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c 2015 Λυγάτσικας Ζ. Ασκήσεις ΄Αλγεβρας Β Λυκείου Λυγάτσικας Ζήνων Πρότυπο ΓΕΛ Βαρβακείου Σχολής Σχολ. ΄Ετος 2015-2016 11 Ιουλίου 2015

λυγάτσικας ζήνων ασκήσεις άλγεβρας B΄λυκείου 2015-6

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Text of λυγάτσικας ζήνων ασκήσεις άλγεβρας B΄λυκείου 2015-6

  • c 20

    15

    .

    . 2015-2016

    11 2015

  • . . , .

    email: [email protected] , 11 2015

    i

  • . ii

  • 1 31.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    1.1.1 f(x) = x2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 f(x) = x2 + x+ . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    2 112.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    3 153.1 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

    4 194.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 & . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

    5 375.1 & . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

    6 & 516.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

    7 55

    1

  • . 2

  • c 2015

    .

    1

    1.1

    1.1.1 f(x) = x2

    f(x) = x2, R, (. 4.1). . .

    1.1: f(x) = x2.

    > 0 () (x, y) = (0, 0)

    < 0 () (x, y) = (0, 0)

    y = x2 . (0, 0).

    3

  • 1.1. 1.

    1 Geogebra y = x2, y = 1000x2 y = 104x2.

    1.1.2 f(x) = x2 + x+

    1.2: x2 + x+ = 0, > 0. f(x) = x2 + x+

    > 0

    (

    2,

    4

    )

    1.3: x2 + x+ = 0, < 0. f(x) = x2 + x+

    < 0

    (

    2,

    4

    )

    . 4

  • 1. 1.1.

    1.1.3

    2 . f(x) = x2 + x+ , 6= 0:

    1. f(x) Cf -.

    2. < 0 < 0 Cf .

    3. Cf (

    2,

    4

    ).

    4. < 0, Cf Ox Oy.

    5. Cf x =

    2.

    3 y = x2 x 2. :

    1. ,

    2. ,

    3. .

    4 - f(x) = x2 + x+ , < 0 - - f(x).

    -3 -2 -1 1 2 3 4 5 6 7 8 9

    -3

    -2

    -1

    1

    2

    3

    C1 C2

    C3 C4

    1.1.4

    1. : f(x) = x2 + ( + 2)x 3. Geogebra f(x), 4.7.

    2. f(x) = x2 + 2(2 2)x 1. f (,1] [1,).

    3. y = x2 2x 2 .

    5 .

  • 1.1. 1.

    1.4: A f(x).

    4. f(x) = (3 1)x2 + 2x 8 R .

    5. A = (2, 0) B(x, 0), x > 0. A M1 M2 : AM1 = AM2 = OB = . B, OM1 OM2 N1 N2 , 4.7.

    1.5: 5.

    N1 N2 x

    y =1

    2x2 y = 1

    2x2.

    . 6

  • 1. 1.1.

    6. 4AB , AB = 8cm H = 6cm. M AB, x = AM f(x) , 4.9. [0, 8] :

    f(x) =

    3

    4x2 0 x 4

    24 34

    (8 x)2 4 < x 8

    1.6: 6.

    7. .

    () p(x) = x2 + x+ , ( 6= 0).i.

    ,

    p() < 0 (1.1)

    ii. p(x) = 0, : p() = 0.iii. 1 < 2 p(x) = 0,

    < 1 < 2,(1 < 2 <

    ), :

    > 0, p() > 0, < 2,

    ( >

    2

    )(1.2)

    () . 2 :

    p(x) = x2 (+ 1)x+ 3 = 0 (1.3)

    7 .

  • 1.1. 1.

    i. Geogebra p(x) .ii. -

    .iii.

    7.

    8.

    p(x) = (+ 1)x2 4x+ 2+ 3 (1.4)

    () Geogebra.

    () 1 p(x) .

    () .

    9. 1 < 2 .

    () :

    p(x) = x2 + x+ , (a 6= 0) (1.5)

    1 2 :

    p(1) < 0 p(2) < 0 1 < 1 < 2 < 2

    p(1) < 0 p(2) > 0 1 < 1 < 2 < 2

    p(1) > 0 p(2) < 0 1 < 1 < 2 < 2

    p(1)p(2) < 0

    1 < 1 < 2 < 21 < 1 < 2 < 2

    2 4 > 0p(1) > 0p(2) > 0

    1 <

    2, 2 >

    2

    2 <

    2

    1 >

    2

    1 < 1 < 2 < 2

    1 < 2 < 1 < 2

    1 < 2 < 1 < 2

    . 8

  • 1. 1.1.

    () i.

    p(x) = ( 1)x2 2(3+ 1)x+ 9, 6= 1 (1.6)

    Geogebra.ii. , ,

    1 0 p(x) = 0.

    iii. 9.

    10. , , :

    () (x 1)(x+ 2) + (x+ 1)(x 2) (x 1)(x 2) = 0

    (1, 2).

    () A2

    x a+

    B2

    x = 2, (x 6= , )

    .

    11.

    x 4x2 3x 3

    : x 4

    x2 3x 3=

    x2 (3+ 1)x 3+ 4

    9 .

  • 1.1. 1.

    . 10

  • c 20

    15

    .

    2

    2.1

    12. {

    2x y = 17x 4y = .

    () , (x0, y0), R.() 2x0 y0 < 1.

    13. D {

    (D 1)x+ y = 1D x+ 3y = 2 , .

    14. {x+ y = x+ y =

    D 6= 0, Dx + 2Dy = 3D x 2y = 1, x y.

    15. x y :

    D2x +D2y + 5D

    2 2D Dx + 4D Dy = 0

    x y.

    16. , {x y = 2x+ 3y = 1

    ,

    { x+ y = 0

    ( 1) x y = 2

    .

    17. (3, 2) {4x 5y = 2x+ 3y = 3 ,

    {2x 3y = 0

    x = 1, 5y

    11

  • 2.1. 2.

    18. a, b, c, a, bc {() : ax+ by = c() : ax+ by = c

    () A(2, 1) B(2,1) () C(1,3) D(1,2).

    19. 0o < < 90o:{( () ())x+ ( () + ()) y = 1( () + ())x ( () ()) y = 1

    . (a, b) , a b 1 .

    20. {

    (+ 3)x+ y = 5x+ y = 1 3 < < 5. x0 y0

    , 16

    3< x0 < 9 y0 < 0.

    . 12

  • c 20

    15

    .

    2. 2.2.

    2.2

    21. :{

    2x 3y = 5

    3x 5y = 7

    ( X =x Y =

    y.)

    22. :

    () :

    {x y = 8

    x2 y2 = 384 , () :

    {2x+ 5y = 34

    4x2 25y2 = 952

    23. x , y :{(4x+ 3y)2 (3x 4y)2 = 6

    (x y)3 + (x+ y)3 + 3(x y)2(x+ y) + 3(x y)(x+ y)2 = 64 : .

    24. :4

    x y y

    (x y)2= 0

    1 x2 + y2

    x2 y2= 0

    : , . 0 0. , .

    25. :

    x2 (2 1)x 3 = 0 (1) x2 ( 2)x+ 3 = 0 (2) 6= 1. , :

    () ,

    () .

    : . Geogebra , , , .

    , :{x2 (2 1)x 3 = yx2 ( 2)x+ 3 = y

    {x = 3y = 3 + 6

    , A . A 0, xx.

    = 12.

    13 .

  • 2.2. 2.

    6= 1, Geogebra, = 1 .

    26. :{

    5x y = 1x2 + 4xy 2y2 + 8x+ 39 = 0 .

    : y .

    27. :{x+ y + xy = 41xy(x+ y) = 330

    :

    . {x+ y = xy =

    . {+ = 41 = 330

    .

    . (x, y) = (5, 6), (15

    214, 15 +

    214)

    . 14

  • c 20

    15

    .

    3

    3.1 -

    28. : f(x) = x7

    3 6x.

    29. f(x) = ||x 3(x+ 1).

    () Geogebra . ;

    () f(x) - .

    30. h R, : h(x) =h(2x 3) h(x3) h(27) = 0. : :

    f(x1) = f(x2) x1 = x2

    31. f, g : R R . :h(x) = f

    (g(x)

    ) .

    32. f R :

    f

    (2

    )< f

    ( 3

    ), f(2) f(2 + 1)

    33. f R :

    i. f(

    6 3) > f(

    3

    2)

    ii. f(

    4

    34)> f(

    6)

    iii. f(2x2 + 2013) < f(2012) iv. f(2 + ) > f( + 2)

    15

  • 3.1. - 3.

    34. f(x) = x2 +x 1.

    () f .

    () f .

    () .

    () f(10) :

    i. f(x) = 103ii. f(x) < 103

    () f(2012

    2011

    ) 1 > 0.

    35. f(x) =

    4 x2 + 2x

    .

    () f .

    () Geogebra.

    () ;

    () Cf O(0, 0);

    36. f(x) =

    x2013, x < 0

    x2013, x 0. Cf

    yy.

    37. f , R , g(x) = |f(x)| .

    38. f, g : A R f g . h(x) = f(x) g(x) .

    39. f : R R : f(x+y) = f(x)+f(y), x, y R.

    () f(0).

    () f .

    40. f : R R . g(x) =f(|x|) |f(x)|, .

    41. f : R R. :

    () g(x) = f(a+ x) f(a x) ,() h(x) = f(b x) + f(b+ x) .

    42. f : R R :

    f(x) f(x) = [f(x)]2

    x R. f .

    . 16

  • 3. 3.1. -

    43. f : N N - :

    f(1) = 1 m n,

    f(m+ n) = f(m) f(n) + f(n) + f(m)

    () f .

    i. f(0).ii. f(2), f(3), f(6).

    () n f(n+ 1) = 2f(n) + 1.

    () n: g(n) = f(n) + 1. m n: g(n+m) = g(n) g(m).

    () f .

    44. f :

    f(0, y) = y + 1, f(x, 0) = f(x 1, 1), f(x+ 1, y + 1) = f(x, f(x+ 1, y))

    f(2, 1) f(2, 2).

    45. f (0,+), :

    x f(x)2 3, 01033 4, 771245 6, 98976 7, 78157 8, 451089101001000

    1000000109

    : x > 0 y > 0 :

    f(x y) = f(x) + f(y)

    () f(4) -.

    () f(245).

    () f(1).

    () x > 0:

    f

    (1

    x

    )= f(x)

    46. f f(x + y) = f(x) + f(y), x, y. :

    17 .

  • 3.2. 3.

    () f(0) = 0

    () f .

    () f(x) = f(x), N

    47. f(x) 1 4, , f(x) =

    2x+

    x2 + 1.

    3.2

    48. f(x) = x2 x2 x 2 = 0.

    49. 3.1 f(x) = (x) g(x) = (x). ;

    3.1: .

    . 18

  • c 20

    15

    .

    4

    4.1

    50. :

    2(1 + 2), 2(1 + 2), 2 2 + 2 2

    51. :

    +

    1 + +

    1 +

    52. :=1

    :=

    1

    , :

    () 2 2 = 1 2 2 = 2 2 1

    ()1

    =

    1 +

    1

    =

    1 +

    ()( +

    )2= 1 + 2

    (

    )2= 1 2

    () 3 + 3 = ( + )(1 )() 4 + 4 = 1 2 2 2 4 4 = 1 2 2

    ()1 2 2 2

    = +

    ()( + + 1

    )( + 1

    )= 2

    ()(

    1 + + )2

    = 2(1 + )(1 + )

    () 1 2

    1 + =

    () ( + ) = 1

    19

  • 4.1. 4.

    ()

    1 +

    1

    = + 1

    () 3 2 + 2 2 =3 + 2 21 + 2

    () 2 2 = 1 2

    1 + 2

    + 1

    +

    1= 2 2

    () 2 2 = 2 2 = 11 + 2

    11 + 2

    () 2 2 2 2 = 2 2

    () + +

    +

    = 0

    . 20

  • c 20

    15

    .

    4. 4.2. &

    4.2 &

    53. :

    () + (270o + ) + (180o + ) (270o ) = 0() (171o + ) (9o + ) = 0() (203o + ) + (23o + ) = 0() (138o + 2 ) + (48o + 2 ) = 0() (280o + ) + (10o + ) = 0

    () (3

    2

    ) ( )

    () ( )

    (32

    + )

    (

    2

    ) ( + )

    () ( + )

    (32

    )

    (32

    + )

    (2 + )() + + = 180o ( + ) = ()() + + = 180o ( + ) = ()

    () + + = 180o () = (2 + + )() + + = 180o () = (2 + + )

    () + + = 180o + 2

    =

    2

    () + + = 180o 2

    = + 2

    2=

    + 2

    2

    () + + = 180o + 2

    = ()

    54. =2 2

    2 + 2, , R.

    55.

    7 R.

    : 13 :

    0o, 7, 2

    7, 3

    7, 4

    7, 5

    7, 6

    7

    . 7.

    56. 1050o. : 1050o = 2 3600 + 3300 = 2 360o + 3600 30o.

    57. 2( 6x+ 6x) 3( 4x+ 4x).

    21 .

  • 4.2. & 4.

    58. : |2 x+ 3 y| 5. : |2 x+ 3 y| |2 x|+ |3 y| 2 + 3 = 5.

    59. 12.

    :( +

    )2 0 2 + 2 2 1

    2 .

    60. | |+ | | = | + |. : = 1 > 0 |a+ b| = |a|+ |b| a b > 0.

    61. 2< x 0

    T = 3, :

    () .

    () f x f .

    () f yy.

    :

    4.4: 181.

    . 24

  • 4. 4.2. &

    () 2

    2= = 1

    2 2 +

    2= 3 = 5.

    () 0 1 + (2x) 2 5(1 + (2x)2

    2. , 2 x

    , 1 = (2x). , x = k2

    k, 4.4.

    () x = 0 5(1 + (2 0))

    2 2 = 3. , yy

    (0, 3), 4.4.

    25 .

  • c 20

    15

    .

    4.3. 4.

    4.3

    68. : 2( + ) + 2 2 ( + ) = 2.

    69. : 2 + 2 + 2 ( + ) = 2( + ).

    70. : (+ ) ( ) + ( +) ( ) + ( + ) ( ) = 0.

    71. : ( + ) ( ) + ( ) ( + ) = 2.

    72. : ( + + = ) ( = 2 ) =

    73. :

    () (45o + ) (45o ) = 2 2

    () 2 =1

    1 + 2() = 2 2() 78o = 12o + 2 24o + 4 42o

    74.

    2 x+ x

    2. : : x+ x =

    2

    (x+

    4

    ).

    75. 90o, .

    76. ( )

    +

    2 2

    = 1 2 = .

    77.

    2x 2 x (x+ ) + 2(x+ )

    x.

    78. + = , 2 + 2 2 = 2

    79. 2x + 2(120o + x) + 2(120o x) x.

    80. 3 4 2x+ 4x.

    81. (

    2

    2

    )2=

    1

    1 2 2

    82. () x+ y , :

    1 2x 2y

    : 2x+ 2y

    2.

    . 26

  • 4. 4.3.

    83. () : 2 (1 2), = 45o. : , ,

    2 (1 2) = = 2 (1 )(1 + ) (1 + 2)

    27 .

  • c 20

    15

    .

    4.4. 4.

    4.4

    84. : x =

    2

    2.

    :x = 2k 4.

    85. : x =

    6

    2.

    : ,

    6

    2= 1.224744872 < 1.

    86. : (1 x)(2 x

    3) = 0.

    : x = 1 x =

    3

    2=

    3.

    87. : (1 x)(2 x

    2 +

    2

    4) = 0.

    : : x = 1 =

    2 x =

    2 +

    2

    4 x = 0.480181

    (86.4326

    )o.

    /.

    88. : (5x+ 80o) = (3x 40o). : x = 60o + 1 180o x = 17o30 + 1 45o.

    89. :

    () (2x+ 12o) + 5x = 0.

    () (

    3x 3

    )+

    (x

    6

    ).

    () 2 2x+

    3 x+ 1 = 0.

    90. : 4 2x 2(1 +

    3) x+

    3 = 0.

    : x =1

    2 x =

    3

    2, x.

    91. :1

    1 + 2x+ 2x 2 x = 0.

    : x = 3

    + 2.

    92. : (

    3 1) 2x (1 +

    3) x x+ 1 = 0. : 2x

    [3 (1 +

    3) x+ 2x

    ]= 0 x =

    4+ 1, x =

    3+ 2.

    93. , x + x = . :

    x+ x = (

    1 2x)

    = x

    2(

    1 2x)2

    = ( x)2

    (2 + 2) 2x 2 x+ 2 2 = 0

    . 28

  • 4. 4.4.

    0 2(2 + 2 2) 0 2 + 2 2

    94. :

    3 x+ x = 1. : x+ 1

    3 x =

    13 x+ 30o x = 1

    3 (x+ 30o) = 30o . . . .

    x

    2= y x =

    2y

    1 + y2 x =

    1 y2

    1 + y2. :

    3 x+ x = 1

    3

    2y

    1 + y2+

    1 y2

    1 + y2= 1

    : y = 0,

    3. y = x

    2 x...

    95. :

    () x+ x = 1.() x (+ 1) x = , .

    96. () : 3x+ 3x = 1. :

    3x+ 3x = 1 ( x+ x)( 2x x x+ 2x) = 1 ( x+ x)(1 x x) = 1

    = x+ x

    (

    1 2 12

    )= 1

    2

    2 74, . 26

    { = x+ x = 1(),2()

    (x+

    4

    )=

    12

    {x = 21, 1 Zx =

    2+ 22, 2 Z

    97. :

    () x+ x+ x x+ 1 = 0.() ( x+ x) = x x, .() x x x+ x 1 = 0

    98. : 3 x x = 0. : y = x y =

    x

    3,

    . , (0, 0).

    99. :

    () 4 2x 3x 2 = 0.

    29 .

  • 4.4. 4.

    4.5: y = x y =x

    3.

    () x = x( x+ x). : x = x( x+ x) 1

    x=

    x+ x x

    . . . .

    100. : x > . :

    < x < a

    : + 2 < x < + 2, Z.

    4.6: 100.

    101. () : ( x

    )=

    ( x

    ).

    : :

    x = 21 +

    2 x (4.3)

    x = (22 + 1)

    2 x (4.4)

    1, 2 Z. 4.3 : x+ x = 21 +

    1

    2(4.5)

    4.5 , 93, 28:

    12 + 12 (

    21 +12

    )2 1621 + 81 7 0 0.9571067810, 1 0.4571067810 1 = 0

    . 30

  • 4. 4.4.

    , 4.5

    x+ x =1

    2(4.6)

    , 4.4 :

    x x = 12

    (4.7)

    4.6 4.7.

    102. () : 2y2 + y 1 = 0() : 2 2x+ x 1 = 0.

    : : y = 1, 12. : x =

    2

    6.

    103. ()

    () p(x) = x2 + x + . = 2 4 0. p(x) = 0:

    i. [1, 1], p(1)p(1) < 0.ii. 1 (1, 1],

    ,

    + = 0 2 < 2.iii. 1 (1, 1], + = 0

    2 > 2.iv. [1, 1] p(1) > 0 p(1) > 0.v. [1, 1] , p(1) > 0 p(1) < 0 p(1) < 0 p(1) > 0.

    () :

    2x 2( 2) x+ + 2 = 0

    :

    () i. 1 < 2 1 < 1 < 1 < 2 1 < 1 < 2 < 1, p(1) p(1) , p(1)p(1) < 0.

    ii. p(1) = 0, + = 0. =

    1 < < 1

    0 (, 23

    ).

    - p(1) = 6 > 0, p(1) = 4 2 > 0 (

    12 ,

    ). , p(1) = 4 2 = 0 = 12 .

    - p(1) > 0 (0,).

    31 .

  • 4.4. 4.

    4.7: 103.

    - p(1) > 0 (, 0)(

    12 ,

    )- p(1)p(1) > 0

    (12 ,

    ).

    , x = 1 = 12 , p(x) = 0 = 12

    2 2 > 0. < 12 p(x) = 0 [1, 1] p(1)p(1) < 0. . 12 < 0 p(1) > 0.

    1, (1 + 2) (1) = 2 +

    =

    2( 1)

    , 12 < x (4.10)

    , x x 4.9,

    ( x) < x (4.11)

    , 4.10 4.11, : ( x) < x < ( x).

    .

    . 34

  • c 20

    15

    .

    4. 4.5.

    4.5

    107. A, 4.10, d E 15o. B, A 1 km, 30o, d.

    : EA1 EB1 = 1 km, EA1 =d

    15o EB1 =

    d

    30o.

    15o.

    4.10: 107

    108. 4.11.

    () 1 + 2 + 3.

    4.11: 108

    () 4.11, n 3o!

    : , 1 = 45o, (2+ 3). 2+ 3 = 45o, 90o. . . .

    35 .

  • 4.5. 4.

    . 36

  • c 20

    15

    .

    5

    5.1 &

    109. g(x) 6= 0 p1(x), p2(x), . . . , pn(x),n N, :

    F (x) = c1p1(x) + c2p2(x) + + npn(x) (5.1)G(x) = p1(x) p2(x) pn(x) (5.2)

    110. g(x)/p(x) p(x)/g(x) p(x) = c g(x), c R.

    111. , p(x), :

    p(p(x)

    )= x2p(x) xp(x) + 1

    : p(x) = x2 + x+ , 6= 0, :

    (x2 + x+ )2 + (x2 + x+ ) + = x2(x2 + x+ ) x(x2 + x+ ) + 1 . . .

    - . p(x) = x2 + x+ 1.

    112. p(x) = x3 + x2 + x+

    {2 = 33 = 272

    .

    : . p(x) =

    3(x +

    )3, ; . , ;

    37

  • 5.1. & 5.

    113. f(x) =p(x)

    q(x), q(x) 6 0,

    p(x) = x + 1x

    1 + + 0 (5.3)q(x) = x

    + 1x1 + + 0 (5.4)

    i 6= 0. . : f(x) = c R, :

    00

    =11

    = = 11

    =

    = c

    114. p(x), , : p(x) p(x 1) = x3 p(0) = 0. S3 = 13 + 23 + + 3. : p(x) =

    x4

    4+x3

    3+x2

    2.

    p(1) p(0) = 13p(2) p(1) = 23p(3) p(2) = 33. . . = . . .p() p( 1) = 3

    (+)

    S3 = p() p(0) =

    (( + 1)

    2

    )3

    115. , , , :

    ()6(x2 2)

    (x2 1)(x2 4)=

    x 1+

    x+ 1+

    x 2+

    x 2.

    ()3x2 4x+ 2

    (x 1)3=

    x 1+

    (x 1)2+

    (x 1)3.

    :

    () = 1, = 1, = 1, = 1.

    () = 3, = 2, = 1.

    116. 1(x) 1(x) p(x) q(x) 6 0 2(x), 2(x) 1(x) h(x) 6 0. p(x) q(x) h(x). , p(x) q(x) h(x) 1(x) 2(x) 0. : . , p(x) q(x)h(x). .

    117. p(x) q(x) = x2 + x+ 1 h(x) = x2 x+ 1 1(x) = x 1 2(x) = 2x + 5. (x)

    . 38

  • 5. 5.1. &

    p(x) q(x) h(x) = x4 + x2 + 1. :

    f(x) = q(x)1(x) + x 1 (1) f(x) = h(x)3(x) + 2x+ 5 (3)1(x) = h(x)2(x) + x+ (2) 3(x) = h(x)4(x) + x+ (4)

    :f(x) =

    [q(x) h(x)

    ]2(x) +

    [q(x) (x+ ) + x 1

    ](5.5)

    f(x) =[q(x) h(x)

    ]4(x) +

    [h(x) (x+ ) + 2x+ 5

    ](5.6)

    5.5 5.6, p(x) q(x) h(x) = x4 + x2 + 1. , , :

    q(x) (x+ ) + x 1 h(x) (x+ ) + 2x+ 5 (5.7)

    q(x) h(x) , 5.7 = 3, = 72, = 3, = 5

    2.

    118. p(x) +1 , . : 1, 2, . . . , +1 p(x) = 0, . p(x) 6= 0 . 6= 0 :

    p(x) = (x 1)(x 2) (x +1)

    do(p(x)) = + 1, .

    119. :

    p(x) =(x )(x )( )( )

    +(x )(x )( )( )

    +(x )(x )( )( )

    6= 6= 6= . p(x) 1. : P (x) = p(x) 1. P () = P () = P () = 0 do(P(x)) = 2. 118, P (x) = 0 p(x) = 1.

    120. p(x) = (x )2( ) + (x )2( ) + (x )2( ) + ( )( )( ), 6= 6= 6= , .

    121. p(x) = xn n q(x) = xm m, 6= 0, m = .n. : m = n + v, 0 v < n, q(x) = xm m = (xn)xv (n)v. v = 0. q(x) p(x). y = xn, p(x) = y n p(x) =yxv (n)v. U(x) = (n)xv (n)v. ,U(x) = 0 (n)xv (n)v = 0 xv v = 0

    (x

    )v= 1 v = 0. , m = .n. ,

    m = kn p(x)/q(x).

    122. p(x) = nxn + n1xn1 + + 1x + 0, n 6= 0, 1 = +

    , , Q

    Qa, ,

    2 = .

    : q(x) = (x 1)(x 2) = x2 2x + 2 . p(x) = q(x)(x) + x + . 1 q(x) = 0, p(1) = q(1) = 0.

    39 .

  • 5.1. & 5.

    (+ ) + = 0

    {+ = 0

    = 0

    { = 0 = 0

    . , p(x) = q(x)(x).

    2 p(x) = 0 q(2) = 0 = p(2).

    123. p(x) = x3 x2 (3 +)x+ + 10 (x 2)2. : = 5, = 2.

    124. p(x) = x3 3x+ 2, , R. , R (x)2 p(x). 3 = 2. : Horner p(x)/(x ),

    p(x) = (x )(x2 + x+ 2 3 (x)

    ) + 3 3+ 2

    : p() = 0 3 3 + 2 = 0 () = 0 32 3 = 0. 3 = 2.

    125. p(x) 2p(x) + p(2 x) = x2 1,

    () p(0) p(2),

    () p(x) x2 2x

    :

    () p(0) = 1, p(2) = 3.

    () (x) = 2x+ 1.

    126. p(x) = x5 2x3 + x+ 3 6. , , p(x) x3 x. . : = 12 , = 0, = 2. x

    2.

    127. 6= p(x) (x )(x ) :

    (x) =p() p()

    x+p() p()

    128. p(x) = x + 1x1 + + 1x + a0 % p(0) = 0, % p

    (p(p(x)

    )).

    129. () 1 - : . p(x) = x3 +x2 +x+, 6= 0, 2 < 0, p(x) = 0 .

    1 Bolzano . .

    . 40

  • 5. 5.1. &

    : , R. (x )/p(x). , Horner , :

    (x) = x2 + (+ )x+ 2 + + (5.8)

    5.8 . x

    d(x) = 322 2 4 + 2 (5.9)

    d(x) ,

    d1(x) = 162(3 2) (5.10)

    d1(x) 2 < 0, (), d(x) , 32 < 0. , (x), 5.8, . p(x) .

    130. () p(x) ( ) p(0) p(1) , p(x) = 0 . : Z, (x )/p(x) (x) :

    p(x) = (x) (x ) (5.11)

    x = 05.11 p(0) = (0) (5.12)

    x = 15.11 p(1) = (1 )(1) (5.13)

    (0) (1 )(1) . , , 1 ,(0), (1) (). 1 (). , p(x) = 0.

    131. () , , , p(x) = x3 + x2 + x+ 1 x = 19 2 x = 62. :

    623 + 622 + 62 + (193 + 192 + 19 + ) = 2 1 = 1

    .

    132. () ; : p(x) = nxn + n1xn1 + + a1x+ a0, n N T 6= 0. p(x) 0. p(x) , p(x) = p(x + T ). x x = 0, T, 2T, 3T, . . . , kT, . . . ,k > n, p(0) = p(T ) = p(2T ) = = p(kT ) = . . . , . f(x) = p(x) p(0). :

    f(0) = p(0) p(0) = 0f(T ) = p(T ) p(0) = 0. . .

    f(kT ) = p(kT ) p(0) = 0. . .

    , f(x) n0 . 118 f(x) 0 p(x) = p(0). p(x) .

    41 .

  • 5.1. & 5.

    133. () 2 % p(x) = x+1x1+ +1x+0 :

    |%| < 1 + |a1|+ |a2|+ + |1|+ |0| : || = max{|i|i=0...1}, || |i|., 1 + || < 1 + |a1|+ |a2|+ + |1|+ |0|. , :

    |%| < 1 + || (5.14)

    |p(%)| |%| ||(|%|1 + + |%|+ 1) = |%| || |%| 1|%| 1

    (5.15)

    , 5.14. |%| 1 + ||

    1 || 1|%| 1

    |%| || |%|

    |%| 1

    ,

    |p(%)| |%| || |%| 1|%| 1

    |%| || |%|

    |%| 1+ || 1

    |%| 1 || |%|

    |%| 1 || |%|

    |%| 1+ || 1

    |%| 1 || 1

    |%| 1> 0

    , % p(x) = 0 p(%) = 0. ,

    |%| < 1 + || < 1 + |a1|+ |a2|+ + |1|+ |0|

    134. ()( 2013 3) %

    %3 + 2%2 + 1%+ 0 = 0

    i R |i| < 3, |%| < 4. :

    %3 + 2%2 + 1%+ 0 = 0 %3 = 2%2 1% 0

    %3 | 2%2 1% 0| %3 |2%2|+ |1%|+ |0| %3 < 3

    (|%|2 + |%|+ 1

    ) %3 < 3 |%|

    3 1|%| 1

    () |%| 1 1 < 4.

    2 , 5.14 , . 1, . , 6= 0. .

    . 42

  • 5. 5.1. &

    () |%| 1,

    |%|3 < 3 |%|3

    |%| 1 3 1|%| 1

    < 3|%|3

    |%| 1, |%|3 < 3 |%|

    3

    |%| 1 |%| < 4, .

    :

    %3 + 1%2 + 2%+ 0 = 0 |%|3 3|%|2 + 3|%|+ 3

    |%|3 < 3|%|2 + 3|%|+ 4 (3 < 4) |%|3 3|%|2 3|%| 4 < 0 (|%| 4)(|%|2 + |%|+ 1) < 0

    : |%| < 4.

    135. % x2 + x+ c = 0, 6= 0, , , R, :

    |%| < ||+ ||+ ||||

    : . .

    ||+ ||+ ||||

    = 1 +

    + (5.16)

    5.16 2 . ,

    1 +

    + = 1 + |%+ %|+ |% %|

    :1 + |%+ %|+ |% %| > |%| (5.17)

    ,

    () |%| < 1, 5.17 .() |%| 1,

    1 + |%+ %|+ |% %| 1 + |%| |%|+ |% %|= 1 + |%|+ |%|(|%| 1) 1 + |%| |%|(|%| 1) > 0> |%|

    5.17.

    43 .

  • c 20

    15

    .

    5.2. 5.

    5.2

    136. x :

    ()x 3 = 5

    () x

    25 x2 = 1()xx+ 1 = 2

    () 2

    5 4x = 5 4x()x2 x+ 5 = x 3

    137. x :

    ()x 2 0 x < 1.

    141. p(x) = x3 + x2 2x.

    () p(x) .

    () x p(x) =

    x 1.

    () x p(x) >

    x 1.

    :

    () p(x) = x(x 1)(x 2)() x = 1, 1

    2.

    () x (1,+).

    142. () x x+

    2(x+ 1) = 11.

    () x x+

    2(x+ 1) < 11. :

    . 44

  • 5. 5.2.

    () x = 7.

    () x (1, 7).

    143. x :

    () (2 x 1)4 + 6(2 x 1)2 7 = 0

    () 2 3x+ 5 2x+ 5 x+ 2 = 0

    () 2 4x 5 3x+ 5 x 2 = 0

    : 1 . .

    () x = 0,1

    2.

    () x = 12.

    () x = 0, ,1

    3.

    144. x: x3 ( + 2)x2 + 2x 1 = 0, , R x. : p(x) = x3 (+ 2)x2 + 2x 1.

    () p(1) = 0 = 43

    p(x) = x3 23x2 8

    3x 1 x = 1, 5 +

    61

    6,

    5

    61

    6.

    () p(1) = 0 = 2, p(x) = x3 4x2 + 4x 1 x = 1, 3 +

    5

    2,

    3

    5

    2.

    145. x : x3 2x2 + 22x+ 60 = 0. : x3 2x2 + 22x+ 60 = (x+ 2)(x2 4x+ 10). x = 2.

    146. x 7x3 13x2 + 3x+ 3 < 0. : 7x3 13x2 + 3x+ 3 = (x 1)(7x2 6x 3) = (x 1)

    (x 3 +

    30

    7

    )(x 3

    30

    7

    ).

    , :

    x (, 3

    30

    7

    )(

    1,3 +

    30

    7

    )

    147. p(x) = x154x14+2x13+2015, p(2 +

    2) + p(2

    2)

    2.

    : p(x) = x13(x2 4x + 2) + 2015 x2 4x + 2 = 0, x = 2

    2. ,p(2 +

    2) + p(2

    2)

    2= 2015.

    45 .

  • 5.2. 5.

    148. x : (x 1)4 + (x 5)4 = 32. :

    x 1 + x 52

    = y, : x 1 = y + 2 x 5 = y 2. ,

    (y + 2)4 + (y 2)4 = 32 y4 + 12y2 = 0

    : y = 0, x = 3 .

    149. x x2 + x = x .

    :

    x2 + x = x x & x2 + x = (x )2

    x & 3x = 2

    () 6= 0, x & x =

    3

    3& x =

    3 < 0, x =

    3

    () = 0, x 0.

    :

    = 0 x x 0 > 0 < 0 x =

    3

    150. x x2 3x+ 2

    x2 + x 6 = x 2.

    : :x2 3x+ 2 0 & x2 + x 6 0 x 3 2 x

    : x2 3x+ 2 = (x 2)(x 1) x2 + x 6 = (x 2)(x+ 3).

    () x 2 :x 2

    [x 1

    x+ 3

    x 2

    ]= 0 (5.18)

    i. 5.18 x = 2.ii.

    x 1

    x+ 3

    x 2 = 0 (). , x > 2 :

    x 1 =

    x+ 3 =

    x 2 x+ 4 = 2

    (x 3)(x 2)

    {x = 4

    3 = 1 +

    2,

    x < 4 () (x 4)2 = 4(x 3)(x 2) x > 2 & x < 4 & x = 2 2

    3

    3

    () x 3 : 2 x

    [1 x

    x 3

    2 x

    ]= 0 (5.19)

    i. 5.19 x = 2 x 3.ii.

    1 x

    x 3

    2 x = 0 ( ). ,

    1 x =

    x 3 +

    2 x 2 + x = 2

    (x 2)(x+ 3)

    x 3, 2 + x < 0 ( )

    . 46

  • 5. 5.2.

    : x = 2.

    151. x x4 2x3 10x2 + 4x+ 16

    x 4> 2x+ 1.

    : x ( 1 +

    21

    2,1

    )(1 +21

    2, 4) (4,+).

    47 .

  • c 20

    15

    .

    5.3. 5.

    5.3

    Euler

    5.3.1 x, ;

    fung.ggb. x .

    5.1: Fung

    Euler, p(n) = n2+n+41. n = 0, . . . , 39 . 40, 40 . . Euler, p(n) = n2 +n+41 = n(n+1)+41. , .

    p(n) = n2 +n+ . - n = 0, 1, 2, . . . , :

    () .

    () p(1) ;

    :

    () , p(0) = .

    () , . p(1) = 1 + + . . , 1 + 1 + + . p(1) , .

    . 48

  • 5. 5.3.

    = 3 = 1. .

    http : //blog.lucaswillems.com/1144/project euler probleme 27

    Python.

    Euler , , . : p(x) = (x 41)2 + (x 41) + 41 =x281x+1681 41 , 5.2.

    5.2: Euler

    - . Maple .

    49 .

  • 5.3. 5.

    . 50

  • c 20

    15

    .

    6

    &

    6.1

    152. x : 3 2x4 2x1 = 5x2 6 5x3. : x = 4.

    153. : (x2 3x+ 2)x22x = 1. : x2 3x+ 2 = 1 x2 2x = 0 x2 3x+ 2 6= 0. , x = 0, 3

    5

    2.

    154.

    f(x) = 6 91

    x 13 61

    x + 6 41

    x

    :

    () f yy.

    () f xx .

    :

    () Df = R, .()

    6 91

    x 13 61

    x + 6 41

    x 6 91

    x

    6

    1

    x

    = 13 + 6 41

    x

    6

    1

    x

    = 0

    6

    (3

    2

    ) 1x

    + 6

    (2

    3

    ) 1x 13 = 0

    6v + 61v 13 = 0, v =

    (3

    2

    ) 1x

    6v2 13v + 6 = 0 v1 =

    2

    3, v2 =

    3

    2

    51

  • 6.2. 6. &

    , x = 1,1 Gf xx (1, 0), (1, 0).

    155. f(x) = kx, 0 < 6= 1 k R.

    ()

    f(x+ 1)

    f(x),

    f(x+ 2)

    f(x+ 1),

    f(x+ 7)

    f(x+ 6)

    ()

    f(x+ 3)

    f(x),

    f(x+ 6)

    f(x+ 3),

    f(x+ 16)

    f(x+ 13)

    () f(x) - x . ( (x, x + 3), (x +3, x+ 6), (x+ 12, x+ 15) )

    () 6.1 -. 155, -.

    6.1:

    6.2

    156. ln i = R tL

    +ln I i = I eR tL .

    .

    157. > > 0 2 + 2 = 11, :

    ln

    3=

    1

    2

    (ln + ln

    ) . 52

  • 6. & 6.2.

    : 2 + 2 = 11 2 + 2 2 = 9

    ( )2 = 9 = 3

    3

    =

    ln(

    3

    )= ln

    . . .

    158. :

    7

    16log(3 + 2

    2) 4 log(

    2 + 1) =

    25

    8log(

    2 1)

    : : 3 + 2

    2 = (

    2 + 1)2.

    159. xlog y = ylog x. .

    160. : log log log = 1 , , R+ {1}.

    161. logx y = logy z logz x, x, y, z R+ {1}, : x = y x =1

    y.

    162. log 2 = log 15 = , - :

    () log 5

    7, 2

    () log 5

    5

    34

    6

    : :log 23 = 3 log 2 = 3

    log 3 = log15

    5= log 15 log 10

    2= + 1

    :

    ()

    log 5

    7, 2 =1

    5log

    72

    10=

    1

    5(log 72 1) = 1

    5(log 8 + log 9 1)

    =1

    5(3 log 2 + 2 log 3 1) = 1

    5(5+ 2 3)

    ()

    log 5

    5

    34

    6 =1

    5

    (log 5 log 3 + 1

    4log(2 3)

    )=

    4 log 5 3 log 3 + log 220

    =7 6 3

    20

    53 .

  • 6.2. 6. &

    163. x2 2(1 + log )x+ 1 (log )2 = 0. : = 2 log (log + 1) = 0 = 1, 1

    10.

    164. x2 2(1 + ln )x+ 1 ln2 = 0

    x R > 0.

    ()

    ()

    :

    () = ln (ln + 1) 0 ln 1, 0.() 0 P > 0, 1 < e

    .

    165. log(x2y2) = log x log y = log x log y . . : log x =

    + 3

    5, log y =

    25

    .

    166. x :xlog

    x = 10.

    : x > 0. log x = 4.

    167. : log(2x + 2 3x) + log 81 = x log 3 + log 178. : x = 4.

    . 54

  • c 20

    15

    .

    7

    168. x > 0 y > 0 xx yy xy yx. : x = y . x 6= y. (x y) (log x log y) 0 .

    169. : f(x) = log(11x2 7x+ 10) log x2 1.

    () f xx .

    () A = (0, 4) B, xx AB.

    : , , . xx x = 2 5 f(x) = 0, 7.1. 6 ...

    x1 2 3 4 5 6 7 8

    y

    K0,01

    0

    0,01

    0,02

    0,03

    7.1: 169.

    170. x: x

    = , , , R+ 6= 1, 6= 1. : :

    x log = log

    (log

    log

    ) x = 1

    log log

    (log

    log

    )(7.1)

    7.1, log

    log> 0 log log , , > 1

    , < 1.

    55

  • 7.

    171. :{xlog y+1 = ylog x+2

    yx+2 = xy2

    : x = y = 1. x, y 6= 1 x, y > 0. x = y2. y

    y2+2 = y2(y2)

    y 6= 1,y2 + 2 = 2(y 2) y > 2. y = 8

    3+

    1

    3

    22. , x =

    86 + 16

    22

    9.

    (x, y) = (1, 1)

    (86 + 16

    22

    9,

    8 +

    22

    3

    )

    172. , R+,

    log[

    log(x2 + x log + 110)]

    = 0

    {ylog z + zlog y = 20logyz = 1

    : 159, 53, ylog z = xlog y, , ylog z + zlog y = 20 2zlog y = 20 log z log y = 1. z = y = 10.

    log[

    log(x2 + x log + 110)]

    = 0 log(x2 + x log + 110) = 1 x2 + x log + 110 = 10 x2 + x log + 100 = 0

    x2 + x log + 100 = 0 z = y = 0,

    z + y = log log = 20 = 1020

    173. f(x) = 3x + x.

    () 3x23x + x2 + 5 = 33x5 + 6x.

    () 3x23x + x2 + 5 < 33x5 + 6x.

    : f(x) . , f(x2 3x) = f(3x 5) x = 1 x = 5. . f(x2 3x) < f(3x 5) 1 < x < 5.

    174. :

    (2 +

    3)x22x+1

    +(

    2

    3)x22x1

    =4

    2

    3

    : (

    2

    3)1

    = 2 +

    3. x2 2x = y, (

    2 +

    3)y

    +(

    2 +

    3)y

    = 4(

    2 +

    3)2y 4(

    2 +

    3)y

    + 1 = 0

    (

    2 +

    3)y

    = . ,

    x = 1 x = 1 +

    2 x = 1

    2.

    . 56

  • 7.

    175. , , AB, , , 6= 1

    ( + )log

    =( + )

    log =( + )

    log (1)

    : = = .

    : , , ; (1) 1

    ,

    : log = + 2

    log = + 2log = + 2

    :

    log + log = log + log = log + log

    .

    176. x R e5x20 + 7x12 + 4x2

    3x14 + 9x10 + 7 x 0.

    : | x| 1 5x20 + 7x12 + 4x2

    3x14 + 9x10 + 7> 0 e

    5x20 + 7x12 + 4x2

    3x14 + 9x10 + 7 > e0 =

    1.

    177. x, y > 0 2x 5y = 7 :

    (x+ 2)(y + 1) log2 140

    log 4 log 25

    : 2x 5y = 7 2x+2 5y+1 = 140 [(x+ 2) log 2 + (y + 1) log 5

    ]2= log2 140 (1)

    (+ )2 4 , .

    178. 10(x+1)(3x+4) 2 10(x+1)(x+2) = 101xx2

    .

    : 101xx2

    103 102(2x2+2x) 20 102x2+2x 1 = 0. ,

    1 +

    11

    100. x = 1

    1

    2log(

    1 +

    11).

    179. N :2x

    2+x2 2x24 = 992 (1)

    : (1) 2x2

    (4 2x 1) = 29 31. : {

    2x2

    = 29

    4 2x 1 = 31

    ; x = 3.

    180. p(x) = x3 + x2 + x+ 3 + 1. () p(x) (x+ ).

    57 .

  • 7.

    () R p(x) (x+ ) .

    () () .

    () , :

    4x+ 9x = 34x (7.2)

    : :() = 2 + 1, = 12

    7.2 :

    4x+12 9x 12 = 32x 6 22x 32x = 3 32x

    6 22x = 4 32x

    (

    2

    3

    )2x=

    2

    3

    x = 12

    181. p(x) = (2 ln 1)x3 + x2 (e 2)x 2e.

    () 2 R.

    () p(x) < 0.

    () x f(x) = e2x (e 2)ex 2e xx.

    () f(x).

    7.2: 181. : = e1/2. p(x) < 0

    x (

    1

    2e 1 1

    2

    e2 + 4e+ 4,

    1

    2e 1 + 1

    2

    e2 + 4e+ 4

    )

    f(x) = e2x (e 2)ex 2e xx

    x (, ln

    (1

    2e 1 + 1

    2

    e2 + 4e+ 4

    )) f(x) x = ln(2) + ln(e 2)

    14e2 e 1

    . 58

  • 7.

    182. p(x) =(1 e2 +1

    )x4 + x3 ln2 + x2 ln + x 1. (0, ),

    3 x 1.

    () =2

    3 =

    1

    e.

    () p(x) x2.() () : y = 5x 5 p(x).

    : p(x)x2 x 1. p(x) (5x 5) (x 1)(x 2)(x+ 2).

    7.3: 182.

    183. : f(x) = e2 ln x x2 ln + 2 ln. f(x) (x 1) 1,

    () f(x) = x2 x+ 1, x > 0.

    () 4 = 2f( 2) 4 2 1 (

    0,

    4

    ).

    () f( 2) = 1 + 4

    2,

    (0,

    4

    ).

    : : f(1) = 1 =e. f(x) =

    e : f(x) = x2 x + 1.

    184, . , 184, : =

    6.

    184. f(x) = log(2 25x 5 4x) () : y = x+ log 3.

    () f(x).

    () () f(x).

    () A =(

    3

    2, f

    (3

    2

    )) ()

    3

    2.

    : (

    1

    2,+

    ).

    , :

    log(2 25x 5 4x) = x+ log(3) log(2 52x 5 22x) = x log(10) + log(3) log(2 52x 5 22x) = log((5 2) (2 5)x) 2 52x + 5x 2x+1 = 2x 5x+1 + 5 22x 25 =

    (25

    )x x = 1

    59 .

  • 7.

    7.4: 184.

    184, f(

    3

    2

    )

    3

    2+ log 3. 7.4, f

    (3

    2

    )>

    3

    2+ log 3.

    185. () : log = log.

    () 52 log x < 5 + 4xlog 5.

    () f(x) = 100log(x) + xlog(100) x. : f( ) = 2 2

    2,

    [0,

    2

    ).

    () : {(2x)log(y) + ylog(2x) = 8x2y

    4x2= ylog(2x)

    : . . . , , : y2 < 4y+5 1 < y < 5, y = xlog 5. , x < 10. 185, : f(x) = 2x2 x. =

    4.

    ylog(2x) = 4x2. :{y = y2 log(2x)

    16x4 = y {x = 1

    2, y = 1} {x = 1

    2

    10, y = 100}.

    186. f(x) = (lnx2).

    () .

    () f(1) = 2 ().

    () f(x) = 0.

    : f(x) 0, 7.5. 0 , . . Df = R. 2 = . x =

    e+

    2 .

    187. p(x) = lnx3 + 4x2 ln + 24 q(x) = ex3 + x2 ln( e) + 24.

    . 60

  • 7.

    7.5: 186.

    () .

    () p(x) ;

    () x p(x) : y = 12ex.

    7.6: 187. : p(x) q(x) = e, p(x) = ex3 + 2x2 + 24, 7.6, (3).

    p(x) < 12ex x < 2e. 7.6, p(x)+(12ex)

    x x , (3) (2).

    188. :

    ) x < 11 log(x) ) 3x + log(x) 3

    : ) f(x) = x 11 + log(x) x R+. f(x) , x1 < x2 f(x1) < f(x2). f(10) = 10 11 + log(10) = 0. , x < 10 f(x) < f(10) x 11 + log(x) < 0 x < 11 log(x). 0 < x < 10. ).

    189. f(x) = x3 + log(x).

    () f(x) .

    61 .

  • 7.

    () :

    (2x2 + 1)3 (x2 + x+ 3)3 log x2 + x+ 3

    2x2 + 1(7.3)

    :

    () 0 < x1 < x2 0 < x31 + log(x1) < x32 + log(x2) f(x)

    () 7.3 :(2x2 + 1)3 + log(2x2 + 1) (x2 + x+ 3)3 + log(x2 + x+ 3)f(2x2 + 1) f(x2 + x+ 3)2x2 x2 + x+ 3x2 x 2 01 x 2

    , x [1, 2] x (0,+) x (0, 2].

    190. log2 3 log3 5. :

    3

    2.

    log2 3 3

    2. 2x x2 x > 0.

    .

    (2log2 3

    )2= 32 = 9 > 8 = 23 =

    (2

    32

    )2, log3 5

    3

    2. 2x x2 x > 0

    (3log3 5

    )2= 52 = 25 < 27 = 33 =

    (3

    32

    )2, log2 3 > log3 5.

    191. p(x) = x3 ( + 3)x2 + 8x 2, R. p(x) (x = 1) 18,

    () ,

    () = 2

    i. p(x) (x2 + 1).ii. p(x) 7x+ 1.

    192. A = ln(x2 6x+ 9).

    () x R A.() A = 4 ln2 |x 3|.

    . 62

  • 7.

    () x R

    (1

    9

    )14

    (A2 A ln 4) ex2 + ex+1 + ln e2

    63 .

  • 7.

    () 2ef(6)2f(4) ex < e2x

    199. f(x) ={x3 x2 5x+ 6 x < 1x 1 + ln x x 1 .

    () f(1) = f(2) .() = 2

    i. f(x) 0 x 1.ii. f(x) = 0.iii. f(x) < 0 x < 1.

    iv. f

    ((12

    )x)= x ln 2 x 0.

    v. f( x) x+ 1 = 0, x [0, ].

    200. f(x) = log(

    x2 + 1 x).

    () .

    () f .

    () y = 2.

    . 64