tengsizlikni isbotlang. 
Yechilishi

tengsizlikni isbotlang.
Yechilishi.
Ravshanki, tengsizligimiz
0
x
≥
da bajariladi, shuning uchun
x
manfiy bo’lgan qiymatlarni qarash etarli.
1
x
≤ −
bo’lsin, holda
10
5
0
x
x
+
≥
,
6
3
0
x
x
+
≥
,
2
0
x
x
+ ≥
va
1
tengsizliklarni qo’shib, izlanayotgan tengsizlikni
olamiz.
0
>
Endi 1
0
x
− < <
bo’lsin. Qism hollarni qaraymiz:
a)
5
1 0
x
x
+ + >
. U holda
10
6
5
3
2
10
6
2
3
5
1
1
(
x
x
x
x
x
x
x
x
x
x
x
x
+
+
+
+
+ + =
+
+
+ +
+
+ + >
1) 0
.
b)
5
1 0
x
x
+ + ≤
. U holda
10
6
5
3
2
5
5
2
3
5
5
1
(
1) (
) (1
)
(
1) 0
x
x
x
x
x
x
x x
x
x
x
x
x x
x
+
+
+
+
+ + =
+ + +
+
+ +
>
+ + ≥
.
18-masala
. 1,
1
x
y
> −
> −
va bo’lsin.
1
z
> −
2
2
2
2
2
1
1
1
2
1
1
1
x
y
z
S
y z
z x
x y
+
+
+
=
+
+
+ +
+ +
+ +
2
≥
tengsizlikni isbotlang.
2
2
2
1
1
1
1
2
x
x
y z
y
z
+
+
≥
+ +
+
+
tengsizlik yordamida ,
x y
va
z
ning manfiy
bo’lmagan qiymatlarini qarash etarli.
2
2
2
2
2
2
2
2
1
1
1
(
)
1
1
1
1
1
1
z x
x y
y z
z
x
y
S
y z
z x
x y
y z
z x
x y
+ +
+ +
+ +
=
+
+
−
+
+
+ +
+ +
+ +
+ +
+ +
+ +
2
≥
2
2
2
3
2
2
2
2
2
1
1
1
3
(
1
1
1
1
1
1
z x
x y
y z
z
x
y
y z
z x
x y
y z
z x
x y
+ +
+ +
+ +
≥
⋅
⋅
−
+
+
+ +
+ +
+ +
+ +
+ +
+ +
2
)
≥
2
2
2
3 (
).
1
1
1
z
x
y
y z
z x
x y
≥ −
+
+
+ +
+ +
+ +
30
Endi
1
2
2
1
1
1
1
z
x
y
S
y z
z x
x y
=
+
+
+ +
+ +
+ +
2
≤
ekanligini isbotlaymiz.
0
x
=
holni
qaraymiz. U holda. Demak,
0
xyz
=
da
1
1
S
≤
.
0
xyz
≠
holda
1
1
1
1
1
1
1
1
1
(
)
(
)
(
)
2
2
2
S
z
y
x
z
y
x
z
y
1
x
x
x
z
z
y
y
x
z
=
+
+
≤
+
+
+
+
+
+
+
+
+
+
+
y
.
,
x
y
a
y
z
=
=
b
va
z
c
x
=
deb belgilab olamiz.
1
abc
=
va
1
1
1
1
2
2
2
S
a
b
c
=
+
+
+
+
+
ekanligi ravshan. U holda
1
1
1
(2
)(2
) (2
)(2
) (2
)(2
)
2
2
2
(2
)(2
)(2
)
b
c
a
c
a
b
a
b
c
a
b
c
+
+ + +
+ + +
+
+
+
=
+
+
+
+
+
+
=
12 4(
) (
)
8 4(
) 2(
)
a b c
ab bc ac
a b c
ab bc ac
abc
+
+ + +
+
+
=
≤
+
+ + +
+
+
+
2 2 2
3
12 4(
) (
)
12 4(
) (
)
1.
12 4(
) (
)
8 4(
) (
) 3
a b c
ab bc ac
a b c
ab bc ac
a b c
ab bc ac
a b c
ab bc ac
a b c
abc
+
+ + +
+
+
+
+ + +
+
+
≤
=
+
+ + +
+
+
+
+ + +
+
+
+
+
=
Demak,
.
1
1
S
≤
19-masala
. Ixtiyoriy musbat ,
(
1,2,..., )
j
j
a b
j
n
=
sonlar uchun
1
1
1
1
...
...
(
)...(
)
n
n
n
n
n
n
a a
b b
a
b
a
b
+
≤
+
+
n
tengsizlik o’rinli ekanligini isbotlang.
Yechilishi.
Gyuygens tengsizligiga asoslanib
1
1
1
1
1
... 1
1
...
n
n
n
n
n
n
a
a
a
a
b
b
b
b
⎛
⎞
⎛
⎞
⎛
⎞
+
+
≥ +
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠ ⎝
⎠ ⎝
⎠
yoki
(
)
1
1
1
1
(
)...(
)
...
...
n
n
n
n
n
n
n
a
b
a
b
a a
b b
+
+
≥
+
ni olamiz. Bu yerdan esa
1
1
1
1
...
...
(
)...(
)
n
n
n
n
n
n
a a
b b
a
b
a
b
+
≤
+
+
n
kelib chiqadi.
31
20-masala
. Ixtiyoriy
musbat sonlar uchun
1
2
,
, ... ,
n
a a
a
2
1
2
1
2
1
1
1
(
...
)
..
n
n
a
a
a
n
a
a
a
⎛
⎞
+
+ +
+
+ +
≥
⎜
⎝
⎠
⎟
tengsizlik o’rinli bo’lishini isbotlang.
Yechilishi.
Koshi–Bunyakovskiy–Shvarts tengsizligiga ko’ra
(
)
2
2
1
2
1
2
2
2
2
2
2
2
1
2
2
1
2
1
1
1
...
1
1
1
(
)
(
)
... (
)
..
n
n
n
n
n
a
a
a
a
a
a
a
a
a
a
a
a
a
⎛
⎞
=
⋅
+
⋅
+ +
⋅
≤
⎜
⎟
⎜
⎟
⎝
⎠
⎛
⎞
⎛
⎞
⎛
⎞
⎛
⎞
⎜
⎟
≤
+
+ +
⋅
+
+ + ⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
⎝
⎠
⎝
⎠
ni olamiz.
21-masala.
(
)
2
1
2
1
2
2
2
2
1
2
1
3
3
4
1
...
...
...
n
n
n
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
+
+ +
≤
+
+ +
+
+ +
+
+
+
2
n
tengsizlikni isbotlang, bu yerda
.
0 (
1,2,..., )
k
a
k
≥
=
Yechilishi.
Koshi–Bunyakovskiy–Shvarts tengsizligiga ko’ra
2
2
1
1
2
1
1
3
1
2
1
3
1
2
(
...
)
(
) ...
(
)
n
n
n
a
a
a
a
a
a a
a
a a
a
a
a
a
a
⎛
⎞
+
+ +
=
⋅
+
+ +
⋅
+
⎜
⎟
⎜
⎟
+
+
⎝
⎠
≤
(
)
1
2
1
1
3
1
2
1
3
3
4
1
2
...
(
) ... (
)
n
n
a
a
a
a a
a
a a
a
a
a
a
a
a
a
⎛
⎞
≤
+
+ +
+
+
+
≤
⎜
⎟
+
+
+
⎝
⎠
2
2
2
2
1
2
1
2
1
3
1
3
3
4
1
2
2
2
2
2
2
2
2
2
1
1
1
2
1
1
...
(
)
(
)
...
2
2
1
1
1
1
(
)
(
)
(
)
(
)
2
2
2
2
n
n
n
n
n
n
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
−
⎛
⎞⎛
⎞
≥
+
+ +
+
+
+
+
⎜
⎟⎜
⎟
+
+
+
⎝
⎠
⎝
⎠
⎛
⎞ ⎛
+
+
+
+
+
+
+
+
=
⎜
⎟ ⎜
⎝
⎠ ⎝
+
⎞
⎟
⎠
2
2
1
2
1
1
3
3
4
1
2
...
(2
... 2 )
n
n
a
a
a
a
a
a
a
a
a
a
a
⎛
⎞
=
+
+ +
+ + +
⎜
⎟
+
+
+
⎝
⎠
.
32
Mashqlar
1.
Agar
, bo’lsa, u holda
tengsizlik o’rinli bo’lishini isbotlang.
, , ,
0,
,
a b c d
c d a c d b
>
+ ≤
+ ≤
ab bc ab
+
≤
2.
Agar
, ,
x y z
lar xaqiqiy sonlar to’plamiga tegishli bo’lsa,
2
2
2
x
y
z
xy yz xz
+
+
≥
+
+
tengsizlikni isbotlang.
3.
Agar
1
x y z
+ + =
bo’lsa,
2
2
2
1
3
x
y
z
+
+
≥
ni isbotlang.
4.
Agar
bo’lsa,
0
ab
>
2
a b
b
a
+ ≥
tengsizlikni isbotlang.
5.
Xar qanday
(
2
n
≥
n N
∈
) larda
2
2
2
1
1
1
1.
2
3
n
+
+ +
<
…
tengsizlik o’rinli
bo’lishini isbotlang.
6.
Xar qanday
(
n
) larda
2
n
≥
N
∈
1 1
1
1
2
2 3
2
n
n
< + + + +
−
…
1
tengsizlik
o’rinli bo’lishini isbotlang
7.
bo’lsa,
n N
∈
(
)
2
1
1
1
9 25
2
1
n
+
+ +
+
…
tengsizlikni isbotlang.
8.
bo’lsa,
n N
∈
1
1
1
1
2
1
2
2
n
n
n
<
+
+ +
<
+
+
…
3
4
2
tengsizlikni isbotlang.
9.
Agar
bo’lsa,
tengsizlikni isbotlang.
2
2
2
2
2
2
1
2
1
2
1
n
n
a
a
a
b
b
b
+
+ +
=
+
+ +
=
…
…
1 1
2 2
1
1
n n
a b
a b
a b
− ≤
+
+ +
≤
…
10.
Agar
bo’lsa,
tengsizlikni isbotlang.
1 2
1
2
1,
, ,
0
n
n
a a
a
a a
a
⋅ ⋅
=
>
…
…
(
)(
) (
)
1
2
1
1
1
n
n
a
a
a
+
+
+
≥
…
11.
Agar
bo’lsa,
1
a b
+ ≥
4
4
1
8
a
b
+
≥
tengsizlikni isbotlang.
33
12.
musbat sonlar va birdan farqli bo’lsa,
,
a b
log
log
2
a
b
b
a
+
≥
tengsizlikni isbotlang.
13.
2
1
1
2
log
log 2
π
π
+
>
tengsizlikni isbotlang.
14. Agar
bo’lsa,
n N
∈
1
1
1
1
1! 2!
!
n
3
+ +
+ +
<
…
tengsizlikni isbotlang.
15.
Agar
bo’lsa,
n N
∈
(
)
1
1
1
2
1 1
1
2
2
3
n
n
n
+ − < +
+
+ +
<
…
tengsizlikni isbotlang.
16. Agar
(
1
2
k
k
k
a
a
a
−
=
+
−
3, 4,
.
k
=
…
) bo’lsa,
2
3
4
5
1
1
2
3
5
2
2 2
2
2
2
2
n
n
a
+
+
+
+
+ +
<
…
tengsizlikni isbotlang.
17. Agar
n
bo’lsa,
N
∈
1
1
1
1
2
1
2
3
1
n
n
n
<
+
+ +
<
+
+
+
…
tengsizlikni
isbotlang.
18. Agar
bo’lsa,
0,
1,2, ,
i
a
i
>
=
…
n
1
2
1 2
n
n
n
a
a
a
a a
a
n
+
+ +
≥
⋅ ⋅
…
…
tengsizlikni isbotlang.
19. Agar
bo’lsa,
0,
1,2, ,
i
a
i
>
=
…
n
(
)
2
1
2
1
2
1
1
1
n
n
a
a
a
n
a
a
a
⎛
+
+ +
+
+ +
≥
⎜
⎝
⎠
…
…
⎞
⎟
tengsizlikni isbotlang.
20.
Agar
bo’lsa,
0
a
>
2
2
2
2 1
1
n
n
a a
a
n
n
a a
a
−
1
+ +
+ +
+
≥
+
+ +
…
…
tengsizlikni isbotlang.
34
21. Agar
1
2
0
2
n
π
α α
α
<
<
<
<
<
…
bo’lsa,
1
2
1
1
2
sin
sin
sin
cos
cos
cos
n
n
n
tg
tg
α
α
α
α
α
α
α
α
+
+ +
<
+
+ +
…
…
<
tengsizlikni isbotlang.
22. Agar
n
bo’lsa, quyidagi tengsizlikni isbotlang.
N
∈
( ) (
)
(
)
2! 4! 6!
2 !
1 !
n
n
n
⋅ ⋅ ⋅ ⋅
≥
+
…
23. Agar
0
2
π
ϕ
< <
bo’lsa,
1
2
ctg
ctg
ϕ
ϕ
≥ +
tengsizlikni isbotlang.
24.
butun sonlar va
,
k l
−
2
n
α
β
π
≠ ± +
bo’lsa,
2
2
cos
cos
cos
cos
cos
cos
k
l
l
k
k
l
α
β
α
β
α
β
−
≤
−
−
tengsizlikni isbotlang.
25. Agar
bo’lsa,
( )
tengsizlikni isbotlang.
2
n
>
2
!
n
n
>
n
26. Agar
va
, , ,
0
a b p q
>
,
p q
ratsional sonlar
1
1
1
p q
+ =
shartni
qanoatlantirsa
p
q
a
b
ab
p
q
≤
+
tengsizlikni isbotlang.
27. Agar
n
bo’lsa,
N
∈
1
2
1
n
n
⎛
⎞
3
< +
<
⎜
⎟
⎝
⎠
tengsizlikni isbotlang.
28. Agar
bo’lsa, quyidagi tengsizlikni isbotlang.
0
n
>
!
3
n
n
n
⎛ ⎞ <
⎜ ⎟
⎝ ⎠
29.Agar
bo’lsa, quyidagi tengsizlikni isbotlang.
( )
0
n
>
( )
3
3 !
n
n
n
>
30. Agar
1
2
;
0,
1,2,
n
i
,
s a
a
a a
i
= +
+ +
>
=
…
…
n
bo’lsa,
(
)(
)
(
)
2
1
2
1
1
1
1
1! 2!
!
n
n
s
s
a
a
a
n
+
+
⋅ ⋅ +
≤ + +
+ +
…
s
…
tengsizlikni isbotlang.
35
31. a)
2
1
3
1
a
a
a
≤
+ +
; b)
2
1
2
4
9
a
a
a
≤
−
+
;
s)
2
2
1
a
a
a
a
+
≥ +
1
; d)
4
2
16
2
4
a
a
a
+
≥
+
.
32. a)
2
9
30
25
a
a
−
+
≥
0
; b)
2
25 10
b
b
+
≥
;
s)
2
4
5 2
2
a
a
a
−
+ ≥
−
; d)
2
2
10 6
b
b
b
1
−
+
≥
−
.
33. a)
; b)
4
4
3
a
b
a b ab
+
≥
+
3
0
4
4
2
2
(
)
a
b
ab a
b
+
+
+
≥
5
;
s)
; d)
6
6
4 2
2 4
a
b
a b
a b
+
≥
+
6
6
5
a
b
a b ab
+
≥
+
.
34. Agar
va
bo’lsa, u holda quyidagilarni isbotlang:
0
a
≥
0
b
≥
a)
; b)
;
3
3
2
a
b
a b ab
+
≥
+
2
3
)
4
3
3
3
(
)
4(
a b
a
b
+
≤
+
s)
; d)
5
5
4
a
b
a b ab
+
≥
+
5
5
3 2
2
a
b
a b
a b
+
≥
+
.
35. Agar
va
bo’lsa, u holda quyidagilarni isbotlang:
0
a
≥
0
b
≥
a)
; b)
3
3
2
a
b
a b ab
−
≥
−
2
)
3
3
3 (
a
b
ab a b
−
≥
−
;
s)
; d)
3
3
2
2
a
b
ab
a
−
≥
−
b
4
5
5
4
a
b
a b ab
−
≥
−
.
36. va sonlarning ixtiyoriy qiymatlarida tengsizlik o’rinli
a
b
bo’lishini isbotlang:
a)
;
4
3
2 2
3
4
2
2
2
a
a b
a b
ab
b
−
+
−
+
≥
0
2
)
2
)
b)
.
4
3
2 2
3
4
4
8
16
16
0
a
a b
a b
ab
b
−
+
−
+
≥
37. Ixtiyoriy
, va sonlar uchun tengsizlik
, ,
a b c
d
a)
;
2
2
2
2
(
)(
) (
a
b
c
d
ac bd
−
−
≤
−
b)
.
2
2
2
2
(
)(
) (
a
b
c
d
ac bd
+
+
≥
+
o’rinli bo’lishini isbotlang, jumladan tenglik bajariladi shu
holda va faqat shu holdaki, qachonki
ad bc
=
.
36
38.
shartni qanoatlantiruvchi ixtiyoriy
a
va
b
sonlar uchun
tengsizlik o’rinli bo’lishini isbotlang.
0
ab
≥
2
2 2
(
)
(
a
b
a b
−
≥
−
4
)
39. Agar
bo’lsa,
a b
<
2
a b
a
b
+
<
<
bo’lishini isbotlang.
40. Agar
bo’lsa,
a b c
< <
3
a b c
a
b
+ +
<
<
bo’lishini isbotlang.
41. Agar
ekanligi ma’lum.
0,
0,
0,
0
a
b
c
d
>
>
<
<
,
,
,
,
,
,
,
,
ab ac
c
b
ac
abd
abc bcd
abcd
c
d
ad cd
bd
c
ifodalar qanday ishoralarga ega bo’ladi ?
42. Agar
a) va
b
bir xil ishorali sonlar;
a
b) va
b
turli ishorali sonlar ekanligi ma’lum bo’lsa,
a
ab
ko’paytma va
a
b
kasr qanday ishoralarga ega bo’ladi ?
43. Agar
a)
; b)
0
ab
>
0
a
b
>
; s)
0
ab
<
; d)
0
a
b
<
; e)
;
2
0
a b
>
f)
; g)
2
0
a b
<
2
0
a
b
<
ekanligi ma’lum bo’lsa, va
b
sonlarning ishorasini
toping.
a
44.
ekanligi ma’lum bo’lsa, ifodaning ishorasi qanaqa ?
2
a
>
a)
3
; b)
10
; s)
6
2
a
−
5
a
−
2
a
−
; d)
(
2)(1
a
a
)
−
−
; e)
2
1
a
a
−
−
;
f)
; g)
2
(
3) (
1
a
a
−
−
)
5
2
a
−
−
; h)
(
1)(2
(5
)
a
a
a
)
−
−
+
.
45.
ekanligi ma’lum bo’lsa, ifodaning ishorasi qanday bo’ladi ?
3
a
<
37
a)
; b)
12
; s)
2
2
a
−
6
8
4
a
−
a
−
; d)
(
5)(
3
a
a
)
−
−
; e)
4
3
a
a
−
−
;
f)
; g)
2
(
1) (
2
a
a
−
−
)
2
3
a
−
; h)
1
(
2)(3
)
a
a
a
−
−
−
.
46. Agar a)
; b)
; s)
1
1
a
<
4
a
>
4
a
< <
; d)
ekanligi ma’lum
bo’lsa
(
1
ifoda qaysi ishoraga ega bo’ladi ?
5
a
>
)(
4
a
a
−
−
)
47. Agar
va
bo’lsa, u holda
1
a
>
1
b
>
1
ab
a b
+ > +
ekanligini isbotlang.
48. Agar
va
bo’lsa, u holda
a b
>
2
b
<
2
(
2)
2
b a
b
a
+
>
+
ekanligini
isbotlang.
49. Agar
bo’lsa, u holda
1
a b
> >
2
2
2
2
a b b
a ab
a
b
+
+ >
+
+
ekanligini
isbotlang.
50. Agar
bo’lsa, u holda
2
a b
< <
2
2
2
2
2
4
2
4
a b
b
a ab
a
b
+
+
<
+
+
ekanligini
isbotlang.
51. Agar
1
bo’lsa, u holda
2
0
a b
< < <
2
2
2
2
2
2
2
a b ab
a
ab
b
a
b
−
−
−
+
+
−
>
ekanligini isbotlang.
52. Agar
bo’lsa, u holda
a b c
≥ ≥
2
2
2
(
)
(
)
(
)
a b c
b c a
c a b
0
− +
−
−
− ≥
ekanligini
isbotlang.
53.
3
sin
(
0)
6
x
x x
x
> −
>
tengsizlikni isbotlang.
54. Sonlarni taqqoslang.
a)
ln 2004
ln 2005
va
ln 2005
ln 2006
b)
va
cos(sin 2006)
sin(cos 2006)
55. 0
x
>
uchun
2
1 2ln
x x
+
≤
tengsizlik o’rinli bo’lishini isbotlang.
56.
1
2
,
, ... ,
n
x x
x
musbat sonlar bo’lsin.
38
1
1
1
...
,
0
( )
...
,
0
n
n
n
x
x
f
n
x
x
α
α
α
α
α
α
⎧
⎛
⎞
+ +
⎪
≠
⎪⎜
⎟
= ⎨⎝
⎠
⎪
⋅ ⋅
=
⎪⎩
funtsiyaning monoton o’suvchi bo’lishini isbotlang. Bundan tashqari
f
funktsiya
qat’iy o’suvchi bo’ladi, faqat va faqat shu holdaki, qachonki
j
x
sonlarning hammasi
o’zaro teng bo’lmasa.
57.
3 3
sin sin sin
8
α
β
γ
≤
tengsizlikni isbotlang, bu yerda ,
α β
va
γ
biror
uchburchakning ichki burchaklari.
58.
lar
1
2
,
, ... ,
n
a a
a
1
(0 :
) (
1,..., )
2
k
a
k
n
∈
=
1
2
...
1
n
a
a
a
va
+ + +
=
xossalarga ega
bo’lgan sonlar bo’lsin.
2
2
2
2
1
2
1
1
1
1
1 ...
1
(
1)
n
n
n
a
a
a
⎛
⎞
⎛
⎞⎛
⎞
−
−
− ≥
−
⎜
⎟
⎜
⎟⎜
⎟
⎝
⎠⎝
⎠ ⎝
⎠
ekanligini isbotlang.
59. Ixtiyoriy
musbat sonlar uchun
, ,
a b c
2
2
2
2
2
2
3
3
2
2
2
a
b
b
c
a
c
a
b
c
a b c
c
a
b
bc ac
+
+
+
+ + ≤
+
+
≤
+
+
3
ab
tengsizlik bajarilishini isbotlang.
60. Agar 1
bo’lsa, u holda
a b c
< ≤ ≤
3
3
1
1
1
1
(
)
ln
ln
ln
3
ln
ln
ln
ln
ln
ln
a
b
c
a
b
c a
b
c
a b c
a
b
c
a
b
c
c
b
c bc
ac
ab
⎛
⎞
+
+
≤
+ +
+
+
≤
+
+
+
+
⎜
⎟
⎝
⎠
3
?
bo’lishini isbotlang.
61.
2
3
2
3 2, (
2, 3, 4,...
n
n
n
−
+
+
≥
=
)
ekanligini isbotlang.
62. Ixtiyoriy
nomanfiy sonlar uchun
, ,
a b c
(
)(
)(
) 8
a b b c c a
abc
+
+
+
≥
tengsizlik o’rinli bo’lishini isbotlang.
39
63. Ixtiyoriy
musbat sonlar uchun
1
2
,
, ... ,
n
a a
a
1
2
2
3
1
1
3
4
1
2
...
2
n
n
n
a
a
a
a
a
a
a
a
n
a
a
a
a
−
+
+
+
+
+
+ +
+
≥
tengsizlik o’rinli bo’lishini isbotlang.
Yensen tengsizligi:
64.
(
) (
)
2
2
2
2
1
2
3
( ,
0)
i
i
i
i
i
i
a
a
a
b
a
b
a b
a
+
+
+
≤
+
∑
∑
∑
>
tengsizlikni
isbotlang. (Ko’rsatma:
2
1
y
x
=
+
).
65.
1
1
n
i
i
i
a
S a
n
=
≥
−
−
∑
n
tengsizlikni isbotlang, bu yerda
1
2
... ,
0
n
i
S a
a
a
a
= +
+
>
.
66.
tengsizlikni isbotlang.
1
1
1
(
)
,
1,
n
n
p
p
p
i
i
i
i
x
n
x
p
x
−
=
=
≥
⋅
>
>
∑
∑
0
i
2
36
c
Koshi-Bunyakovskiy tengsizligi:
67.
tengsizlikni isbotlang, bu yerda
lar
uchburchakning tomonlari;
lar uchburchakning shu tomonlarga
tushirilgan balandliklari; uchburchakning yuzi.
2
2
2
2
2
2
(
)(
)
a
b
c
a
b
c
h
h
h
S
+
+
+
+
≥
, ,
a b c
, ,
a
b
h h h
S
66.
2
2
(1
)(1
) 1,
1,
1
ab
a
b
a
b
+
−
−
≤
≤
≤
|