Ασκήσεις Άλγεβρας B΄Λυκείου 2015-6
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Transcript of Ασκήσεις Άλγεβρας B΄Λυκείου 2015-6
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c 20
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pi
. 2015-2016
11 2015
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pi . pi pi pi. pi, pi pi pi .
email: [email protected] pi , 11 2015
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1 pi pi 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 pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 pi 55
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c 2015
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1
pi pi
1.1
1.1.1 f(x) = x2
f(x) = x2, R, (. 4.1). pi. pi .
1.1: f(x) = x2.
> 0 () (x, y) = (0, 0)
< 0 () (x, y) = (0, 0)
y = x2 pi . pi (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
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1. 1.1.
1.1.3
2 pi pi . f(x) = x2 + x+ , 6= 0:
1. f(x) Cf pi -.
2. < 0 < 0 Cf pi .
3. Cf (
2,
4
).
4. < 0, Cf Ox Oy.
5. Cf x = 2
.
3 pi y = x2 x 2. :1. pi,
2. ,
3. pi .
4 - f(x) = x2 + x+ , < 0 pipi pi pi- pi -pipi f(x).
-3 -2 -1 1 2 3 4 5 6 7 8 9
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-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. pi pi pi 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.
pi A M1 M2 : AM1 = AM2 = OB = . B, OM1 OM2 N1 N2 , 4.7.
1.5: 5.
pi N1 N2 x pi y = 1
2x2 y = 1
2x2.
. 6
-
1. 1.1.
6. 4AB , AB = 8cm H = 6cm. M pi AB, x = AM f(x) pi, 4.9. [0, 8] :
f(x) =
3
4x2 0 x 4
24 34
(8 x)2 4 < x 8
1.6: 6.
7. pi pi pi .
() pi p(x) = x2 + x+ , ( 6= 0).i. pi
,
p() < 0 (1.1)
ii. p(x) = 0, : p() = 0.iii. 1 < 2 pi p(x) = 0,
< 1 < 2,(1 < 2 <
), :
> 0, p() > 0, < 2,
( >
2
)(1.2)
() pi pi pipi. pi 2 :
p(x) = x2 (+ 1)x+ 3 = 0 (1.3)
7 .
-
1.1. 1.
i. Geogebra p(x) pi .ii. pi pi pi pi-
.iii. pi pi pi pi
pi 7.
8. pi pi
p(x) = (+ 1)x2 4x+ 2+ 3 (1.4)() Geogebra.() 1 pi p(x)
.() pi pi pi .
9. pi pi pi 1 < 2 pi .
() pi :
p(x) = x2 + x+ , (a 6= 0) (1.5) 1 2 pi pi pi pipi:
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. pi
p(x) = ( 1)x2 2(3+ 1)x+ 9, 6= 1 (1.6)
Geogebra.ii. , pi ,
pi pi 1 0 pi p(x) = 0.
iii. pi pi pi 9.
10. pi, pi , :
() (x 1)(x+ 2) + (x+ 1)(x 2) (x 1)(x 2) = 0
pi pi (1, 2).()
A2
x a +B2
x = 2, (x 6= , )
pi .
11.
x 4x2 3x 3
pi : x 4x2 3x 3 =
x2 (3+ 1)x 3+ 4
9 .
-
1.1. 1.
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c 20
15
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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. pi (3, 2) {4x 5y = 2x+ 3y = 3 ,
{2x 3y = 0
x = 1, 5y
11
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2.1. 2.
18. pi a, b, c, a, bc {() : ax+ by = c() : ax+ by = c
() pi A(2, 1) B(2,1) () pi C(1,3) D(1,2).
19. pi 0o < < 90o:{( () ())x+ ( () + ()) y = 1( () + ())x ( () ()) y = 1
. (a, b) , a b pi pi pi 1 .
20. {
(+ 3)x+ y = 5x+ y = 1 3 < < 5. x0 y0
, 163< x0 < 9 y0 < 0.
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c 20
15
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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 = 344x2 25y2 = 952
23. x , pi 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 = 64pi : pipi pi pi .
24. pi :4
x y y
(x y)2 = 0
1 x2 + y2
x2 y2 = 0pi : , pi pi . pi 0 0. , pi .
25. :
x2 (2 1)x 3 = 0 (1) x2 ( 2)x+ 3 = 0 (2) 6= 1. , :() ,() .
pi : pi pi pi pi . pi Geogebra pi , , pi , ., :{
x2 (2 1)x 3 = yx2 ( 2)x+ 3 = y
{x = 3y = 3 + 6
pi pi , A . pi pi A 0, xx. pi = 1
2.
13 .
-
2.2. 2.
pi 6= 1, Geogebra, = 1 pi .
26. :{
5x y = 1x2 + 4xy 2y2 + 8x+ 39 = 0 .
pi : y pi pi .
27. :{x+ y + xy = 41xy(x+ y) = 330
pi : pi
. {x+ y = xy =
. {+ = 41 = 330
.
. (x, y) = (5, 6), (15214, 15 +214)
. 14
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c 20
15
.
3
3.1 -
28. pi : f(x) = x7 3 6x.29. f(x) = ||x 3(x+ 1).
() Geogebra pi pi . pi ;
() f(x) pi pi- .
30. h R, : h(x) =h(2x 3) h(x3) h(27) = 0.pi : :
f(x1) = f(x2) x1 = x2
31. f, g : R R . :h(x) = f
(g(x)
) .
32. f R :
f
(2
pi
)< f
( pi3
), f(2) f(2 + 1)
33. f R :
i. f(
6 3) > f(
32)
ii. f(
4
34)> f(
6)
iii. f(2x2 + 2013) < f(2012) iv. f(2 + ) > f( + 2)
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3.1. - 3.
34. f(x) = x2 +x 1.
() pi f .() f .() pi .() f(10) :
i. f(x) = 103ii. f(x) < 103
() f(2012
2011
) 1 > 0.
35. f(x) =
4 x2 + 2x
.
() pi f .() Geogebra.() pi ;() Cf O(0, 0);
36. f(x) =
x2013, x < 0
x2013, x 0. Cf
yy.
37. f pi pi , R pi, g(x) = |f(x)| .
38. f, g : A R pi f pi g . h(x) = f(x) g(x) pi.
39. f : R R pi : f(x+y) = f(x)+f(y), x, y R.() f(0).() f pi.
40. f : R R pi pi. g(x) =f(|x|) |f(x)|, .
41. f : R R. :() g(x) = f(a+ x) f(a x) pi,() h(x) = f(b x) + f(b+ x) .
42. f : R R pi :f(x) f(x) = [f(x)]2
x R. f .
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3. 3.1. -
43. pi f : N N pi pi pi - :
f(1) = 1 m n,
f(m+ n) = f(m) f(n) + f(n) + f(m)
() pi f pi.i. pi f(0).ii. pi 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 pi pi pi.
44. f pi pi :
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. pi f (0,+), pi pi pi :
x f(x)2 3, 01033 4, 771245 6, 98976 7, 78157 8, 451089101001000
1000000109
: pi x > 0 y > 0 :
f(x y) = f(x) + f(y)
() f(4) pi pi pi-.
() pi f(245).
() pi f(1).
() x > 0:
f
(1
x
)= f(x)
46. f pi f(x + y) = f(x) + f(y), pi x, y. pi :
17 .
-
3.2. 3.
() f(0) = 0() f pi.() f(x) = f(x), N
47. f(x) 1 4, , f(x) = 2x+
x2 + 1.
3.2 pi
48. pi f(x) = x2 x2 x 2 = 0.
49. 3.1 pi f(x) = (x) g(x) = (x).pi pi pi pi ;
3.1: pi pi pi .
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4
4.1
50. pipi pi :
2(1 + 2), 2(1 + 2), 2 2 + 2 2
51. pi :
+ 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
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4.1. 4.
() 1 +
1 = + 1
() 3 2 + 2 2 = 3 + 22
1 + 2
() 2 2 = 1 2
1 + 2
+ 1 +
1 = 22
() 2 2 = 2 2 = 11 + 2
1
1 + 2() 2 2 2 2 = 2 2() + + +
= 0
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15
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4. 4.2. &
4.2 &
53. :
() + (270o + ) + (180o + ) (270o ) = 0() (171o + ) (9o + ) = 0() (203o + ) + (23o + ) = 0() (138o + 2 ) + (48o + 2 ) = 0() (280o + ) + (10o + ) = 0()
(3pi2
) (pi )
() (pi )
(3pi2
+ )
(pi
2
) (pi + )
() (pi + )
(3pi2
)
(3pi2
+ )
(2pi + )() + + = 180o ( + ) = ()() + + = 180o ( + ) = ()
() + + = 180o () = (2 + + )() + + = 180o () = (2 + + )() + + = 180o +
2=
2
() + + = 180o 2
= + 22
= + 22
() + + = 180o + 2
= ()
54. = 2 22 + 2
, , R.
55. pi pi pi pi7
R.pi : pi 13 pi :
0o, pi7, 2pi
7, 3pi
7, 4pi
7, 5pi
7, 6pi
7
pi . 7.
56. 1050o.pi : 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.pi : |2 x+ 3 y| |2 x|+ |3 y| 2 + 3 = 5.
59. pi 12.
pi :( +
)2 0 2 + 2 2 1
2 .
60. | |+ | | = | + |.pi : pi = 1 > 0 pi pi pi |a+ b| = |a|+ |b| a b > 0.
61. pi2< x 0
pi T = pi 3, :
() .() f x pi f pi
.() pipi pi
f yy.
pi :
4.4: 181.
. 24
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4. 4.2. &
() pi 2pi2
= pi = 1 2 2 +
2= 3 = 5.
() 0 1 + (2x) 2 5(1 + (2x)2
2. , 2 x pi , 1 = (2x). , x = kpi
2 k
pi, 4.4.
() x = 0 5(1 + (2 0))2
2 = 3. , yy (0, 3), 4.4.
25 .
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c 20
15
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4.3. 4.
4.3
68. : 2( + ) + 2 2 ( + ) = 2.69. : 2 + 2 + 2 ( + ) = 2( + ).70. : (+ ) ( ) + ( +) ( ) + ( + ) ( ) = 0.71. : ( + ) ( ) + ( ) ( + ) = 2.72. pi pi: ( + + = pi) ( = 2 ) = 73. :
() (45o + ) (45o ) = 2 2() 2 = 1
1 + 2() = 2 2() 78o = 12o + 2 24o + 4 42o
74. 2 x+ x 2.pi : : x+ x =
2
(x+
pi
4
).
75. pi 90o, .
76. pi ( ) + 2 2 = 1
2 = .
77. pi
2x 2 x (x+ ) + 2(x+ )
x.
78. + = , 2 + 2 2 = 279. pi 2x + 2(120o + x) + 2(120o x)
x.
80. pipi pi 3 4 2x+ 4x.
81. pi (
2
2
)2=
1
1 2 282. () x+ y , pi :
1 2x 2y
pi : pi 2x+ 2y2
.
. 26
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4. 4.3.
83. () pi : 2 (1 2), = 45o.pi : pi , ,
2 (1 2) = = 2 (1 )(1 + ) (1 + 2)
27 .
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c 20
15
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4.4. 4.
4.4
84. : x =
2
2.
pi :x = 2kpi pi4.
85. : x =
6
2.
pi : ,
6
2= 1.224744872 < 1.
86. : (1 x)(2 x3) = 0.pi : x = 1 x =
3
2= pi
3.
87. : (1 x)(2 x
2 +
2
4) = 0.
pi : : x = 1 = pi2
x =
2 +
2
4 x = 0.480181
(86.4326pi
)o.
pi /.
88. : (5x+ 80o) = (3x 40o).pi : x = 60o + 1 180o x = 17o30 + 1 45o.
89. :
() (2x+ 12o) + 5x = 0.
() (
3x pi3
)+
(x pi
6
).
() 2 2x+
3 x+ 1 = 0.90. : 4 2x 2(1 +3) x+3 = 0.
pi : x = 12
x =
3
2, x.
91. : 11 + 2x +
2x 2 x = 0.pi : x = pi
3+ 2pi.
92. : (
3 1) 2x (1 +3) x x+ 1 = 0.pi : 2x
[3 (1 +3) x+ 2x
]= 0 x = pi
4+ 1pi, x =
pi
3+ 2pi.
93. pi pipi pi , x + x = .pi :
x+ x = (1 2x
)= x
2(1 2x
)2= ( x)2
(2 + 2) 2x 2 x+ 2 2 = 0
. 28
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4. 4.4.
0 2(2 + 2 2) 0 2 + 2 2
94. :
3 x+ x = 1.pi : x+ 1
3 x = 1
3 x+ 30o x = 1
3 (x+ 30o) = 30o . . . .
pi pi x2
= y x = 2y1 + y2
x = 1 y2
1 + y2. :
3 x+ x = 1
3
2y
1 + y2+
1 y21 + y2
= 1
: y = 0,
3. pi y = x2
x...
95. :
() x+ x = 1.() x (+ 1) x = , pi .
96. () : 3x+ 3x = 1.pi :
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(pi),2(pipi)
(x+
pi
4
)=
12
{
x = 21pi, 1 Zx =
pi
2+ 22pi, 2 Z
97. :
() x+ x+ x x+ 1 = 0.() ( x+ x) = x x, pi .() x x x+ x 1 = 0
98. : 3 x x = 0.pi : pi y = x y = x
3,
. pi , (0, 0).
99. :
() 4 2x 3x 2 = 0.
29 .
-
4.4. 4.
4.5: y = x y = x3.
() x = x( x+ x).pi : x = x( x+ x) 1
x=
x+ x x . . . .
100. : x > .pi : pi
< x < pi a: + 2pi < x < pi + 2pi, Z.
4.6: 100.
101. () : (pi x
)=
(pi x
).
pi : pi :
pi x = 21pi +pi
2 pi x (4.3)
pi x = (22 + 1)pi pi2 pi x (4.4)
1, 2 Z. pi 4.3 : x+ x = 21 +
1
2(4.5)
4.5 pipi, 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)
pi 4.6 4.7.
102. () : 2y2 + y 1 = 0() : 2 2x+ x 1 = 0.
pi : pi : y = 1, 12. : x = pi
2 pi
6.
103. ()() p(x) = x2 + x + . pi = 2 4 0.
p(x) = 0:i. [1, 1], p(1)p(1) < 0.ii. 1 (1, 1], pi
,
+ = 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
pi :
() i. pi 1 < 2 1 < 1 < 1 < 2 1 < 1 < 2 < 1, p(1) p(1) pi, p(1)p(1) < 0.
ii. p(1) = 0, + = 0. =
1 < < 1
0 (, 23
).
- p(1) = 6 > 0, p(1) = 4 2 > 0 (12 ,
). pi, 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 pi
2 2 > 0. < 12 p(x) = 0 [1, 1] p(1)p(1) < 0. pi . 12 < 0 p(1) > 0.
pi pi
1, pi (1 + 2) (1) = 2 +
=2( 1)
, pi 12 < 0 ( pi2 , pi2 ), pi pi
( pi2 , pi2 ).
pi, ( x) ( x) , yy, pi pi
(0, pi2
).
, ( x) = ( x) ( x) = (pi2 x) x+ x = pi2 (0, pi2 ). x x = 12 + pi
2
8 . pi x+ x , x x pi x = x = 45o =
22
12 . :
x x < 12
+pi2
8
, pi x : ( x) = ( x). pi . pi ,. 6. pi pi pi pi I =
(0, pi2
).
( x) < ( x)
33 .
-
4.4. 4.
4.9: ( x) = ( x).
pi x I | x| < |x| pi pi x , , pi x. ,
x < x, x I (4.9)
pi x, x (0, 1) I. x I, :
x < x ( x) > x (4.10)
pi, pi x x 4.9, pi
( x) < x (4.11)
, pi 4.10 4.11, pi : ( x) < x < ( x).pi .
. 34
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c 20
15
.
4. 4.5.
4.5
107. pi A, 4.10, d pi pi E pi 15o. B, pi pi pi Api 1 km, pi pi 30o, pi d.pi : pi EA1 EB1 = 1 km, pi EA1 = d
15o EB1 =
d
30o.
pi 15o.
4.10: 107
108. pi 4.11.
() pi 1 + 2 + 3.
4.11: 108
() pi 4.11, n 3o!
pi : pi, 1 = 45o, pi (2+ 3). 2+ 3 = 45o,pi 90o. . . .
35 .
-
4.5. 4.
. 36
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c 20
15
.
5
5.1 &
109. pi g(x) 6= 0 pi pi p1(x), p2(x), . . . , pn(x),n N, pi:
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. pi , p(x), pi pi :
p(p(x)
)= x2p(x) xp(x) + 1
pi : p(x) = x2 + x+ , 6= 0, :
(x2 + x+ )2 + (x2 + x+ ) + = x2(x2 + x+ ) x(x2 + x+ ) + 1 . . .
pi pi pi- pi . p(x) = x2 + x+ 1.
112. pi p(x) = x3 + x2 + x+
pi pi {2 = 33 = 272
.
pi : pi pi pi. pi pi p(x) = 3
(x +
)3, pi ; pi pi. , ;
37
-
5.1. & 5.
113. pi f(x) = p(x)q(x)
, q(x) 6 0, pi
p(x) = x + 1x1 + + 0 (5.3)
q(x) = x + 1x1 + + 0 (5.4)
i 6= 0. .pi : f(x) = c R, :
00
=11
= = 11
=
= c
114. pi p(x), , pi : p(x) p(x 1) = x3 p(0) = 0. pi S3 = 13 + 23 + + 3.pi : p(x) = x
4
4+x3
3+x2
2. pi
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. pi , , , :
() 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 .
pi :
() = 1, = 1, = 1, = 1.() = 3, = 2, = 1.
116. pi1(x) 1(x) pi pipi pi p(x) pi q(x) 6 0 pi2(x), 2(x) pi pipi pi pi1(x) h(x) 6 0. pi pipi pi p(x) q(x) h(x). pi , p(x) q(x) h(x) 1(x) 2(x) 0.pi : pi . pi pi pi , pipi p(x) q(x)h(x). pi pi.
117. pi p(x) q(x) = x2 + x+ 1 h(x) = x2 x+ 1 pipi 1(x) = x 1 2(x) = 2x + 5. pipi (x)
. 38
-
5. 5.1. &
pi p(x) q(x) h(x) = x4 + x2 + 1.pi :
f(x) = q(x)pi1(x) + x 1 (1) f(x) = h(x)pi3(x) + 2x+ 5 (3)pi1(x) = h(x)pi2(x) + x+ (2) pi3(x) = h(x)pi4(x) + x+ (4)
:f(x) =
[q(x) h(x)
]pi2(x) +
[q(x) (x+ ) + x 1
](5.5)
f(x) =[q(x) h(x)
]pi4(x) +
[h(x) (x+ ) + 2x+ 5
](5.6)
5.5 5.6, p(x) q(x) h(x) = x4 + x2 + 1. , pi pipi pipi pipi , :
q(x) (x+ ) + x 1 h(x) (x+ ) + 2x+ 5 (5.7)
q(x) h(x) , pi 5.7 = 3, = 72, = 3, = 5
2.
118. pi p(x) +1 pi pi , pi.pi : 1, 2, . . . , +1 pi p(x) = 0, pi. p(x) 6= 0pi . 6= 0 :
p(x) = (x 1)(x 2) (x +1)
do(p(x)) = + 1, .
119. pi:
p(x) =(x )(x )( )( ) +
(x )(x )( )( ) +
(x )(x )( )( )
6= 6= 6= . p(x) 1.pi : pi P (x) = p(x) 1. P () = P () = P () = 0 do(P(x)) = 2. pi 118, P (x) = 0 p(x) = 1.
120. pi p(x) = (x )2( ) + (x )2( ) + (x )2( ) + ( )( )( ), 6= 6= 6= , pi.
121. p(x) = xn n q(x) = xm m, 6= 0, m = pi.n.pi : m = n pi + v, 0 v < n, q(x) = xm m = (xn)pixv (n)piv. v = 0. pipi q(x) p(x). y = xn, p(x) = y n p(x) =ypixv (n)piv. pipi U(x) = (n)pixv (n)piv. ,U(x) = 0 (n)pixv (n)piv = 0 xv v = 0
(x
)v= 1 v = 0. , m = pi.n. ,
m = kn pi p(x)/q(x).
122. pi p(x) = nxn + n1xn1 + + 1x + 0, n 6= 0, 1 = +
, , Q Qa, ,
2 = .
pi : q(x) = (x 1)(x 2) = x2 2x + 2 . p(x) = q(x)pi(x) + x + . 1 q(x) = 0, p(1) = q(1) = 0. pi
39 .
-
5.1. & 5.
pipi (+ ) + = 0
{+ = 0
= 0
{ = 0 = 0
. , p(x) = q(x)pi(x). pi
2 p(x) = 0 q(2) = 0 = p(2).
123. pi pi pi p(x) = x3 x2 (3 +)x+ + 10 pi (x 2)2.pi : = 5, = 2.
124. pi p(x) = x3 3x+ 2, , R. , R (x)2 pi p(x). 3 = 2.pi : Horner p(x)/(x ), pi
p(x) = (x )(x2 + x+ 2 3 pi(x)
) + 3 3+ 2
: p() = 0 3 3 + 2 = 0 pi() = 0 32 3 = 0. pi pi3 = 2.
125. pi p(x) 2p(x) + p(2 x) = x2 1, () p(0) p(2),() pipi p(x) x2 2x
pi :
() p(0) = 1, p(2) = 3.() (x) = 2x+ 1.
126. pi p(x) = x5 2x3 + x+ 3 6. pi , , p(x) pi x3 x. pi pi .pi : = 12 , = 0, = 2. x
2.
127. 6= pipi pi p(x) (x )(x ) :
(x) =p() p() x+
p() p()
128. pi p(x) = x + 1x1 + + 1x + a0 pi % p(0) = 0, % p
(p(p(x)
)).
129. () 1 pi pi pi - : pi pi pi . pi pi p(x) = x3 +x2 +x+, 6= 0, pi 2 < 0, p(x) = 0 pi .
1 pi Bolzano . .
. 40
-
5. 5.1. &
pi : pi pi, pi R. (x )/p(x). , Horner pi, pi pi pi:
pi(x) = x2 + (+ )x+ 2 + + (5.8)
pi 5.8 pi . pi x
d(x) = 322 2 4 + 2 (5.9) d(x) pi pi ,
d1(x) = 162(3 2) (5.10) pi d1(x) pi 2 < 0, pi (pi), pi d(x) , 32 < 0. pi, pi(x), 5.8, pi . pi p(x) pi .
130. () pi p(x) ( pi ) pi p(0) p(1) pi , p(x) = 0 .pi : Z, (x )/p(x) pi pi pi pi(x) :
p(x) = pi(x) (x ) (5.11)
x = 05.11 p(0) = pi(0) (5.12)
x = 15.11 p(1) = (1 )pi(1) (5.13)
pi pi pi(0) (1 )pi(1) pi. , , 1 ,pi(0), pi(1) pi (pi). pi 1 pi pi (pi). , p(x) = 0.
131. () pi pi , , , pip(x) = x3 + x2 + x+ 1 x = 19 2 x = 62.pi :
623 + 622 + 62 + (193 + 192 + 19 + ) = 2 1 = 1 .
132. ()pi pi pi ;pi : p(x) = nxn + n1xn1 + + a1x+ a0, n N pi pi pi T 6= 0. p(x) 0. p(x) pi , p(x) = p(x + T ). x x = 0, T, 2T, 3T, . . . , kT, . . . ,k > n, p(0) = p(T ) = p(2T ) = = p(kT ) = . . . , pi. 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. . .
, pi f(x) n0 pi . pi 118 f(x) 0 p(x) = p(0). p(x) pi pi.
41 .
-
5.1. & 5.
133. () 2 % pi p(x) = x+1x1+ +1x+0 :
|%| < 1 + |a1|+ |a2|+ + |1|+ |0|pi : || = 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) pi % pi
%3 + 2%2 + 1%+ 0 = 0
i R |i| < 3, pi pi |%| < 4.pi :
%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 , pi pi .
pi pi 1, . pi pi , pi pi pi 6= 0. pi .
. 42
-
5. 5.1. &
() |%| 1,|%|3 < 3 |%|
3
|%| 1 31
|%| 1 < 3|%|3|%| 1 , |%|
3 < 3|%|3|%| 1 |%| < 4, pi.
:
%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
pi pi pi : |%| < 4.
135. % x2 + x+ c = 0, 6= 0, , , R, :
|%| < ||+ ||+ ||||pi : pi pi pi pi. pi .
||+ ||+ |||| = 1 +
+ (5.16)
5.16 2 . ,
1 +
+ = 1 + |%+ %|+ |% %|
pi :1 + |%+ %|+ |% %| > |%| (5.17)
pi,
() |%| < 1, 5.17 pi.() |%| 1,
1 + |%+ %|+ |% %| 1 + |%| |%|+ |% %|= 1 + |%|+ |%|(|%| 1) 1 + |%| |%|(|%| 1) > 0> |%|
pi 5.17.
43 .
-
c 20
15
.
5.2. 5.
5.2
136. pi x :
()x 3 = 5
() x25 x2 = 1()xx+ 1 = 2
() 2
5 4x = 5 4x()x2 x+ 5 = x 3
137. pi x :
()x 2 < 2x+ 1
()
4x+ 1 5x+ 9 (x+ 1)(x2 4x+ 9) > 0 x < 1.
141. p(x) = x3 + x2 2x.() p(x) pi pi.() pi x
p(x) =
x 1.
() pi x p(x) >
x 1.
pi :
() p(x) = x(x 1)(x 2)() x = 1, 12.() x (1,+).
142. () pi x x+
2(x+ 1) = 11.() pi x x+
2(x+ 1) < 11.
pi :
. 44
-
5. 5.2.
() x = 7.
() x (1, 7).
143. pi 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 = 0pi : 1 . pi.
() x = 0, 12pi.
() x = 12pi.
() x = 0, pi, 13pi.
144. pi x: x3 ( + 2)x2 + 2x 1 = 0, , R pi x.pi : p(x) = x3 (+ 2)x2 + 2x 1.
() p(1) = 0 = 43
p(x) = x3 23x2 8
3x 1 x = 1, 5 +
61
6,
5616
.
() p(1) = 0 = 2, p(x) = x3 4x2 + 4x 1 x = 1, 3 +
5
2,
352
.
145. pi x : x3 2x2 + 22x+ 60 = 0.pi : x3 2x2 + 22x+ 60 = (x+ 2)(x2 4x+ 10). x = 2.
146. pi x 7x3 13x2 + 3x+ 3 < 0.pi : 7x3 13x2 + 3x+ 3 = (x 1)(7x2 6x 3) = (x 1)
(x 3 +
30
7
)(x 3
30
7
).
, pi pi :
x (, 3
30
7
)(
1,3 +
30
7
)
147. p(x) = x154x14+2x13+2015, pi pi p(2 +
2) + p(22)2
.pi : p(x) = x13(x2 4x + 2) + 2015 x2 4x + 2 = 0, x = 2 2. ,p(2 +
2) + p(22)
2= 2015.
45 .
-
5.2. 5.
148. pi x : (x 1)4 + (x 5)4 = 32.pi : x 1 + x 5
2= y, : x 1 = y + 2 x 5 = y 2. ,
(y + 2)4 + (y 2)4 = 32 y4 + 12y2 = 0
pi : y = 0, pi x = 3 pi.
149. pi x x2 + x = x .
pi : 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.
pi:
= 0 x x 0 > 0 < 0 x =
3
150. pi x x2 3x+ 2x2 + x 6 = x 2.
pi : pi :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 1x+ 3x 2
]= 0 (5.18)
i. 5.18 x = 2.ii.
x 1x+ 3x 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 23
3
() x 3 : 2 x
[1 xx 32 x
]= 0 (5.19)
i. 5.19 x = 2 x 3.ii.
1 xx 32 x = 0 ( ). ,
1 x = x 3 +2 x 2 + x = 2(x 2)(x+ 3)
x 3, 2 + x < 0 ( )
. 46
-
5. 5.2.
pi: x = 2.
151. pi x x4 2x3 10x2 + 4x+ 16
x 4 > 2x+ 1.
pi : x ( 1 +
21
2,1
)(1 +21
2, 4) (4,+).
47 .
-
c 20
15
.
5.3. 5.
5.3
pi Euler
5.3.1 pi pi pi x, pi pi ;
fung.ggb. x pi pi pi .
5.1: pi Fung pi pi Euler, p(n) = n2+n+41. n = 0, . . . , 39 pi . pi 40, 40 pi pi . pipi pi pi pi pi. pi Euler, pi p(n) = n2 +n+41 = n(n+1)+41. , pi. p(n) = n2 +n+ pi. pi pi pi - n = 0, 1, 2, . . . , pi pi:
() pi.() pipi pi p(1) pi ;
pi :
() pi, p(0) = pipi pi.
() pi, pi. p(1) = 1 + + . pipi pi . , 1 + pi 1 + + . p(1) pi, pipipi .
. 48
-
5. 5.3.
= 3 = 1. pi.
http : //blog.lucaswillems.com/1144/project euler probleme 27
pi Python.pi pi Euler , pipi, pi. pi: p(x) = (x 41)2 + (x 41) + 41 =x281x+1681 pi pi 41 , pi 5.2.
5.2: pi Eulerpi pi - . Maple pi pi pi .
49 .
-
5.3. 5.
. 50
-
c 20
15
.
6
&
6.1
152. pi x : 3 2x4 2x1 = 5x2 6 5x3.pi : x = 4.
153. : (x2 3x+ 2)x22x = 1.pi : 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
:
() pi f yy.() pi f xx .
pi :
() pi Df = R, pi .()
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. pi 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)
() pi f(x) pi - x pi pi . ( pi pi (x, x + 3), (x +3, x+ 6), (x+ 12, x+ 15) pi)
() 6.1 pi pi-.pi 155, pi pi -.
6.1:
6.2
156. ln i = R tL
+ln I i = I eR tL . pi
.
157. > > 0 2 + 2 = 11, :
ln
3=
1
2
(ln + ln
) . 52
-
6. & 6.2.
pi : 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)
pi : : 3 + 2
2 = (
2 + 1)2.
159. xlog y = ylog x. pi .
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 = , pi pi- :
() log 5
7, 2
() log 5
5
34
6
pi : :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. pi x2 2(1 + log )x+ 1 (log )2 = 0.pi : = 2 log (log + 1) = 0 = 1, 1
10.
164. x2 2(1 + ln )x+ 1 ln2 = 0
pi x R > 0. pi () pi()
pi :
() = ln (ln + 1) 0 ln 1, 0.() 0 P > 0, 1 < e
.
165. log(x2y2) = log x log y = log x log y . pi .pi : log x = + 3
5, log y = 2
5.
166. pi pi x :xlog
x = 10.
pi : pi x > 0. log x = 4.
167. : log(2x + 2 3x) + log 81 = x log 3 + log 178.pi : x = 4.
. 54
-
c 20
15
.
7
pi
168. x > 0 y > 0 xx yy xy yx.pi : x = y . pipi pi x 6= y. pipi (x y) (log x log y) 0 pi .
169. : f(x) = log(11x2 7x+ 10) log x2 1.
() pi f xx .
() A = (0, 4) B, pipi xx AB.
pi : , pi pi, pi . xx x = 2 5 f(x) = 0, 7.1. 6 ...
x
1 2 3 4 5 6 7 8
y
K0,01
0
0,01
0,02
0,03
7.1: 169.
170. pi pi x: x = , , , R+ 6= 1, 6= 1.pi : :
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. pi :{xlog y+1 = ylog x+2
yx+2 = xy2
pi : x = y = 1. x, y 6= 1 x, y > 0. pi x = y2. y
y2+2 = y2(y2)
pi y 6= 1,y2 + 2 = 2(y 2) y > 2. y = 8
3+
1
3
22. , x = 86 + 16
22
9. pi
(x, y) = (1, 1) (
86 + 16
22
9,
8 +
22
3
)
172. pi , R+, log[
log(x2 + x log + 110)]
= 0
pi {ylog z + zlog y = 20logyz = 1
pi : pi 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, pipi
z + y = log log = 20 = 1020
173. f(x) = 3x + x.
() 3x23x + x2 + 5 = 33x5 + 6x.() 3x23x + x2 + 5 < 33x5 + 6x.
pi : pi f(x) . , pi f(x2 3x) = f(3x 5) pi x = 1 x = 5. . pi f(x2 3x) < f(3x 5) pi1 < x < 5.
174. :
(2 +
3)x22x+1
+(
2
3)x22x1
=4
23pi :
(2 3
)1= 2 +
3. x2 2x = y,
(2 +
3)y
+(
2 +
3)y
= 4(
2 +
3)2y 4(
2 +
3)y
+ 1 = 0
pi pi (
2 +
3)y
= . , x = 1 x = 1 +
2 x = 1
2.
. 56
-
7.
175. , , pi AB, , , 6= 1 ( + )
log=( + )
log =( + )
log (1)
pi : = = .pi : , , pi ; (1) 1
,
: log = + 2
log = + 2log = + 2
:
log + log = log + log = log + log
pi pi .
176. x R e5x20 + 7x12 + 4x2
3x14 + 9x10 + 7 x 0.
pi : pi pi | 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 pi :
(x+ 2)(y + 1) log2 140
log 4 log 25
pi : 2x 5y = 7 2x+2 5y+1 = 140 [(x+ 2) log 2 + (y + 1) log 5
]2= log2 140 (1)
pi (+ )2 4 pi , pi .
178. 10(x+1)(3x+4) 2 10(x+1)(x+2) = 101xx2 .pi : 101xx2 103 102(2x2+2x) 20 102x2+2x 1 = 0. pi , pi 1 +
11
100. x = 1
1
2log(
1 +
11).
179. N :2x
2+x2 2x24 = 992 (1)
pi : (1) 2x2
(4 2x 1) = 29 31. : {
2x2
= 29
4 2x 1 = 31 ; x = 3.
180. pi p(x) = x3 + x2 + x+ 3 + 1. () pipi p(x) (x+ ).
57 .
-
7.
() pi R p(x) (x+ ) .() () .
() pi pipi, :
4x+ 9x = 34x (7.2)
pi : pipi :() = 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. pi p(x) = (2 ln 1)x3 + x2 (e 2)x 2e.
() pi 2 R.() p(x) < 0.
() x pi f(x) = e2x (e 2)ex 2e pi xx.
() f(x).
7.2: 181.pi : = 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 pi 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. pi p(x) =(1 e2 +1)x4 + x3 ln2 + x2 ln + x 1. (0, pi),
pi 3 x 1.
() pi = 2pi3
= 1e.
() pi pipi p(x) x2.() pi () : y = 5x 5 p(x).
pi : pi pipi p(x)x2 x 1. p(x) (5x 5) pi pi (x 1)(x 2)(x+ 2).
7.3: 182.
183. : f(x) = e2 ln x x2 ln + 2 ln. pipi f(x) (x 1) 1,() f(x) = x2 x+ 1, x > 0.() pi 4 = 2f( 2) 4 2 1
(0,pi
4
).
() f( 2) = 1 + 42
, (
0,pi
4
).
pi : : f(1) = 1 = e. f(x) = e : f(x) = x2 x + 1. 184, pi pi pi . , 184, : =
pi
6.
184. f(x) = log(2 25x 5 4x) () : y = x+ log 3.() pi f(x).() () f(x).
() A =(
3
2, f
(3
2
)) () 3
2.
pi : pi (
1
2,+
).
, pipi :
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. pi pi 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,pi
2
).
() : {(2x)log(y) + ylog(2x) = 8x2y
4x2= ylog(2x)
pi : pi . . pi . , pi pi , pi : y2 < 4y+5 1 < y < 5,pi y = xlog 5. , x < 10. 185, : f(x) = 2x2 x. = pi
4. pi
pi ylog(2x) = 4x2. :{y = y2 log(2x)
16x4 = y {x = 1
2, y = 1} {x = 1
2
10, y = 100}.
186. f(x) = (lnx2).
() pi .() f(1) = 2 ().() f(x) = 0.
pi : f(x) pi 0, 7.5. pi pi pi 0 , pi. pi . pi Df = R. 2 = pi. x =
epi+
pi2 .
187. p(x) = lnx3 + 4x2 ln + 24 q(x) = ex3 + x2 ln( e) + 24.
. 60
-
7.
7.5: 186.
() pi .() pi p(x) ;() x pi pi p(x)
pi : y = 12ex.
7.6: 187.pi : pi p(x) q(x) = e, p(x) = ex3 + 2x2 + 24, 7.6, (3). p(x) < 12ex x < 2
e. 7.6, pi pi p(x)+(12ex) pi
x pi x pi , pi (3) (2).
188. :
) x < 11 log(x) ) 3x + log(x) 3pi : ) 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). pi 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)
pi :
() 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.pi : pi 3
2.
log2 3 3
2. 2x x2 x > 0. pi
pi pi pi pi .
(2log2 3
)2= 32 = 9 > 8 = 23 =
(232
)2, log3 5
3
2. 2x x2 x > 0
(3log3 5
)2= 52 = 25 < 27 = 33 =
(332
)2, log2 3 > log3 5.
191. pi p(x) = x3 ( + 3)x2 + 8x 2, R. pipi p(x) (x = 1) 18,() ,() = 2
i. p(x) (x2 + 1).ii. p(x) 7x+ 1.
192. pi 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. pi f(x) ={x3 x2 5x+ 6 x < 1x 1 + ln x x 1 .
() f(1) = f(2) pi .() = 2
i. f(x) 0 x 1.ii. f(x) = 0.iii. f(x) < 0 x < 1.
iv. f((1
2
)x)= x ln 2 x 0.
v. f( x) x+ 1 = 0, x [0, pi].
200. f(x) = log(
x2 + 1 x).
() pi .() f pi.() pi
y = 2.
. 64
