0, 1, 2.. :
f0 = 1,
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