luan van thac si su pham,luan van ths giao duc1 of 141.

❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷

❑❍❖❆ ❚❖⑩◆

❚➔♦ ❚❤à ⑩♥❤ ❚➙♠

❇⑨■ ❚❖⑩◆ ❈Ü❈ ❚❘➚ ❈Õ❆ ❍⑨▼ ❙➮

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈

❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✼

luan van thac si su pham,luan van ths giao duc1 of 141.


luan van thac si su pham,luan van ths giao duc2 of 141.

❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷

❑❍❖❆ ❚❖⑩◆

❚➔♦ ❚❤à ⑩♥❤ ❚➙♠

❇⑨■ ❚❖⑩◆ ❈Ü❈ ❚❘➚ ❈Õ❆ ❍⑨▼ ❙➮
❈❤✉②➯♥ ♥❣➔♥❤✿ ✣↕✐ sè
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈


▲í✐ ❝❛♠ ✤♦❛♥
❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❝❤②➯♥ ♥❣➔♥❤ ✣↕✐ sè ✈î✐ ✤➲ t➔✐ ✧❇➔✐ t♦→♥ ❝ü❝
trà ❝õ❛ ❤➔♠ sè✧ ❧➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ r✐➯♥❣ tæ✐ ❞÷î✐ sü ❤÷î♥❣ ❞➝♥ t➟♥ t➻♥❤
❝õ❛ ❝æ ❚❙✳ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ✈➔ ❦❤æ♥❣ ❝â sü trò♥❣ ❧➦♣ ✈î✐ ❜➜t ❦➻ ❝æ♥❣ tr➻♥❤
❦❤♦❛ ❤å❝ ♥➔♦ ❦❤→❝✳
❍➔ ◆ë✐✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✼
❙✐♥❤ ✈✐➯♥
❚➔♦ ❚❤à ⑩♥❤ ❚➙♠

luan van thac si su pham,luan van ths giao duc4 of 141.


luan van thac si su pham,luan van ths giao duc5 of 141.

▼ö❝ ❧ö❝
▲í✐ ♠ð ✤➛✉

✶

✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✸

✶✳✶ ❚➟♣ ❧ç✐✱ ❤➔♠ ❧ç✐✱ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✳✷ ❚➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷ ❍➔♠ sè ✤ì♥ ✤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳✷ ❚➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



luan van thac si su pham,luan van ths giao duc6 of 141.



õ tốt ồ







Pữỡ
ỷ ử t tự
ỷ ử t tự
ỷ ử t tự
ỷ ử t tự tr tt ố ỡ
Pữỡ tr ừ số
Pữỡ sỷ ử ỗ ó
Pữỡ tồ ở tỡ

















t



t




luan van thac si su pham,luan van ths giao duc6 of 141.


luan van thac si su pham,luan van ths giao duc7 of 141.

❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

▲í✐ ♠ð ✤➛✉
❉↕♥❣ t♦→♥ ✧❚➻♠ ❣✐→ trà ❧î♥ ♥❤➜t✱ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❤➔♠ sè tr➯♥
♠ët t➟♣ ①→❝ ✤à♥❤ ♥➔♦ ✤â✧ ❧➔ ❞↕♥❣ t♦→♥ ❤❛② ✈➔ ❦❤â tr♦♥❣ t♦→♥ sì ❝➜♣✳
❉↕♥❣ t♦→♥ ♥➔② ❝á♥ ✤÷ñ❝ ❣å✐ ❧➔ ❞↕♥❣ t♦→♥ t➻♠ ❝ü❝ trà ❝õ❛ ❤➔♠ sè✳ ◆â


❚æ✐ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷ñ❝ sü ✤â♥❣ ❣â♣ þ ❦✐➳♥ ❝õ❛ ❝→❝ t❤➛② ❝æ ❣✐→♦ ✈➔
❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥✳
❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦
❍➔ ◆ë✐✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✼
❙✐♥❤ ✈✐➯♥
❚➔♦ ❚❤à ⑩♥❤ ❚➙♠

luan van thac si su pham,luan van ths giao duc8 of 141.

✷


luan van thac si su pham,luan van ths giao duc9 of 141.

❈❤÷ì♥❣ ✶
❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✶✳✶ ❚➟♣ ❧ç✐✱ ❤➔♠ ❧ç✐✱ t➼♥❤ ❝❤➜t
✶✳✶✳✶

✣à♥❤ ♥❣❤➽❛

❚➟♣ D ✤÷ñ❝ ❣å✐ ❧➔ t➟♣ ❧ç✐ ♥➳✉ ✈î✐ ♠å✐ x, y ∈ D✱ ♠å✐ λ ∈ [0, 1] t❛ ❝â
λx + (1 − λ)y ∈ D.

❍➔♠

✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ❧ç✐ tr➯♥ ♠✐➲♥ ①→❝ ✤à♥❤
x, y ∈ D✱ ♠å✐ λ ∈ [0, 1] t❛ ❝â


(x) > 0✱
(x) < 0✱

✶✳✷ ❍➔♠ sè ✤ì♥ ✤✐➺✉
✶✳✷✳✶

✣à♥❤ ♥❣❤➽❛

●✐↔ sû ❤➔♠ sè y = f (x) ①→❝ ✤à♥❤ tr➯♥ K ✳ ❑❤✐ ✤â✿
◆➳✉ ✈î✐ ♠å✐ x1, x2 ∈ K, x1 > x2✱ t❛ ❝â f (x1) > f (x2) t❤➻ ❤➔♠ sè
y = f (x) ❣å✐ ❧➔ ✤ì♥ ✤✐➺✉ t➠♥❣ ✭❤❛② ✤ç♥❣ ❜✐➳♥ tr➯♥ K ✮✳
◆➳✉ ✈î✐ ♠å✐ x1, x2 ∈ K, x1 < x2✱ t❛ ❝â f (x1) < f (x2) t❤➻ ❤➔♠ sè
y = f (x) ❣å✐ ❧➔ ✤ì♥ ✤✐➺✉ ❣✐↔♠ ✭❤❛② ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ K ✮✳
✶✳✷✳✷

❚➼♥❤ ❝❤➜t

❍➔♠ sè y = f (x) ❝â ✤↕♦ ❤➔♠ tr➯♥ K ✳ ❑❤✐ ✤â✱
◆➳✉ f (x) > 0✱ ✈î✐ ♠å✐ x ∈ K t❤➻ f (x) ✤ç♥❣ ❜✐➳♥ tr➯♥ K ✳
◆➳✉ f (x) < 0✱ ✈î✐ ♠å✐ x ∈ K t❤➻ f (x) ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ K ✳

✶✳✸ ❈ü❝ trà ❝õ❛ ❤➔♠ sè
✶✳✸✳✶

✣à♥❤ ♥❣❤➽❛ ❝ü❝ trà

❛✳ ✣à♥❤ ♥❣❤➽❛ ✶
luan van thac si su pham,luan van ths giao duc10 of 141.


❝â f (x0) ≥ m✱ ✈î✐ m = min f (x)
x∈D
◆➳✉ f (x) ✤↕t ❝ü❝ ✤↕✐ ✤à❛ ♣❤÷ì♥❣ t↕✐ x0 ∈ D t❤➻ t❛ ❝â✿ f (x0) ≤ M ✱
✈î✐ M = max f (x) ❱➟② ●❚▲◆ ✭●❚◆◆✮ ❝õ❛ ❤➔♠ sè f (x) ❝❤÷❛ ❝❤➢❝
x∈D
trò♥❣ ✈î✐ ❝ü❝ ✤↕✐ ✤à❛ ♣❤÷ì♥❣ ✭❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣✮ tr➯♥ ♠✐➲♥ ①→❝ ✤à♥❤
D ♥➔♦ ✤â✳

◆❤➟♥ ①➨t ✶✳✶✳

luan van thac si su pham,luan van ths giao duc11 of 141.

✺


luan van thac si su pham,luan van ths giao duc12 of 141.



õ tốt ồ


t

số õ ỹ tr ữỡ
số f (x) tr [a, b] 0 [a, b] t f (x) tọ
ởt tr s
f (x) ổ õ t 0
f (x) õ t 0 t f (x0) = 0


❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

✶✳✹ ●✐→ trà ❧î♥ ♥❤➜t ✭●❚▲◆✮✱ ❣✐→ trà ♥❤ä ♥❤➜t
✭●❚◆◆✮ ❝õ❛ ❤➔♠ sè
✶✳✹✳✶

✣à♥❤ ♥❣❤➽❛

❈❤♦ ❤➔♠ sè y = f (x) ①→❝ ✤à♥❤ tr➯♥ D
❙è M ✤÷ñ❝ ❣å✐ ❧➔ ●❚▲◆ ❝õ❛ f (x) ♥➳✉ ✤ç♥❣ t❤í✐ t❤ä❛ ♠➣♥ ✷ ✤✐➲✉
❦✐➺♥ s❛✉
✐✳ f (x) ≤ M ✱ ✈î✐ ♠å✐ x ∈ D✳
✐✐✳ ❚ç♥ t↕✐ x0 ∈ D s❛♦ ❝❤♦ f (x0) = M
❑❤✐ ✤â✱ t❛ ❦➼ ❤✐➺✉ max f (x) = M ❤❛② M = max f (x) ✳
x∈D
x∈D
◆➳✉ D = R t❤➻ t❛ ❦➼ ❤✐➺✉ M = max f (x)✳
❙è m ✤÷ñ❝ ❣å✐ ❧➔ ●❚◆◆ ❝õ❛ f (x) ♥➳✉ ✤ç♥❣ t❤í✐ t❤ä❛ ♠➣♥ ✷ ✤✐➲✉
❦✐➺♥ s❛✉
✐✳ f (x) ≥ m✱ ✈î✐ ♠å✐ x ∈ D✳
✐✐✳ ❚ç♥ t↕✐ x0 ∈ D s❛♦ ❝❤♦ f (x0) = m✳
❑❤✐ ✤â✱ t❛ ❦➼ ❤✐➺✉ m = min f (x) ❤❛② m = min f (x) ✳
x∈D
x∈D
◆➳✉ D = R t❤➻ t❛ ❦➼ ❤✐➺✉ m = min f (x)✳
✶✳✹✳✷

❚➼♥❤ ❝❤➜t

õ tốt ồ
max f (x) max f (x)
xA

xB

min f (x) min f (x)
xA

ỵ

xB

sỷ số f (x) tr D t ổ õ
max f (x) = min(f (x)) .
xD

xD

t t t t t
t t ữủ
ỵ sỷ f (x), g(x) số tr D tọ

f (x) > g(x) ợ ồ x D õ t õ max f (x)



xD

max g(x) .




ứ t t t ờ t t
ừ số tr ởt ự t t
ởt t t ừ số õ tr
ỡ õ t t ồ r
t

ỵ số f1 (x), f2 (x), ..., fn (x) tr
D

t f (x) = f1(x) + f2(x) + ... + fn(x)

luan van thac si su pham,luan van ths giao duc14 of 141.




luan van thac si su pham,luan van ths giao duc15 of 141.



õ tốt ồ

õ tỗ t max fi(x) , max f (x) , min f (x) , min f (x) ợ
xD
xD
xD
xD


t t t r ổ t t
t ừ ởt tờ số t
ừ tứ số ỡ
t

sỷ số f1(x), f2(x), ..., fn(x) tr
D t õ fi(x) > 0 ợ ồ x D ồ i 1, n.
t f (x) = f1(x).f2(x).....fn(x) õ tỗ t max fi(x) ,
xD
max f (x) , min f (x) , min f (x) ợ ồ i 1, n t t õ

ỵ

i

xD

xD

xD

max f (x) (max f1 (x)) .(max f2 (x)) .....(max fn (x) )
xD

xD

xD

xD

x∈D

x∈D

x∈D

min h(x) ≥ min f (x) − min g(x)
x∈D

x∈D

x∈D

✭✶✳✼✮
✭✶✳✽✮

❉➜✉ ✧❂✧ tr♦♥❣ ✭✶✳✼✮ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ tç♥ t↕✐ x0 ∈ D s❛♦ ❝❤♦
max f (x) = f (x0 ); max g(x) = g(x0 ) .
x∈D

x∈D

❉➜✉ ✧❂✧ tr♦♥❣ ✭✶✳✽✮ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ tç♥ t↕✐ x0 ∈ D s❛♦ ❝❤♦
min f (x) = f (x0 ); min g(x) = g(x0 ) .
x∈D

x∈D

✣à♥❤ ❧þ ✶✳✶✶✳ ●✐↔ sû f (x), g(x) ❧➔ ❤❛✐ ❤➔♠ sè ①→❝ ✤à♥❤ tr➯♥ ♠✐➲♥ D


max f (x) = f (x0 ); min g(x) = g(x0 ) .
x∈D

x∈D

✶✵

luan van thac si su pham,luan van ths giao duc16 of 141.


luan van thac si su pham,luan van ths giao duc17 of 141.

❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

❉➜✉ ✧❂✧ tr♦♥❣ ✭✶✳✶✵✮ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ tç♥ t↕✐ x0 ∈ D s❛♦ ❝❤♦
min f (x) = f (x0 ); max g(x) = g(x0 ) .
x∈D

x∈D

✭✣à♥❤ ❧➼ ▲❛❣r❛♥❣❡✮✳ ❈❤♦ ❤➔♠ sè [a, b] → R t❤ä❛ ♠➣♥
✷ ✤✐➲✉ ❦✐➺♥ s❛✉✿
✐✳ f ❧✐➯♥ tö❝ tr➯♥ [a, b]✳
✐✐✳ f ❝â ✤↕♦ ❤➔♠ tr♦♥❣ (a, b)✳
❑❤✐ ✤â✱ tç♥ t↕✐ c ∈ (a, b) s❛♦ ❝❤♦ f (b) − f (a) = f (c).(b − a) ✳

✣à♥❤ ❧þ ✶✳✶✷


luan van thac si su pham,luan van ths giao duc18 of 141.


luan van thac si su pham,luan van ths giao duc19 of 141.

❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝

●✐↔✐

❚❳✣✿ D = R.

y = x2 − x − 2✳

❙✉② r❛ y
❇❇❚

=0

❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = −1 ❤♦➦❝ x = 2✳
x

−∞

−1
+

y


●✐↔✐

❚❳✣✿ D = R\{ 12 }✳
−7
y =
< 0✱ ✈î✐ ♠å✐ x ∈ D✳
(2x − 1)2
❱➟② ❤➔♠ sè ❦❤æ♥❣ ❝â ❝ü❝ trà✳
2x2 − x − 1
❇➔✐ ✸✳ ❚➻♠ ❝ü❝ trà ❝õ❛ ❤➔♠ sè y = x + 1 ✳
●✐↔✐

❚❳✣✿ D = R\{ − 1}✳
2x2 + 4x
y =
2 s✉② r❛ y
(x + 1)

=0

❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = 0 ❤♦➦❝ x = −2✳

✶✸

luan van thac si su pham,luan van ths giao duc19 of 141.


luan van thac si su pham,luan van ths giao duc20 of 141.

❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼

+

+∞

y
−∞

−∞

−1

❱➟② ❤➔♠ sè ✤↕t ❝ü❝ ✤↕✐ t↕✐ ✤✐➸♠ x = −2 ✈➔ ❣✐→ trà ❝ü❝ ✤↕✐ y❈✣ = −9
✈➔ ❤➔♠ sè ✤↕t ❝ü❝ t✐➸✉ t↕✐ ✤✐➸♠ x = 0 ✈➔ ❣✐→ trà ❝ü❝ t✐➸✉ yCT = −1✳
◆❤➟♥ ①➨t ✷✳✶✳ ❱➻ ❤➔♠ sè

❝â ✤↕♦ ❤➔♠ ❦❤æ♥❣ ✤ê✐ ❞➜✉
tr➯♥ t➟♣ ①→❝ ✤à♥❤ ♥➯♥ ❤➔♠ sè ❦❤æ♥❣ ❝â ❝ü❝ trà✳
ax2 + bx + c
❍➔♠ sè mx + n (am = 0) ♥➳✉ ❝â ❝ü❝ trà t❤➻ s➩ ❝â ❤❛✐ ❝ü❝ trà
✈➔ ❣✐→ trà ❝ü❝ ✤↕✐ ❝õ❛ ❤➔♠ sè ❧✉æ♥ ♥❤ä ❤ì♥ ❣✐→ trà ❝ü❝ t✐➸✉ ❝õ❛ ❤➔♠
sè✳
ax + b
(ac = 0)
cx + d

❇➔✐ ✹✳ ❚➻♠ ●❚▲◆✱ ●❚◆◆ ❝õ❛ ❝→❝ ❤➔♠ sè s❛✉✿
❛✳ y = x2 x++x 1+ 1 ✳
1
✳
❜✳ y = (sin x + cos x)2 + sin2xcos

−∞

−2
+

y

+∞

0

✵

✵

−

−∞

+

1

y
−

1
3

0


0

1
−

+

y

+∞

+∞

y
4

6

❱➟② min y = 4 t↕✐ t = −1 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = −π
+ kπ (k ∈ Z) ✈➔
4
❤➔♠ sè ❦❤æ♥❣ ✤↕t ●❚▲◆✳
❇➔✐ ✺✳ ❈❤♦ ❜❛ sè t❤ü❝ x, y, z t❤ä❛ ♠➣♥ x + y + z ≤ 3✳ ❚➻♠ ●❚▲◆
❝õ❛
√
√
x + 1 + x x2 + 1 y + 1 + y y 2 + 1 z + 1 + z z 2 + 1
√
√

+
x2 + 1
z2 + 1
y2 + 1
x+1
y+1
z+1
+
+ x + y + z.
=√
+√
x2 + 1
z2 + 1
y2 + 1
1
✣➦t T = √x +
2

y+1

z+1
.
+√
x +1
z2 + 1
y2 + 1
1
❳➨t ❤➔♠ sè f (t) = √t +
❝â t➟♣ ①→❝ ✤à♥❤ D = R✳
t2 + 1

1

❉ü❛ ✈➔♦ ❜↔♥❣ ❜✐➳♥ t❤✐➯♥ t❛ t❤➜②✿ f (x) ≤
❙✉② r❛
√
x+1
√
≤ 2 ✭✶✮
2
x +1
y+1

2✱

√

2 ✭✷✮
y2 + 1
√
z+1
√
≤ 2 ✭✸✮
z2 + 1
≤

√

√

❈ë♥❣ ✭✶✮✱ ✭✷✮✱ ✭✸✮ t❤❡♦ ✈➳✱ t❛ ✤÷ñ❝ T ≤ 3

sin4 x + cos4 x = 1 − sin2 2x✳
2
3
sin6 x + cos6 x = 1 − sin2 2x✳
4

1
1 − sin2 2x
2 − sin2 2x
2
y=
= 2.
✳
2
3 2
4
−
3sin
2x
1 − sin 2x
4

❱✐➳t ❧↕✐ ❤➔♠ sè ✤➣ ❝❤♦ ❞÷î✐ ❞↕♥❣

✣➦t t = sin22x ✈î✐ 0 ≤ t ≤ 1✳
❳➨t ❤➔♠ sè f (t) = 2. 42−−3tt ✈î✐ 0 ≤ t ≤ 1✳
❙✉② r❛ f (t) = 4 2 > 0✱ ✈î✐ 0 ≤ t ≤ 1✳
(4 − 3t)
❙✉② r❛ f (t) ❧➔ ❤➔♠ ✤ç♥❣ ❜✐➳♥ tr➯♥ [0; 1]✳
❱➟② max y = max f (t) = f (1) = 2 t↕✐ t = 1 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = π4 + kπ

r ởt số ỳ tở [a; b]
f (x) ỗ tr [a; b] t
x[a;b]

= max{f (a), f (b), f (xi )}

min f (x) = f (a), max f (x) = f (b).
x[a;b]

x[a;b]

f (x) tr [a; b] t
max f (x) = f (a), min f (x) = f (b).
x[a;b]



x[a;b]

Pữỡ

ỡ s ỵ Pữỡ tữớ ữủ sỷ ử ố ợ
số t tữỡ ố õ
Pữỡ

y ữỡ tr y = 0 xi(i = 1, 2, ...)
ừ õ
y (x) y (xi) rỗ t
y (xi ) < 0 t số t ỹ t xi


y = − cos x − 2 cos 2x✳

❚❛ ❝â y (kπ) = − cos(kπ) − 2 cos(2kπ) = ±1 − 2 < 0✳ ❙✉② r❛ ❤➔♠ sè
✤↕t ❝ü❝ ✤↕✐ t↕✐ x = kπ (k ∈ Z)✳
±2π
±2π
±4π
1
3
y (
+ k2π) = − cos(
) − 2 cos(
) = + 1 = > 0✳ ❙✉②
3
3
3
2
2
±2π
r❛ ❤➔♠ sè ✤↕t ❝ü❝ t✐➸✉ t↕✐ x = 3 + k2π (k ∈ Z)✳
❱➟② ❤➔♠ sè ✤↕t ❝ü❝ ✤↕✐ t↕✐ x = kπ (k ∈ Z) ✈➔ ✤↕t ❣✐→ trà ❝ü❝ ✤↕✐
1
±2π
yC ❉ = y(kπ) = cos(kπ) − ✱ ❤➔♠ sè ✤↕t ❝ü❝ t✐➸✉ t↕✐ x =
+ k2π
2
3
−7
(k ∈ Z) ✈➔ ❣✐→ trà ❝ü❝ t✐➸✉ yCT =
✳