Ôn tập giải tích thi Olympic sinh viên

ữỡ

ợ ừ số
Pữỡ t ỡ
ỵ tt é ú t sỷ ử ỵ s t ợ ừ ởt số õ
ỵ số {a } ỡ t tr t tỗ t ợ lim
n

an

n+

ỳ

ỡ

lim an = supan .

n+

nN

số {an } ỡ ữợ t tỗ t ợ n+
lim an ỡ ỳ
lim an = inf an .

n+

nN

ử

ử số {u } ữủ ữ s
n



u1 = 1

un+1 = un +

ợ
lim

n+

u2n
1999 .

u2
un
u1
+
+ ããã +
.
u2
u3
un+1

ứ tt t õ
ứ õ s r

un
u2n
1999(un+1 un )
=
=
= 1999
un+1
un+1 un
un+1 un

1
1

un
un+1

u1
u2
uk
+
+ ããã +
= 1999
u2
u3
uk+1

.

1
1

u1
uk+1

.





t {un } ỡ t ỡ ỳ un 1, n {un } tr
õ t ỵ ở tử ỡ tỗ t ợ n+
lim un = L. ứ ổ tự tr ỗ
L
ừ q ợ n t ữủ L = L + 1999
L = 0 ổ ỵ
un 1. {un } ổ tr õ n+
lim un = +.
k tr ú ỵ tr t ữủ
2

lim

k+

u1
u2
uk

+
+ ããã +
u2
u3
uk+1

= lim 1999
k+

1
1

u1
uk+1

= 1999 lim

ởt số t ử

số {u } ữủ ữ s
n



u1 = 2

un+1 =

u2n +1999un
,

2000

{Sn } ợ

n

Sn =
k=1

ợ

n N.

uk
.
uk+1 1

lim Sn .

n+

số {u } ữủ ữ s
n



u1 N

un+1 =

1
2

ln 1 + u2n ,

n N.

ự r số {xn } ở tử
số {xn} ữủ ữ s


0 < xn < 1

xn+1 (1 xn ) 1 ,
4

ự r
lim xn =

n+

n N.

1
.
2

f : [0, ) [0, ) tử sỷ r

f () = ,



f () = ,




, 0,

õ t = = a.
ự r {xn+1 = f (xn )} ợ x0 > 0 ở tử a.

1

k+ uk+1

= 1999.


✸

❇➔✐ ✺✳ ❈❤♦ ❞➣② sè {a } ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿
n



an =

1
2

an−1 +

2015
an−1

,

n ≥ 2,


a1 = 2016.
√
lim an = 2015.

❈❤ù♥❣ ♠✐♥❤ r➡♥❣ n→+∞
❇➔✐ ✻✳ ●✐↔ sû r➡♥❣ ❞➣② {an} ❜à ❝❤➦♥ ✈➔ t❤ä❛ ♠➣♥
an+2 ≤

1
2
an+1 + an
3
3

✈î✐ n ≥ 1.

❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❞➣② tr➯♥ ❤ë✐ tö✳

✶✳✷ ◆❣✉②➯♥ ❧þ ❦➭♣
▲þ t❤✉②➳t✳ ✣è✐ ✈î✐ ❞↕♥❣ ❜➔✐ t➟♣ ♥➔② t❛ sû ❞ö♥❣ ✤à♥❤ ❧þ ❞÷î✐ ✤➙② ✤➸ ❧➔♠ ❜➔✐ t➟♣✳
✣à♥❤ ❧þ ✶✳✸✳ ❈❤♦ ❜❛ ❞➣② sè {a , b , c } t❤ä❛ ♠➣♥ a ≤ b ≤ c . ●✐↔ sû r➡♥❣
n

lim cn = A.

n→+∞

n

n

n

n

n

❑❤✐ ✤â t❛ ❝â n→+∞
lim bn = A.

lim an =

n→+∞

✣è✐ ✈î✐ ❜➔✐ t➟♣ ②➯✉ ❝➛✉ t➼♥❤ ❣✐î✐ ❤↕♥ ❝õ❛ ♠ët ❞➣② n→+∞
lim bn ♥➔♦ ✤â✱ t❛ ❝â t❤➸ t➻♠ ❝→❝ ❞➣② an , cn

s❛♦ ❝❤♦ an ≤ bn ≤ cn ✈î✐ n ✤õ ❧î♥✱ ✈➔ ❤ì♥ ♥ú❛ ❝→❝ ❣✐î✐ ❤↕♥ n→+∞
lim an , lim cn ♣❤↔✐ ❞➵ t➼♥❤✱ ✈➔
n→+∞
♣❤↔✐ ❜➡♥❣ ♥❤❛✉✳ ❑❤✐ ✤â →♣ ❞ö♥❣ ✤à♥❤ ❧þ tr➯♥ t❛ s✉② r❛ ✤÷ñ❝ ❣✐î✐ ❤↕♥ ❝õ❛ ❞➣② n→+∞
lim bn .
❇➯♥ ❝↕♥❤ ✤â✱ ❝❤ó♥❣ t❛ ❝ô♥❣ ❝➛♥ tî✐ ♠ët sè ❜➜t ✤➥♥❣ t❤ù❝ s❛✉✳
❱î✐ x > 0 t❛ ❝â
✭❛✮
1−

x2
x4
x2
< cos x < 1 −
+ .
2
2
4!

x−

x3
x3
x5
< sin x < x −
+ .
3!
3!
5!

✭❜✮
✭❝✮
✭❞✮

√
1
1
1
1
1
1 + x − x2 < 1 + x < 1 + x − x2 + x3 .
2
8
2
8
16
x−

⑩♣ ❞ö♥❣✳

x2
x3
x4
x2
x3
+
−
< ln(1 + x) < x −
+ .
2

3
4
2
3

❱➼ ❞ö ✶✳✹✳ ❈❤♦ ❜✐➳t r➡♥❣

n

Sn =

❈❤ù♥❣ ♠✐♥❤ r➡♥❣

( 1+
k=1

k
− 1).
n2

lim Sn =

n→+∞

1
.
4


✹

❈❤ù♥❣ ♠✐♥❤✳ ❚r÷î❝ t✐➯♥✱ t❛ ❝❤ó þ r➡♥❣ ✈î✐ x > −1, t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ s❛✉
√
x
x
< 1+x−1< .
2+x
2

⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥✱ t❤❛② x ❧➛♥ ❧÷ñt ❜ð✐ nk , ❝❤ó♥❣ t❛ ♥❤➟♥ ✤÷ñ❝
2

k
<
2n2 + k

1+

k
k
− 1 < 2.
n2
2n

▲➜② tê♥❣ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ t❤❡♦ k tø 1 ✤➳♥ n t❛ ♥❤➟♥ ✤÷ñ❝
n

k=1

❚❛ ❝â

n

k=1

k
1
= 2
2n2
2n

k
< Sn <
2
2n + k

n

n

k=1

❦❤✐ n → ∞.

1
n(n + 1)
→
4n2
4

k=

k=1

✭✶✳✷✮

k
.
2n2

✭✶✳✸✮

▼➦t ❦❤→❝✱ t❛ ❧↕✐ ❝â
lim {

n→+∞

▲↕✐ ❝â

n

k=1

1
2n2

n

n

k−
k=1

k2
<
2
2n (2n2 + k)

k=1

n

k=1

❉♦ ✤â
lim {

n→+∞

k
} = lim
n→+∞
2n2 + k

n

k=1

k2
2n2 (2n2

k2

n(n + 1)(2n + 1)
=
→0
4
4n
24n4

1
2n2

n

n

k−
k=1

❈❤ó þ ✭✶✳✸✮ t❛ ♥❤➟♥ ✤÷ñ❝

k=1

n

lim

n→+∞

k=1

+ k)

.

❦❤✐ n → ∞.

k
} = 0,
+k

2n2

k
1
= .
2
2n
4

✭✶✳✹✮

▲➜② ❣✐î✐ ❤↕♥ tr♦♥❣ ✭✶✳✷✮ ❦❤✐ n → ∞ ✈➔ ❝❤ó þ ✭✶✳✸✮ ✈➔ ✭✶✳✹✮ t❤❡♦ ♥❣✉②➯♥ ❧þ ❦➭♣ t❛ ♥❤➟♥ ✤÷ñ❝
lim Sn =

n→+∞

1
.
4

▼ët sè ❜➔✐ t➟♣ →♣ ❞ö♥❣✳

❇➔✐ ✶✳ ❈❤♦ ❞➣② sè {x } ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐
n

xn = (1 +

❚➻♠ n→+∞
lim lnxn .

1
2
n
)(1 + 2 ) · · · (1 + 2 ).
2
n
n
n




t sỹ ở tử ừ số s
xn =

n2
n2
n2
+
+ ããã +
.

6
6
n +1
n +2
n6 + n

t sỹ ở tử ừ số s
xn =

[] + [2] + ã ã ã + [n]
,
n2

tr õ [x] số ợ t ổ ữủt q x, ởt số tỹ t
ợ s

n
lim

n+


lim

(

lim

n+

{a } {b }
n

1+

k=1

n

k2
1).
n3

12 + 22 + ã ã ã + n2 .

n+



3

1
2
n
+
+ ããã + 2
.
n2 + 1 n2 + 2
n +n

n

a1 = 3, b1 = 2, an+1 = an + 2bn

ỡ ỳ

bn+1 = an + bn .

an
, n N.
bn


|cn+1 2| < 21 |cn 2|, n N.
cn =

ự r
n+
lim cn .
ợ

ln2 n
n+ n

n2

lim

k=2

1
.
ln k ln (N k)

số tờ qt
ỵ tt ố ợ t tổ tữớ s t ợ ừ ởt số
lim an

n

tr õ {an } ữủ ữợ tr ỗ
t t sỷ ử ữỡ s t số tờ qt ừ
số tự ữ số {an } af (n) s õ sỷ ử tự ồ t ợ
lim f (n).

n




ử
ử (0, 2). ợ

ừ số {xn } ữủ

lim xn

n+

xn+1 = xn + (1 )xn1

t , x0 , x1 .
tt t õ xn xn1
n1

xn xn1 = ( 1)

n = 1, 2, ã ã ã

= ( 1)(xn1 xn2 ).

(x1 x0 ).

õ

n

xn x0 =

q t ự ữủ
n

( 1)k1

(xk xk1 ) = (x1 x0 )
k=1



k=1

(0, 2) | 1| < 1. t t õ
n

( 1)k1 =
k=1

1 ( 1)n
1 ( 1)n
=
.
1 ( 1)
2

ợ n tr ú ỵ tự tr t ữủ
lim xn = x0 + lim (x1 x0 )

n+

n+

1 ( 1)n
(1 )x0 + x1
=
.
2
2

ởt số t ử

ữủ ữ s f

1

ự r ợ

lim

n+

= 1, f2 = 2, fn+1 = fn + fn1 ,

fn+1
fn

, n 2.

tỗ t ợ tr
x0 = a, x1 = b xn+2 = 13 (xn + 2xn+1), n N. ự r ợ n+
lim xn
tỗ t ợ õ
ọ tữỡ tỹ tr
x0 = a, x1 = b, xn = (1

sỷ r b R, a

n

aR

xn+1 = an + bxn . ự r


xn


xn

1
1
)xn1 + xn2 , n N.
n
n

a
1b

a
1b

|b| < 1;

|b| > 1 x1 +

ak
= 0.
bk


✼

✶✳✹ P❤÷ì♥❣ ♣❤→♣ t➼❝❤ ♣❤➙♥
▲þ t❤✉②➳t✳
⑩♣ ❞ö♥❣✳
❱➼ ❞ö ✶✳✻✳ ❚➼♥❤ ❣✐î✐ ❤↕♥
1
1
1
+
+ ··· +
.
n+1 n+2
3n

lim

n→+∞

❚❛ ❝â
Sn :=

❳➨t ❤➔♠ sè f (x) =
✤♦↕♥ ♥➔②✳ ❉♦ ✤â

1
1
2 +x

1
1

1
1
+
+ ··· +
=
n+1 n+2
3n
2n

1
k=1 2

1
.
k
+ 2n

tr➯♥ ✤♦↕♥ [0, 1]. ❍➔♠ f ❧➔ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ [0, 1] ♥➯♥ ❦❤↔ t➼❝❤ tr➯♥

1

1
n→+∞ 2n

2n

f(

f (x)dx = lim
0

▲↕✐ ❝â

2n

1

k=1

1
k
) = lim
n→+∞ 2n
2n

2n
1
k=1 2

1
.
k
+ 2n

1

f (x)dx =
0

1

2

0

❉♦ ✤â✱ t❛ ❝â

1
3
1
1
dx = ln ( + x)|10 = ln − ln = ln 3.
2
2
2
+x

1
1
1
+
+ ··· +
n+1 n+2
3n

lim

n→+∞

= ln 3.

▼ët sè ❜➔✐ t➟♣ →♣ ❞ö♥❣✳

❇➔✐ ✶✳ ❚➼♥❤ ❝→❝ ❣✐î✐ ❤↕♥ s❛✉✳
✐✳
lim n2

n→+∞

n3

✐✐✳
lim

n→+∞

✐✐✐✳

1
1
1
+ 3
+ ··· + 3
.
3
3
+1
n +2
n + n3
1 k + 2 k + · · · + nk
,

nk+1

1
n→+∞ n

(n + 1)(n + 2) · · · (n + n).

lim

✐✈✳
lim

n→+∞

sin

k ≥ 0.

n
n
n
.
+ sin 2
+ · · · + sin 2
n2 + 1 2
n + 22
n + n2

✈✳

1

lim

n→+∞

2

n

2n
2n
2n
+
+ ··· +
1
n+1 n+ 2
n + n1

.




ự r ợ

sin n+1

lim

1

n+

+

2
sin n+1

+ ããã +

2

n
sin n+1

n

ởt số ữỡ
ự r f tử tr [0, 1] t

lim n

n+



1

n

i
f( )
n
i=1

f (x)dx =

f (1) f (0)
.
2

0

ỷ ử t q tr t
lim n

n+

1k + 2k + ã ã ã + nk
1

,
nk+1
k+1

ợ k 0, t
lim

n+

k 0.

1k + 3k + ã ã ã + (2n 1)k
.
nk+1

Pữỡ ợ tr ữợ
ỵ tt r ố ự ợ

ừ ởt số tỗ t ổ t ự
ợ tr ừ số õ ợ ữợ ừ số õ ổ tự t
tr õ tự s ổ
lim an

n+

lim inf an = lim sup an .
n

n

ử
ử sỷ r {a } ởt số tỹ s
n

tỹ k số ữỡ s

lim an = 1

n+

{bn } số

lim (bn an bn+k ) = l,

n+

ự r l = 0.
ự t
b = lim inf bn ,
n

B = lim sup bn .
n

{bn } số b, B số ỳ ừ lim inf n an , lim supn an
tỗ t {bp }, {bq } ừ {bn } s bp b, bq B r . an 1
bn an bn+k l n {bp +k }, {bq +k } ừ {bn } tữỡ ự t tợ b l
B l r . q b l l B l l. õ l = 0.
r

r

r

r

r

r


✾
▼ët sè ❜➔✐ t➟♣ →♣ ❞ö♥❣✳

❇➔✐ ✶✳ ❈❤♦ ❞➣② sè t❤ü❝ {a } s❛♦ ❝❤♦ a
n

n

≥ 1,

∀n ≥ 1

✈➔ ❞➣② {an + a−1
n } ❤ë✐ tö✳ ❈❤ù♥❣ ♠✐♥❤

r➡♥❣ ❞➣② {an } ❤ë✐ tö✳
❇➔✐ ✷✳ ❈❤♦ ❞➣② sè t❤ü❝ {an} s❛♦ ❝❤♦ n→+∞
lim (2an+1 − an ) = l. ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ lim an = l.
n→+∞
❇➔✐ ✸✳ ❈❤♦ ❞➣② sè ❞÷ì♥❣ {an} s❛♦ ❝❤♦ n→+∞
lim an = L. ❈❤ù♥❣ ♠✐♥❤ r➡♥❣
1

lim (a1 a2 · · · an ) n = L.

n→+∞

❈❤÷ì♥❣ ✷

❚➼❝❤ ♣❤➙♥
✷✳✶ ❚➼♥❤ t➼❝❤ ♣❤➙♥
✣è✐ ✈î✐ ❞↕♥❣ ❜➔✐ t➟♣ ♥➔② t❤➻ ❝❤õ ②➳✉ ❧➔ ❞ò ♣❤➨♣ ✤ê✐ ❜✐➳♥ ✈➔ t➼❝❤ ♣❤➙♥ ✤➸ ✤÷❛ t➼❝❤ ♣❤➙♥ ❜❛♥ ✤➛✉
✈➲ t➼❝❤ ♣❤➙♥ ❞➵ t➼♥❤ ❤ì♥✳ ❇➯♥ ❝↕♥❤ ✤â ♠ët sè ❜➔✐ t➟♣ s➩ →♣ ❞ö♥❣ ❝→❝ ❦➳t q✉↔ s❛✉ ✤➸ t➼♥❤ t➼❝❤ ♣❤➙♥✳

▼➺♥❤ ✤➲ ✷✳✶✳ ❈❤♦ f : [−a, a] → R ❧➔ ❤➔♠ ❧✐➯♥ tö❝ ✈➔ ❝❤➤♥ ✭a > 0.✮ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣
a

a

f (x)
dx =
1 + ex
−a

✭✷✳✶✮

f (x)dx
0

▼➺♥❤ ✤➲ ✷✳✷✳ ❈❤♦ f : [−a, a] → R ❧➔ ❤➔♠ ❧✐➯♥ tö❝ ✭a > 0.✮ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣
✶✳

a

a

f (x)dx = 2
−a

✷✳

♥➳✉ f ❧➔ ❤➔♠ ❝❤➤♥ .

f (x)dx

✭✷✳✷✮

0
a

f (x)dx = 0

♥➳✉ f ❧➔ ❤➔♠ ❧➫ .

✭✷✳✸✮

−a

▼➺♥❤ ✤➲ ✷✳✸✳ ❈❤♦ ❤➔♠ f : R → R ❧➔ ❤➔♠ ❧✐➯♥ tö❝ ✈➔ t✉➛♥ ❤♦➔♥ ❝❤✉ ❦➻ T > 0. ❈❤ù♥❣ ♠✐♥❤
r➡♥❣

✶✳ ✈î✐ ♠é✐ sè t❤ü❝ a t❛ ❝â

a+T

T

f (x)dx =
a

f (x)dx,

✭✷✳✹✮

0

✷✳ ✈î✐ ♠é✐ sè t❤ü❝ a < b t❛ ❝â
b

lim

n→+∞

b−a
f (nx)dx =
T

a

T

f (x)dx.
0

✶✵

✭✷✳✺✮




f : [0, 1] R tử ự r





xf (sin x)dx =
2
0



f (sin x)dx.
0

ử t



2

I1 =

2

cos2014 (x)

dx.
1x+ x2 +1

ự ố ợ t ú t s ũ ờ t t st
ữợ t t t số õ ủ õ trữợ t t tỷ
ủ ú t ữủ õ
1x+

1




(1 x) + x2 + 1
(1 x) + x2 + 1


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

t t tự t s õ ũ t tứ t
õ õ tự tr ỏ ự õ tự
t ữủ t tr t ỷ ữủ số t
ữ ữợ t xn cos2014 (x) tr õ n ởt số ữỡ õ

tỷ t s t f (x) = 1x+1 x +1 t tỷ

1

= f (x).
f (x) = x + x2 + 1. õ t q st t f (x) = x+1x +1 =
(x)+ (x) +1
õ t ủ ỵ ú t õ tr ữ t t
t ỡ é t q st t f (x) õ ủ ỵ t ờ
t = x. ợ ờ t s ữủ t q
2

2


2


2

2014

cos
(x)
dx =
1 + f (x)

I1 =

2

2

2

=

2

2

=

2

cos2014 (t)

dt
1 + t + t2 + 1

cos2014 (t)

dt
1 + t + t2 + 1
f (x) cos2014 (x)
dx.
1 + f (x)

2

ứ õ s r

2

2I1 =

2

cos2014 (x)
dx +
1 + f (x)


2


2

f (x) cos2014 (x)
dx =
1 + f (x)


2

cos2014 (x)dx.

2

ợ sỷ ử ổ tự tr ỗ t tứ t t õ t t
ữủ t I1 . ỏ ồ


✶✷
✷✳

π
2014

I2 =
0
π
2

✸✳

I3 =

✹✳

I4 =

0
1
0

1

1+ecos 2014x dx.

ex sin x
(cos x+sin x)2 dx.

x2014
2
2014
1+x+ x2 +···+ x2014!

dx.

✷✳✷ ❇➜t ✤➥♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥
✣à♥❤ ❧þ ✷✳✻✳ ❈❤♦ f, g ❧➔ ❝→❝ ❤➔♠ ①→❝ ✤à♥❤ ✈➔ ❦❤↔ t➼❝❤ tr➯♥ ✤♦↕♥ [a, b] s❛♦ ❝❤♦ f ≥ g. ❑❤✐ ✤â t❛ ❝â
b

b

f (x)dx ≥

f (x)dx.

a

a

✣à♥❤ ❧þ ✷✳✼ ✭❇➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤② ✲ ❙❝❤✇❛r③✮✳ ❈❤♦ f, g ❧➔ ❝→❝ ❤➔♠ ①→❝ ✤à♥❤ ✈➔ ❦❤↔ t➼❝❤ tr➯♥
✤♦↕♥ [a, b]. ❑❤✐ ✤â t❛ ❝â



2

b

b

a

b
2

f (x)g(x)dx ≤



g 2 (x)dx.

f (x)dx
a

a

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â✱ ✈î✐ ♠å✐ x ∈ R t❤➻
b

b
2

0≤

(xf (t) + g(t)) dt =x
a

2

b

b

2

(f (t)) dt + 2x
a

g 2 (t)dt

f (t)g(t)dt +
a

a

=Ax2 + Bx + C,

tr♦♥❣ ✤â

b

b
2

A=

(f (t)) dt,

B=

a

❚❛♠ t❤ù❝ ❜➟❝ ❤❛✐✱ Ax

2

b

f (t)g(t)dt,
a

+ Bx + C


a

❦❤æ♥❣ ➙♠ ✈î✐ ♠å✐ x ∈ R ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ B 2 − AC ≤ 0, tù❝ ❧➔
2

b

b

a

b
2

f (t)g(t)dt ≤



g 2 (t)dt.

(f (t)) dt
a

✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐

a

b
2

(xf (t) + g(t)) dt = 0

❤❛②

g 2 (t)dt.

C=

a

xf (t) = g(t),

t ∈ [a, b].


✶✸

❱➼ ❞ö ✷✳✽✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ f ❧➔ ❝→❝ ❤➔♠ ①→❝ ✤à♥❤ ✈➔ ❦❤↔ t➼❝❤ tr➯♥ ✤♦↕♥ [a, b], t❤➻


2

b



a

b

f 2 (x)dx.

f (x) cos xdx ≤ (b − a)

f (x) sin xdx + 



2

b

a

a

❈❤ù♥❣ ♠✐♥❤✳ ⑩♣ ❞ö♥❣ ✣à♥❤ ❧þ ✷✳✼ ✈î✐ g(x) = sin x ✈➔ g(x) = cos x t❛ ♥❤➟♥ ✤÷ñ❝


2

b

b



a

2

b

a

b

a

b
2

f (x) cos xdx ≤



sin2 (x)dx

f (x)dx

a

✈➔

b
2

f (x) sin xdx ≤



cos2 (x)dx.

f (x)dx
a

a

❉♦ ✤â



b

2



a

b

a

b

f (x)dx
a

b
2

2

f (x) cos xdx ≤

f (x) sin xdx + 



2

b

sin (x)dx +
a

a

sin2 (x)dx +

a

a

(sin2 x + cos2 x)dx
a

b

b
2

=

cos2 (x)dx

b

f 2 (x)dx

=



b

a
b

a

b

f 2 (x)dx 

=

cos2 (x)dx

f (x)dx
a



b

b
2

f (x)dx
a

dx
a

b

f 2 (x)dx.

=(b − a)
a

❱➼ ❞ö ✷✳✾✳ ●✐↔ sû r➡♥❣ f : [a, b] → [m, M ] ✈➔

b

f (x)dx = 0.
a

❈❤ù♥❣ ♠✐♥❤ r➡♥❣

b

f 2 (x)dx ≤ −mM (b − a).

✭✷✳✼✮

a

P❤➙♥ t➼❝❤✿ Ð ✤➙② t❛ ❝è ❣➢♥❣ →♣ ❞ö♥❣ ♠ët tr♦♥❣ ❤❛✐ ✤à♥❤ ❧þ✿ ✣à♥❤ ❧þ ✷✳✻ ❤♦➦❝ ✣à♥❤ ❧þ ✷✳✼ ✤➸ ❝❤ù♥❣

♠✐♥❤ ❜➔✐ t♦→♥ tr➯♥✳ ◆➳✉ t❛ →♣ ❞ö♥❣ ✣à♥❤ ❧þ ✷✳✼ ✤➸ ❝❤ù♥❣ ♠✐♥❤✱ ❦❤✐ ✤â t❛ ❝❤÷❛ t❤➜② ①✉➜t ❤✐➺♥ ❤➔♠ g,
tr♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔② ♠✉è♥ ①✉➜t ❤✐➺♥ ❤➔♠ g t❛ ❝â t❤➸ ♣❤➙♥ t➼❝❤ f t❤➔♥❤ t➼❝❤ ❝õ❛ ❤❛✐ ❤➔♠ ♥➔♦ ✤â✳ ✣➸
❝â ♣❤➙♥ t➼❝❤ f t❤➔♥❤ t➼❝❤ ❝õ❛ ❤❛✐ ❤➔♠ t❤➻ f ❝❤➾ ❝â t❤➸ ❝â ♣❤➙♥ t➼❝❤ ❞↕♥❣ f = αf α, α = 0 ✭t↕✐ s❛♦
❄❄❄❄❄❄❄❄❄✮✳ ❑❤✐ ✤â t❛ ❝â


✶✹


2

b

0=



b

f (x)dx = 
a

2
f (x)
αdx
α

a

b

b

f (x) 2
(
) dx
α

≤
a

α2 dx
a

b

=α2 (b − a)

(

f (x) 2
) dx
α

a
b

f 2 (x)dx.

=(b − a)
a

✣➳♥ ✤➙②✱ ❝❤ó♥❣ t❛ ❝❤➥♥❣ ❣✐↔✐ q✉②➳t ✤÷ñ❝ ✈➜♥ ✤➲ ❣➻✳
◆➳✉ ♥❤÷ t❛ ❦❤æ♥❣ ♣❤➙♥ t➼❝❤ f = αf × α ♠➔ t➻♠ ♠ët ♣❤➙♥ t➼❝❤ ❦❤→❝ ✈➔ →♣ ❞ö♥❣ ✣à♥❤ ❧þ ✷✳✼ ✤➸
❝❤ù♥❣ ♠✐♥❤ ✭✷✳✼✮ ♥â✐ ❝❤✉♥❣ ❧➔ r➜t ❦❤â✳
✣➳♥ ✤➙②✱ t❛ t❤û ♥❣❤➽ tî✐ ✈✐➺❝ →♣ ❞ö♥❣ ✣à♥❤ ❧þ ✷✳✻ ①❡♠ s❛♦✳ ✣➸ →♣ ❞ö♥❣ ✤à♥❤ ❧þ ♥➔②✱ t❛ ❝➛♥ t➻♠
❤➔♠ g(x) ≥ 0, ∀x ∈ [a, b]. ❚❛ ❝❤ó þ r➡♥❣ f : [a, b] → [m, M ]. ❉♦ ✤â✱ t❛ ❝â f (x) − m ≥ 0, M − f (x) ≥
0, ∀x ∈ [a, b]. ◆❤÷ ✈➟② ❤➔♠ g tr♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔② ❝â t❤➸ ❧➔ f (x) − m ❤♦➦❝ M − f (x) ❤♦➦❝ ♠ët ❜✐➸✉
t❤ù❝ ♥➔♦ ✤â ❝õ❛ ♠ët ❤♦➦❝ ❝↔ ❤❛✐ t❤ø❛ sè tr➯♥✳ ▼➦t ❦❤→❝✱ t❛ ❧↕✐ ❝❤ó þ tr♦♥❣ ✭✷✳✼✮ ❝â ①✉➜t ❤✐➺♥ ❝↔ m
✈➔ M, ❝❤♦ ♥➯♥ t❛ ❞ü ✤♦→♥ g ❝➛♥ ①✉➜t ❤✐➺♥ ❝↔ ❤❛✐ t❤ø❛ sè f (x) − m ✈➔ M − f (x). ◆❤÷ ✈➟② t❛ ❞ü ✤♦→♥
❤➔♠ g ❧➔ (f (x) − m)(M − f (x)). ❙❛✉ ✤â t❛ t❤û →♣ ❞ö♥❣ ✣à♥❤ ❧þ ✷✳✻✱ ✈➔ t❛ ❝â ❝❤ù♥❣ ♠✐♥❤ ♥❤÷ s❛✉✳
❈❤ù♥❣ ♠✐♥❤✳ ✣➦t g(x) = (f (x) − m)(M − f (x)),
→♣ ❞ö♥❣ ✣à♥❤ ❧þ ✷✳✻ t❛ ❝â
b

0≤

∀x ∈ [a, b].

❑❤✐ ✤â g(x) ≥ 0,

∀x ∈ [a, b].

b

(f (x) − m)(M − f (x))dx

g(x)dx =
a

a
b

−f 2 (x) + (n + M )f (x) − mM dx

=
a

b

b

f 2 (x)dx +

=−
a

b

a
b

f 2 (x)dx − mM (b − a).

=−
a

❉♦ ✤â✱ t❛ ❝â ✭✷✳✼✮✳

−mM dx

(n + M )f (x)dx +
a

❑❤✐ ✤â


✶✺

❱➼ ❞ö ✷✳✶✵✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✱ ♥➳✉ f ❧➔ ❝→❝ ❤➔♠ ①→❝ ✤à♥❤✱ ❞÷ì♥❣ ✈➔ ❦❤↔ t➼❝❤ tr➯♥ ✤♦↕♥ [a, b], t❤➻
b

b
2

(b − a) ≤

1
dx.
f (x)

f (x)dx
a

a

❍ì♥ ♥ú❛✱ ♥➳✉ 0 < m ≤ f (x) ≤ M, t❤➻
b

b

f (x)dx

1
(m + M )2
dx ≤
(b − a)2 .
f (x)
4mM

a

a

❱➼ ❞ö ✷✳✶✶✳ ❈❤♦ f ∈ C ([a, b]), f (a) = f (b) = 0 ✈➔

b

1

f 2 (x)dx = 1.
a

❈❤ù♥❣ ♠✐♥❤ r➡♥❣

b

xf (x)f (x)dx = −

1
2

✭✷✳✽✮

a

✈➔

b

1
≤
4

b
2

x2 f 2 (x)dx.

(f (x)) dx
a

a

❍÷î♥❣ ❞➝♥✿ ⑩♣ ❞ö♥❣ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ ❝❤♦ ✈➳ tr→✐ ❝õ❛ ✭✷✳✽✮ ✈➔ sû ❞ö♥❣ ❣✐↔ t❤✐➳t✳
P❤➛♥ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝✱ t❛ →♣ ❞ö♥❣ ✣à♥❤ ❧þ ✷✳✼ ❝❤♦ ❤➔♠ f (x) ❧➔ ❤➔♠ f (x) ✈➔ ❤➔♠ g(x)
❧➔ ❤➔♠ xf (x).

❱➼ ❞ö ✷✳✶✷✳ ❚➻♠

1

(1 + x2 )f 2 (x)dx

min
f ∈A

0

tr♦♥❣ ✤â

1

A = {f ∈ C[0, 1] :

f (x)dx = 1}
0

✈➔ t➻♠ ❤➔♠ f s❛♦ ❝❤♦ t❛ ♥❤➟♥ ✤÷ñ❝ ❣✐→ trà ♥❤ä ♥❤➜t✳
❍÷î♥❣ ❞➝♥✿ ⑩♣ ❞ö♥❣ ✣à♥❤ ❧þ ✷✳✼ ✈î✐ ❤➔♠ f (x) ❝❤➼♥❤ ❧➔ ❤➔♠ √1 + x2f (x) ✈➔ ❤➔♠ g(x) ❝❤➼♥❤ ❧➔
1
❤➔♠ √1+x
.
2

❱➼ ❞ö ✷✳✶✸✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ f ❧➔ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ [0, 1], ❦❤↔ ✈✐ tr➯♥ (0, 1), f (0) = 0 ✈➔
0 < f (x) ≤ 1

tr➯♥ (0, 1) t❤➻



1

2

1

f 3 (x)dx.

f (x)dx ≥


0

0

❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f (x) = x.

❱➼ ❞ö ✷✳✶✹✳ ❚➻♠

1

f 2 (x)dx

A := min
f ∈A

0

✶✻
tr♦♥❣ ✤â

1

A = {f ∈ R[0, 1] :

1

f (x)dx = 3,
0

xf (x)dx = 2}
0

✈➔ t➻♠ ❤➔♠ f s❛♦ ❝❤♦ t❛ ♥❤➟♥ ✤÷ñ❝ ❣✐→ trà ♥❤ä ♥❤➜t✳
P❤➙♥ t➼❝❤✿ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔②✱ ❞♦ ✈➲ ❝ì ❜↔♥ t❛ ❦❤æ♥❣ ❝â t❤æ♥❣ t✐♥ ❣➻ ✈➲ ❤➔♠ f ❧î♥ ❤ì♥ ❤❛② ❜➨
❤ì♥ ❤➔♠ ❤♦➦❝ ❤➡♥❣ sè ♥➔♦ ♥➯♥ ✤➸ →♣ ❞ö♥❣ ✣à♥❤ ❧þ ✷✳✻ ♥â✐ ❝❤✉♥❣ s➩ r➜t ❦❤â✳ ❚r÷í♥❣ ❤ñ♣ ♥➔② t❛ t❤û
→♣ ❞ö♥❣ ✣à♥❤ ❧þ ✷✳✼ ✤➸ ❧➔♠✳ ❚❤æ♥❣ t❤÷í♥❣ ❝â t❤➸ ❝â ♠ët sè ❜↕♥ s✐♥❤ ✈✐➯♥ s➩ ❧➔♠ ♥❤÷ s❛✉✳ ❚❛ ❝â


2

1

9=



f (x)dx = 
0

2

1

f (x) . . . 1dx
0

1

1
2

≤

12 dx

f (x)dx
0

0
1

f 2 (x)dx.

=
0

❉♦ ✤â A = 9. ❚✉② ♥❤✐➯♥ ❝❤ó þ r➡♥❣ ❝→❝❤ ❧➔♠ ♥➔② s➩ ❦❤æ♥❣ ✤ó♥❣✳ ▲þ ❞♦ ♥❤÷ s❛✉✿ ❚❛ ❦❤æ♥❣ t❤➸ t➻♠
✤÷ñ❝ ❤➔♠ f ♥➔♦ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✤➛✉ ❜➔✐ ✤➸ ❝❤♦ A = 9. ❱➻ ♥➳✉ ❦❤æ♥❣ t❤➻ ❦❤✐ ✤â t❛ ♣❤↔✐ ❝â



f (x) = α,




1
f (x)dx = 3,

0


1



 xf (x)dx = 2,
0

✤✐➲✉ ♥➔② ❧➔ ✈æ ❧þ✳
✣è✐ ✈î✐ ❜➔✐ t♦→♥ t➻♠ ❣✐→ trà ❧î♥ ♥❤➜t ❤❛② ♥❤ä ♥❤➜t✱ ❝❤ó♥❣ t❛ ❝➛♥ ♣❤↔✐ t➻♠ ✤÷ñ❝ ❤➔♠ s❛♦ ❝❤♦ t❛
t❤➟t sü ♥❤➟♥ ✤÷ñ❝ ❣✐→ trà ♥❤ä ♥❤➜t ❤♦➦❝ ❧î♥ ♥❤➜t✳
✣è✐ ✈î✐ ❜➔✐ t♦→♥ tr➯♥✱ t❛ ❧➔♠ ♥❤÷ s❛✉✳
❈❤ù♥❣ ♠✐♥❤✳ ⑩♣ ❞ö♥❣ ✣à♥❤ ❧þ ✷✳✼✱ t❛ ❝â ✈î✐ ♠å✐ b ∈ R t❤➻


2

1

2

0

1

f 2 (x)dx ≥

1
2

(x + b)f (x)dx ≤

(2 + 3b) ≤ 

❉♦ ✤â

1

0

3(2 + 3b)2
,
3b2 + 3b + 1

0

∀b ∈ R.

0

❱➻ ✈➟②

1

f 2 (x)dx ≥ max
b∈R

0

f 2 (x)dx.

(x + b) dx

3(2 + 3b)2
= 12.
3b2 + 3b + 1


✶✼
✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐



3(2+3b)2


max 3b

2 +3b+1 = 12,


b∈R





f (x) = α(x + b),
1



f (x)dx = 3,



0


1



 xf (x)dx = 2,
0

●✐↔✐ ❤➺ tr➯♥ t❛ t➻♠ ✤÷ñ❝ f (x) = 6x.

❱➼ ❞ö ✷✳✶✺✳ ❚➻♠

1

(f (x))2 dx

min
f ∈A

0

tr♦♥❣ ✤â
A = {f ∈ C 2 [0, 1] : f (0) = f (1) = 0, f (0) = a}

✈➔ t➻♠ ❤➔♠ f s❛♦ ❝❤♦ t❛ ♥❤➟♥ ✤÷ñ❝ ❣✐→ trà ♥❤ä ♥❤➜t✳
❍÷î♥❣ ❞➝♥✿ ⑩♣ ❞ö♥❣ ✣à♥❤ ❧þ ✷✳✼ ✈î✐ f (x) = 1 − x ❝á♥ g(x) = f

✷✳✸ ✣à♥❤ ❧þ ❣✐→ trà tr✉♥❣ ❜➻♥❤
✣è✐ ✈î✐ ❞↕♥❣ ❜➔✐ t➟♣ ♥➔②✱ ❝❤ó♥❣ t❛ ❝❤ó þ tî✐ ❝→❝ ✤à♥❤ ❧þ s❛✉✳

✣à♥❤ ❧þ ✷✳✶✻ ✭✣à♥❤ ❧þ ✮✳ ●✐↔ sû r➡♥❣✿
✭✐✮ ❤➔♠ f (x) ❧✐➯♥ tö❝ tr♦♥❣ [a, b],
✭✐✐✮

f (a)f (b) < 0.

❑❤✐ ✤â tç♥ t↕✐ c ∈ (a, b) s❛♦ ❝❤♦ f (c) = 0.

✣à♥❤ ❧þ ✷✳✶✼ ✭✣à♥❤ ❧þ ❘♦❧❧❡✮✳ ●✐↔ sû r➡♥❣
✭✐✮ ❤➔♠ f (x) ❧✐➯♥ tö❝ tr♦♥❣ [a, b],
✭✐✐✮ ❤➔♠ f (x) ❦❤↔ ✈✐ tr♦♥❣ (a, b).
✭✐✐✐✮

f (a) = f (b).

❑❤✐ ✤â tç♥ t↕✐ c ∈ (a, b) s❛♦ ❝❤♦ f (c) = 0.

✣à♥❤ ❧þ ✷✳✶✽ ✭✣à♥❤ ❧þ ▲❛❣r❛♥❣❡✮✳ ●✐↔ sû r➡♥❣

(x).



f (x) tử tr [a, b],
f (x) tr (a, b).
õ tỗ t c (a, b) s f (b) f (a) = f (c)(b a).

ỵ ỵ số f, g : [a, b] R. sỷ r
f (x), g(x) tử tr [a, b].
f (x), g(x) tr (a, b).


g (x) = 0

x (a, b).

õ tỗ t c (a, b) s

f (b)f (a)
g(b)g(a)

=

f (c)
g (c) .

t f số tử tr [a, b]. õ số F (x) =

x

số
tr [a, b]. r trữớ ủ f t ởt số ỳ t F (x)
tử tr [a, b].
f (t)dt

a

ử ự r f : [a, b] R số t õ tỗ t [a, b] s


b

f (t)dt.

f (t)dt =
a

P t ổ tữớ ố ợ t ú t sỷ ử ỵ

r ú t tỷ ử ỵ t ử ỵ
t t ỹ F ởt tr số t r tứ f
x
t õ t tợ t F (x) = f (t)dt, t tr trữớ ủ f t
a
F tử õ ử ỵ trỹ t F ổ t
t tử t t õ t ỹ õ t tứ F t ợ ởt
số s số ứ t t t õ t õ tr
t ởt ữ tỷ t s tỷ sỷ ử ỵ ờ
b

b



f (t)dt

f (t)dt =
a



ứ tt t s r

f (t)dt.
a

b



f (t)dt = 2
a

f (t)dt.
a

õ ự tr t ỗ t (a, b) s
b



f (t)dt = 2
a

f (t)dt.
a


✶✾
❚❤❛② θ ❜ð✐ x, ✈➔ t❛ ①➨t ❤➔♠

b

h(x) = 2F (x) −

f (t)dt,
a

tr♦♥❣ ✤â

x

F (x) =

✭✷✳✾✮

f (t)dt.
a

❚❛ ❝❤ó þ r➡♥❣




b

h(a)h(b) = 2F (a) −

f (t)dt 2F (b) −
a


= 0 −

b

a

a

f (t)dt
a

2

b

=−



b

f (t)dt −

f (t)dt 2



f (t)dt
a



b



b

f (t)dt ≤ 0.
a

◆❤÷ ✈➟②✱ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ✣à♥❤ ❧þ ✷✳✶✻ ❝â t❤➸ →♣ ❞ö♥❣ ✤÷ñ❝✳ ❚ø ✤â✱ t❛ ❝â ❝→❝❤ ❣✐↔✐ ♥❤÷ s❛✉✳
❈❤ù♥❣ ♠✐♥❤✳ ✣➦t

b

h(x) = 2F (x) −

f (t)dt
a

tr♦♥❣ ✤â

x

F (x) =

x ∈ [a, b].

f (t)dt,
a

❑✐➸♠ tr❛ ✤÷ñ❝ h ❧➔ ❤➔♠ sè ❧✐➯♥ tö❝ tr➯♥ [a, b] ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥

b

−h(a) = h(b) =

f (t)dt.
a

❉♦ ✤â✱


2

b

h(a)h(b) = − 

f (t)dt ≤ 0.
a

⑩♣ ❞ö♥❣ ✣à♥❤ ❧þ ✷✳✶✻✱ tç♥ t↕✐ θ ∈ (a, b) s❛♦ ❝❤♦ h(θ) = 0, tù❝ ❧➔
θ

b

f (t)dt =
a

f (t)dt.
θ

ử f : [a, b] R số tử sỷ r

tỗ t (a, b) s

b

f (t)dt = 0.
a

ự r



f (t)dt = f ().
a

P t ữỡ tỹ tr t tỷ ử ỵ r ự ụ

ợ F ữ tr t t õ F (x) = f (x) F (a) = F (b) õ ừ
ỵ ữủ tọ ử trỹ t ỵ t t ữủ tỗ
t (a, b) s F () = 0, tr t tỗ t (a, b) s F () = F ().
r trữớ ủ t t r ởt F (x) tt õ t ỹ tr
F (x). õ t ử ữủ ỵ t tỷ F (x) F (x) ợ ởt h(x) ữỡ
s tỗ t G(x) s G (x) = (F (x) F (x)) h(x). r trữớ ủ t õ t t
ữủ h(x) = ex , x [a, b]. õ ữ s

ự t G(x) = F (x)ex , tr õ F (x) ữủ ữ tr õ tứ tt

t õ



G(x)

tử tr [a, b],



G(x)

tr (a, b).



G(a) = G(b) = 0.

ử ỵ tỗ t (a, b) s G () = 0. t
G (x) = (F (x) F (x)) ex .

õ F () F ()



f (t)dt = f ().
a

ử f : [a, b] R số tử a > 0 sỷ r
r tỗ t (a, b) s

f (t)dt = f ().
a

ự ồ F (x) = x1

x

f (t)dt.
a

b

f (t)dt = 0.
a

ự


✷✶

❱➼ ❞ö ✷✳✷✹✳ ❈❤♦ f, g : [a, b] → R ❧➔ ❤❛✐ ❤➔♠ sè ❧✐➯♥ tö❝✳❈❤ù♥❣ ♠✐♥❤ r➡♥❣✱ tç♥ t↕✐ θ ∈ (a, b) s❛♦ ❝❤♦
b

b

a

a

❈❤ù♥❣ ♠✐♥❤✳ ❈❤å♥ ❤➔♠ F (x) =

g(t)dt.

f (t)dt = f (θ)

g(θ)

x

f (t)dt
a

✈➔ G(x) =

x

g(t)dt.
a

⑩♣ ❞ö♥❣ ✣à♥❤ ❧þ ✷✳✶✾✳

❱➼ ❞ö ✷✳✷✺✳ ❈❤♦ f, g : [a, b] → R ❧➔ ❤❛✐ ❤➔♠ sè ❧✐➯♥ tö❝✳❈❤ù♥❣ ♠✐♥❤ r➡♥❣✱ tç♥ t↕✐ θ ∈ (a, b) s❛♦ ❝❤♦
b

θ

f (t)dt = f (θ)

g(θ)
a

❈❤ù♥❣ ♠✐♥❤✳ ❈❤å♥ ❤➔♠ F (x) =

θ
b

x

g(t)dt.

f (t)dt
x

a

g(t)dt.

⑩♣ ❞ö♥❣ ✣à♥❤ ❧þ ✷✳✶✼✳

❱➼ ❞ö ✷✳✷✻✳ ❈❤♦ f, g : [a, b] → R ❧➔ ❤❛✐ ❤➔♠ sè ❞÷ì♥❣✱ ❧✐➯♥ tö❝✳❈❤ù♥❣ ♠✐♥❤ r➡♥❣✱ tç♥ t↕✐ θ ∈ (a, b)
s❛♦ ❝❤♦

f (θ)

−

θ

g(θ)
g(t)dt

f (t)dt
a

❈❤ù♥❣ ♠✐♥❤✳ ❈❤å♥ ❤➔♠ F (x) = e−x

= 1.

b
θ

x

b

f (t)dt g(t)dt.
a

x

⑩♣ ❞ö♥❣ ✣à♥❤ ❧þ ✷✳✶✼✳

▼ët sè ❜➔✐ t➟♣ →♣ ❞ö♥❣✳
✶✳ ❈❤♦ f : [0, 1] → [−1, 1] ❦❤↔ ✈✐ ❧✐➯♥ tö❝ tr➯♥ [0, 1] ✈➔ f (0) = f (1) = 1. ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ tç♥ t↕✐
c ∈ (0, 1) s❛♦ ❝❤♦ f (x) = 2c tan f (c).
✷✳ ❈❤♦ ❤➔♠ sè

❧✐➯♥ tö❝ tr➯♥
(2013, 2015) s❛♦ ❝❤♦
f

[2013, 2015]

✈➔

2015

f (t)dt = 0.
2013

❈❤ù♥❣ ♠✐♥❤ r➡♥❣ tç♥ t↕✐

c ∈

c

2014

f (t)dt = cf (c).

2013
1

1

✸✳ ❈❤♦ f : [0, 1] → R ❧✐➯♥ tö❝ s❛♦ ❝❤♦ f (t)dt = tf (t)dt. ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ tç♥ t↕✐ c ∈ (0, 1) s❛♦
0

0
❝❤♦
c
f (c) = 2014

f (t)dt.
0
b

✹✳ ❈❤♦ ❤➔♠ f ❧✐➯♥ tö❝ tr➯♥ [a, b] ✈➔ ❦❤↔ ✈✐ tr➯♥ (a, b) ✈î✐ a > 0 ✈➔ f (t)dt = 0.❈❤ù♥❣ ♠✐♥❤ r➡♥❣
a
tç♥ t↕✐ c ∈ (a, b) s❛♦ ❝❤♦
c

c

f (t)dt − 2013cf (c) + 2012

2014c
a

f (t)dt = 0.
a


✷✷
✺✳ ❈❤♦ ❤➔♠ sè f : [−1, 1] → R ❧✐➯♥ tö❝✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤
xf 2014 (x) − 2014f (x) = −2013x

❝â ♥❣❤✐➺♠ tr➯♥ ✤♦↕♥ [−1, 1].

✷✳✹ ◗✉② t➢❝ ▲✬❍♦s♣✐t❛❧
✣à♥❤ ❧þ ✷✳✷✼✳ ❈❤♦ I ❧➔ ❦❤♦↔♥❣ ♠ð ❦❤→❝ tr♦♥❣ R, ❣✐↔ sû x

✈➔ f, g : I → R ❧➔ ❤❛✐ ❤➔♠
∀x ∈ I. ●✐↔ sû r➡♥❣ ♠ët tr♦♥❣ ❤❛✐ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷ñ❝ t❤ä❛

❦❤↔ ✈✐✱ ✈î✐ g ✤ì♥ ✤✐➺✉ t➠♥❣ ✈➔ g (x) =
♠➣♥
✶✳

f (x), g(x) → 0

✷✳

g(x) → ±∞

0

∈ R ∪ {±∞}

❦❤✐ x → x0 ,

❦❤✐ x → x0 . ❑❤✐ ✤â✱ ♥➳✉ ❣✐î✐ ❤↕♥
lim

x→x0

t❤➻ t❛ ❝â

f (x)
= l ∈ R ∪ {±∞}
g (x)
f (x)
= l.
x→x0 g(x)
lim

❱➼ ❞ö ✷✳✷✽✳ ❈❤♦ ❤➔♠ f ❧✐➯♥ tö❝ tr➯♥ (0, +∞) ✈➔

lim f (x) = 2016.

x→+∞

❚➼♥❤

x

1
lim
x→+∞ x

f (t)dt.
0

❈❤ù♥❣ ♠✐♥❤✳ ✣➸ →♣ ❞ö♥❣ q✉② t➢❝ ▲✬❍♦s♣✐t❛❧✱ t❛ ❝➛♥ ❝â ♣❤➨♣ ❝❤✐❛ ♠ët ❤➔♠ ❝❤♦ ♠ët ❤➔♠✳ ❚r♦♥❣
x
t❤÷í♥❣ ❤ñ♣ ♥➔②✱ t❛ ❝â t❤➸ t❤➜② ✤÷ñ❝ ♠ët ❤➔♠ ❧➔ f (t)dt ✈➔ ❤➔♠ ❝á♥ ❧↕✐ ❧➔ x. ⑩♣ ❞ö♥❣ ✣à♥❤ ❧þ ✷✳✷✼

❝❤♦ ❤➔♠ f (x) ð tr➯♥ ❝❤➼♥❤ ❧➔ ❤➔♠

0

x

f (t)dt,
0

❝á♥ ❤➔♠ g(x) ð ✣à♥❤ ❧þ tr➯♥ ❝❤➼♥❤ ❧➔ ❤➔♠ x. ❑❤✐ ✤â ❝→❝

✤✐➲✉ ❦✐➺♥ ❝õ❛ ✣à♥❤ ❧þ ✷✳✷✼ ✤➲✉ ✤÷ñ❝ t❤ä❛ ♠➣♥✱ ❤ì♥ ♥ú❛ ✈î✐ G(x) = x, F (x) =
F (x)
f (x)
=
= f (x).
G (x)
1

❉♦ ✤â✱ t❤❡♦ ❣✐↔ t❤✐➳t ✈➔ ✣à♥❤ ❧þ ✷✳✷✼ t❛ ❝â
x

1
lim
x→+∞ x

f (t)dt = lim f (x) = 2016.
x→+∞

0

❱➼ ❞ö ✷✳✷✾✳ ❈❤♦ f : (0, +∞) → R ❧➔ ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ❝➜♣ ✷ s❛♦ ❝❤♦
|f (x) + 2xf (x) + (x2 + 1)f (x)| ≤ 1,

∀x ∈ R.

x

f (t)dt
0

t❤➻


✷✸
❈❤ù♥❣ ♠✐♥❤ r➡♥❣
lim f (x) = 0.

x→+∞

❱➼ ❞ö ✷✳✸✵✳
❱➼ ❞ö ✷✳✸✶✳