✣❸■ ❍➴❈ ✣⑨ ◆➂◆●
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
❑❍❖❆ ❚❖⑩◆
−−− −−−

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P

Ù◆● ❉Ư◆●
P❍❺◆ ▼➋▼ ▼❆❚❍❊▼❆❚■❈❆
❈❍❖ P❍×❒◆● P❍⑩P ❚■➌P ❚❯❨➌◆

●✐↔♥❣ ữợ r
tỹ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠
▲ỵ♣✿ ✶✶❈❚❯❉✶

✣➔ ◆➤♥❣✱ ✵✺✴✷✵✶✺


✷

▼ư❝ ❧ư❝
▲í✐ ❝↔♠ ì♥✦

✹

▲í✐ ♥â✐ ✤➛✉

✺

✶

❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✽

✶✳✶

❈→❝ ❦❤→✐ ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✷

Þ ♥❣❤➽❛ ❤➻♥❤ ❤å❝ ❝õ❛ ♥❣❤✐➺♠

✽

✶✳✸

❙ü tỗ t tỹ ừ ữỡ tr t







Pữỡ t t

ữợ ú ữỡ tr ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳





ữợ sỡ ở





ữợ t







✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻

✷✳✷✳✶

◆ë✐ ❞✉♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻

✷✳✷✳✷

❙ü ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥

✶✾

✷✳✷✳✸

✣→♥❤ ❣✐→ s❛✐ sè ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥

✷✳✷✳✹

❱➼ ❞ư →♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥ ◆❡✇t♦♥

P❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥

✳ ✳ ✳ ✳ ✳ ✳
✳ ✳ ✳

✷✷

✳

✷✸

×✉ ✤✐➸♠ ✈➔ ❤↕♥ ❝❤➳ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥ ✳ ✳ ✳ ✳ ✳

✷✹

✷✳✸✳✶

×✉ ✤✐➸♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✹

✷✳✸✳✷

❍↕♥ ❝❤➳

✷✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

Ù♥❣ ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ❝❤♦ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣
t✉②➳♥

✷✻

✸✳✶

❚ê♥❣ q✉❛♥ ✈➲ ♥❣æ♥ ♥❣ú ❧➟♣ tr➻♥❤ ▼❛t❤❡♠❛t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳

✷✻

✸✳✷

P❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥ tr♦♥❣ ▼❛t❤❡♠❛t✐❝❛

✷✼

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

✳ ✳ ✳ ✳ ✳ ✳ ✳

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠







trú tr tt ố ợ ữỡ
t t✉②➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✼

▼ët sè ❜➔✐ t➟♣ →♣ ❞ö♥❣

✷✾

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❱➔✐ ♥➨t ✈➲ ■s❛❛❝ ◆❡✇t♦♥ ✈➔ ❏♦s❡♣❤ ❘❛♣❤s♦♥

✹✶

❑➳t ❧✉➟♥

✹✷

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✹✹

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠


✹

▲í✐ ❝↔♠ ì♥✦
❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❚❤➛② ❣✐→♦ r t
ữợ ợ t t t ữợ t➟♥ t➻♥❤
tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❡♠ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ❝õ❛ ♠➻♥❤✳ ❊♠ ①✐♥ ❣û✐ ❧í✐ ❝↔♠
ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❇❛♥ ❈❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥✱ ❝ị♥❣ ❝→❝ ❚❤➛② ❈ỉ tr♦♥❣
❦❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ✣➔ ◆➤♥❣✱ ✤➣ t↕♦ ✤✐➲✉
❦✐➺♥✱ ❣✐ó♣ ✤ï ❡♠ ❤♦➔♥ t❤➔♥❤ tèt ▲✉➟♥ ✈➠♥ ♥➔②✳ ❈↔♠ ì♥ ❝→❝ ❜↕♥ ❝ị♥❣
❧ỵ♣ ✤➣ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï ❡♠ ❤♦➔♥ t❤➔♥❤ tèt ▲✉➟♥ ✈➠♥ ♥➔②✳

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠


✺

▲í✐ ♥â✐ ✤➛✉
✶✳ ▲➼ ❞♦ ❝❤å♥ ✤➲ t➔✐
◆❤ú♥❣ ♣❤÷ì♥❣ tr➻♥❤ ①✉➜t ❤✐➺♥ tr♦♥❣ ❝→❝ ❜➔✐ t♦→♥ t❤ü❝ t➳✱ ♥â✐ ❝❤✉♥❣
❝â t❤ỉ♥❣ t✐♥ ✤➛✉ ✈➔♦ ❝❤➾ ❧➔ ❣➛♥ ✤ó♥❣✳ ❱➻ ✈➟②✱ t
ụ ổ õ ỵ tỹ t ợ r õ ợ ữỡ
✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤✱ t❛ t❤÷í♥❣ ❝â ❝ỉ♥❣ t❤ù❝ ✤→♥❤ ❣✐→ ✤ë ❝❤➼♥❤
①→❝ ❝õ❛ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ✈➔ ❝â t❤➸ t➻♠ ở t ý
trữợ ữỡ ú ữỡ tr õ ỵ rt
q trå♥❣ tr♦♥❣ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐ t♦→♥ t❤ü❝ t➳ ♥â✐ ❝❤✉♥❣ ✈➔ ♣❤÷ì♥❣
tr➻♥❤ ♣❤✐ t✉②➳♥ ♥â✐ r✐➯♥❣✳
❚r↔✐ q✉❛ ❜➲ ❞➔② ❧à❝❤ sû ❚♦→♥ ❤å❝✱ ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ t➼♥❤ t♦→♥ ❤✐➺✉
q✉↔ ✤➣ ✤÷đ❝ ❝→❝ ♥❤➔ ❚♦→♥ ❤å❝ ①➙② ❞ü♥❣ ✤➸ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❝→❝ ♣❤÷ì♥❣
tr➻♥❤ ♣❤✐ t✉②➳♥✱ ♥❤÷ ✿ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ✤ỉ✐✱ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣✱ ♣❤÷ì♥❣
♣❤→♣ t✐➳♣ t✉②➳♥ ✭♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥❘❛♣❤s♦♥✮✱ ♣❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣
✳ ✳ ✳ ◆ê✐ ❜➟t ❧➯♥ tr số õ ữỡ t t ợ tt t♦→♥
✤ì♥ ❣✐↔♥✱ ❧↕✐ ❤ë✐ tư ✤➳♥ ♥❣❤✐➺♠ ❦❤→ ♥❤❛♥❤✳
❙ü ♣❤→t tr✐➸♥ ❝õ❛ ❝→❝ ❝ỉ♥❣ ❝ư t✐♥ ❤å❝ ✤➣ ❣✐ó♣ ❝❤♦ ữỡ
ú ởt ữợ ự ợ õ ỵ ỡ rt
ữỡ t t ❝ơ♥❣ ❦❤ỉ♥❣ ♥➡♠ ♥❣♦➔✐ ♣❤↕♠ ✈✐ ✤â✳ ❱✐➺❝ sû ❞ư♥❣
❝→❝ ♣❤➛♥ ♠➲♠ t♦→♥ ❤å❝ t➼♥❤ t♦→♥ ♥❤÷ ▼❛t❤❧❛❜✱ ▼❛♣❧❡✱ ▼❛t❤❡♠❛t✐❝❛ ✳
✳ ✳ ❝❤♦ ♣❤➨♣ ♥❣÷í✐ ❞ị♥❣ t❤ỉ♥❣ q✉❛ ❧➟♣ tr➻♥❤✱ ổ ọ q tr tỹ
ữợ ♣❤÷ì♥❣ tr➻♥❤✱ ❣✐ó♣ rót ♥❣➢♥ tè✐ ✤❛ t❤í✐ ❣✐❛♥
✈➔ t❤❛♦ t ữớ ồ ỷ ỵ t tø ❞➵ ✤➳♥ ♣❤ù❝ t↕♣✳

✣➸ ♠ð rë♥❣ ✈è♥ ❤✐➸✉ ❜✐➳t ➼t ä✐ ❝õ❛ ♠➻♥❤✱ t❤➯♠ ✈➔♦ ✤â ð ❝❤÷ì♥❣ tr➻♥❤
✣↕✐ ❤å❝✱ s✐♥❤ ✈✐➯♥ ❝❤÷❛ ❝â ✤✐➲✉ ❦✐➺♥ t✐➳♣ ❝➟♥ ♥❤✐➲✉ ✈ỵ✐ ✈✐➺❝ ✈➟♥ ❞ư♥❣
♣❤➛♥ ♠➲♠ t♦→♥ ❤å❝ ▼❛t❤❡♠❛t✐❝❛ ✲ ♠ët ♣❤➛♥ ♠➲♠ r➜t ❤❛② ✈➔ ✤❛♥❣ ✤÷đ❝
❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠



sỷ ử tr ồ ữợ tr➯♥ t❤➳ ❣✐ỵ✐ ✈ỵ✐ q✉② ♠ỉ ♥❣➔②
❝➔♥❣ rë♥❣ ✲ ♥➯♥ tỉ✐ ❝❤å♥ ♥❣❤✐➯♥ ❝ù✉ ✤➲ t➔✐

✑ Ù♥❣ ❞ư♥❣ ♣❤➛♥ ♠➲♠

▼❛t❤❡♠❛t✐❝❛ ❝❤♦ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥ ✑ ✳
✷✳ ▼ư❝ ✤➼❝❤ ♥❣❤✐➯♥ ự
ỗ ữỡ


ữỡ r ởt số ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✱ tr♦♥❣ ✤â ❝â✿

❦❤→✐ ♥✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ t ỵ ồ ừ sỹ tỗ
t ♥❣❤✐➺♠ t❤ü❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♣❤✐ t✉②➳♥ ✈➔ ✤à♥❤ ♥❣❤➽❛



ữỡ

õ ữủ ữợ ú ởt ữỡ tr

f (x) = 0



ợ t❤✐➺✉ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥✿ ♥ë✐ ❞✉♥❣ ♣❤÷ì♥❣ ♣❤→♣✱ sü ở tử
s số ởt ử ỵ ❝â ❧✐➯♥ q✉❛♥ ✳✳✳
✰ ◆❤ú♥❣ ÷✉ ✤✐➸♠✱ ❤↕♥ ❝❤➳ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥✳
✲

❈❤÷ì♥❣ ✸✿ ❚r➻♥❤ ❜➔② ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤➛♥ ♠➲♠ ❤é trđ ▼❛t❤❡♠❛t✐❝❛

❞ị♥❣ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ♣❤✐ t ỗ t số ữỡ
tr tt t ố ợ ữỡ t t t ❞ư♥❣ ✳✳✳

✸✳ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉
✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➔ t➻♠ ❤✐➸✉ ❝õ❛ ▲✉➟♥ ✈➠♥ ♥➔② ❝❤➼♥❤ ❧➔ Ù♥❣ ❞ö♥❣
❝õ❛ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ❝❤♦ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥ ✭✤➸ t➻♠ ♥❣❤✐➺♠
❣➛♥ ✤ó♥❣✮ ✳

✹✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉
✲ ◆❣❤✐➯♥ ❝ù✉ ❧➼ rữợ t ồ t q ✤➳♥ ♥ë✐
❞✉♥❣ ✤➲ t➔✐✳ ❉ü❛ ✈➔♦ ❝→❝ t➔✐ ❧✐➺✉ ✤➣ õ t tờ ủ rỗ rút r
t
ọ ỵ ừ ữợ

ợ ❤↕♥ ✤➲ t➔✐
✣➲ t➔✐ ❦❤æ♥❣ ✤✐ s➙✉ ✈➔♦ ♥❣❤✐➯♥ ❝ù✉ t➜t ❝↔ ❝❤ù❝ ♥➠♥❣ t➼♥❤ t♦→♥ ❝õ❛ ♣❤➛♥
♠➲♠ ▼❛t❤❡♠❛t✐❝❛✱ ♠➔ ợ t ữỡ t t ụ ữ
ỹ ❝→❝ ❝❤÷ì♥❣ tr➻♥❤ ❝♦♥✱ t❤✉➟t t♦→♥ ❝❤♦ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥
tr♦♥❣ ♠ỉ✐ tr÷í♥❣ ▼❛t❤❡♠❛t✐❝❛ ✤➸ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣

tr➻♥❤ ♣❤✐ t✉②➳♥✳

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠



r ởt tớ tữỡ ố ợ ✈➲ ♠➦t ❦✐➳♥ t❤ù❝ ❝ơ♥❣
♥❤÷ ❦✐♥❤ ♥❣❤✐➺♠ t❤ü❝ t✐➵♥ ❝á♥ ❝❤÷❛ ♥❤✐➲✉✱ ♥➯♥ ❦❤ỉ♥❣ t❤➸ tr→♥❤ ❦❤ä✐
♥❤ú♥❣ t❤✐➳✉ sât✳ ❱➻ ✈➟②✱ ❡♠ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü ❝❤➾ ❜↔♦✱ ❞↕② ộ ừ
ổ sỹ õ ỵ t t ừ ❝→❝ ❜↕♥ ✤➸ ❦✐➳♥ t❤ù❝ t❤➯♠ ❤♦➔♥
❝❤➾♥❤ ✈➔ ❦❤ä✐ ❜ï ù ữợ tỹ t
t ✷✵✶✺
❙✐♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥

◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠


✽

❈❤÷ì♥❣ ✶
❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✶✳✶ ❈→❝ ❦❤→✐ ♥✐➺♠
P❤÷ì♥❣ tr➻♥❤ ♣❤✐ t✉②➳♥ ✭❤❛② ❝á♥ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè ❤❛② s✐➯✉
✈✐➺t✱ ♣❤÷ì♥❣ tr➻♥❤ ♠ët ❜✐➳♥ sè t❤ü❝ ✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣✿

f (x) = 0,
tr♦♥❣ ✤â

f

✭✶✳✶✮

❧➔ ♠ët ❤➔♠ sè số s t trữợ ừ

x

ừ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ ❝â t❤➸ ❧➔ sè t❤ü❝ ❤♦➦❝ sè ♣❤ù❝✱ ♥❤÷♥❣
ð ✤➙② t❛ ❝❤➾ ❦❤↔♦ s→t ❝→❝ ♥❣❤✐➺♠ t❤ü❝ ✭x

∈R

✮✳

◆❣❤✐➺♠ t❤ü❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ ❧➔ sè t❤ü❝
❧➔ ❦❤✐ t❤❛② ✈➔♦

x

ð ✈➳ tr→✐ t❛ ✤÷đ❝

f (α) = 0

❦❤ỉ♥❣ ✤✐➸♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮✳

α

t❤ä❛ ♠➣♥ ✭✶✳✶✮✱ tù❝

✳ ❑❤✐ ✤â✱

α

❝á♥ ✤÷đ❝ ồ

ị ồ ừ
ỗ t❤à ❝õ❛ ❤➔♠ sè✿

y = f (x)

✭✶✳✷✮

tr♦♥❣ ❤➺ tå❛ ✤ë ✈✉æ♥❣ ❣â❝ ❖①② ✭❤➻♥❤ ✶✳✶✮✳

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠





sỷ ỗ t t trử t ởt ✤✐➸♠ ▼ t❤➻ ✤✐➸♠ ▼ ♥➔② ❝â tå❛
✤ë

(α; 0)

✳ ❚❤❛② ✈➔♦ ✭✶✳✷✮ t❛ ✤÷đ❝✿

0 = f (α).
❱➟② ❤♦➔♥❤ ✤ë

α

❝õ❛ ❣✐❛♦ ởt ừ

rữợ ỗ t t ụ õ t ờ ♣❤÷ì♥❣ tr➻♥❤
✭✶✳✶✮ ✈➲ ❞↕♥❣ t÷ì♥❣ ✤÷ì♥❣✿

g(x) = h(x)

✭✶✳✸✮

y = g(x),
y = h(x).



rỗ ỗ t ừ số

õ tèt ♥❣❤✐➺♣

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠

sỷ ỗ t t t ✤✐➸♠ ▼ ❝â ❤♦➔♥❤ ✤ë

x=α

t❤➻ t❛ ❝â

g(α) = h(α).
❱➟② ❤♦➔♥❤ ở



ừ ừ ỗ t ởt

ừ tự ừ

ỹ tỗ t tỹ ừ ữỡ tr
t
rữợ t t➼♥❤ ❣➛♥ ✤ó♥❣ ♥❣❤✐➺♠ t❤ü❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
✭✶✳✶✮✱ t❛ ❝➛♥ tỹ õ tỗ t ổ tr ớ t õ
t ũ ữỡ ỗ t ữ ử tr ụ õ t ũ
ỵ s❛✉✿

◆➳✉ ❤➔♠ f (x) ❧✐➯♥ tö❝ tr➯♥
✤♦↕♥ [a, b] ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ f (a).f (b) < 0 t❤➻ ♣❤÷ì♥❣ tr➻♥❤ f (x) = 0
❝â ➼t ♥❤➜t ♠ët ♥❣❤✐➺♠ tr (a, b)

ỵ ỵ

ị ồ ừ ỵ ró r ỗ t ừ ởt
số tử

y = f (x)

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

t↕✐

a≤x≤b

❧➔ ♠ët ✤÷í♥❣ ❝♦♥❣ ❧✐➯♥ tư❝ ✭❧✐➲♥

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠


✶✶
♥➨t✮ ♥è✐ ✷ ✤✐➸♠

❆ ✈➔ ❇✱ ❦❤✐ ❝❤✉②➸♥ tø ✤✐➸♠ ❆ ✭❛✱ ❢✭❛✮✮ s❛♥❣ ✤✐➸♠ ❇ ✭❜✱

❢✭❜✮✮ ♥➡♠ ð ❤❛✐ ♣❤➼❛ ❦❤→❝ ♥❤❛✉ ❝õ❛ trư❝ ❤♦➔♥❤✱ ✤÷í♥❣ ❝♦♥❣ ♥➔② ♣❤↔✐
❝➢t trö❝ ❤♦➔♥❤ t↕✐ ➼t ♥❤➜t ♠ët ✤✐➸♠ ✭❝â t❤➸ t↕✐ ♥❤✐➲✉ ✤✐➸♠✮✳ ✭❍➻♥❤ ✶✳✸✮✳

❍➻♥❤ ✶✳✸✿

❈❤ù♥❣ ♠✐♥❤✳

a+b
[a, b] t❤➔♥❤ ✷ ♣❤➛♥ ❜➡♥❣ ♥❤❛✉ ❜ð✐ ✤✐➸♠

✳
2
a+b
a+b
✰ ◆➳✉ f (
).f (a) > 0 t❤➻ ✤➦t a1 =
❀ b1 = b✳
2
2
a+b
a+b
✰ ◆➳✉ f (
).f (a) < 0 t❤➻ ✤➦t a1 = a❀ b1 =
✳
2
2
a1 + b 1
❈❤✐❛ [a1 , b1 ] t❤➔♥❤ ✷ ♣❤➛♥ ❜➡♥❣ ♥❤❛✉ ❜ð✐ ✤✐➸♠
✳
2
a1 + b 1
a1 + b 1
✰ ◆➳✉ f (
).f (a) > 0 t❤➻ ✤➦t a2 =
❀ b2 = b1 ✳
2
2
a1 + b 1
a1 + b1
✰ ◆➳✉ f (

).f (a) < 0 t❤➻ ✤➦t a2 = a1 ❀ b2 =
✳
2
2

❈❤✐❛ ✤♦↕♥

✳ ✳ ✳ ✳ ✳ ✳ ✳

{an }, {bn } ♠➔ {an }
✈ỵ✐ f (a)✱ f (bn ) ❝ị♥❣

❈ù t✐➳♣ tư❝ q✉→ tr➻♥❤ tr➯♥ t❛ ①➙② ❞ü♥❣ ✤÷đ❝ ✷ ❞➣②
❧➔ ❞➣② t➠♥❣✱

{bn }

❧➔ ❞➣② ❣✐↔♠✱ ✈➔

f (an )

❝ò♥❣ ❞➜✉

f (b)✳
=⇒ f (an ).f (bn ) < 0; ∀n ∈ N✳
❉➣② {an } t➠♥❣✱ ❜à ❝❤➦♥ tr➯♥ ❜ð✐ b lim an

ợ

n

{bn }

ữợ

õ tốt

a lim bn
n

tỗ t
tỗ t

▼✐♥❤ ❚r➙♠


✶✷
α = lim an ❀ β = lim bn ✳
n→∞
n→∞
=⇒ lim (bn − an ) = β − α✳
n→∞
b−a
; ∀n ∈ N✳
▼➔ bn − an =
2n
♥➯♥ lim (bn − an ) = 0 ⇒ β − α = 0 ⇒ β = α✳
n→∞

=⇒ lim bn = lim an = α = β ✳
n→∞
n→∞
=⇒ lim f (bn ) = f (α) = lim f (an )✳
✣➦t

n→∞

n→∞

=⇒ 0 ≥ f (α) ≥ 0 f () = 0

ỵ ữủ ự



∈ (a, b)✳

❈❤♦ f (x) ❧➔ ♠ët
❤➔♠ sè ①→❝ ✤à♥❤✱ ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a, b]✳ ❑❤✐ ✤â f (x) ❧➜② ➼t ♥❤➜t ♠ët
❧➛♥ ♠å✐ ❣✐→ trà ♥➡♠ ❣✐ú❛ f (a) ✈➔ f (b) ✳
❍➺ q✉↔ ✶✳✶✳ ✭❍➺ q✉↔ ❝õ❛ ỵ

ự

t ỵ ❣✐↔ sû

f (a) ≤ λ ≤ f (b)✱

❦❤✐ ✤â

c ∈ (a, b) s❛♦ ❝❤♦ f (c) = λ✳
❚❤➟t ✈➟②✱ ✤➦t g(x) = f (x) − λ ⇒ g ❧✐➯♥ tö❝ tr➯♥ [a, b]✳
g(a).g(b) = (f (a) − λ).(f (b) − λ) ≤ 0
✰ ◆➳✉ g(a) = 0 ❤♦➦❝ g(b) = 0 t❤➻ s✉② r❛ f (a) = λ ❤♦➦❝ f (b) = λ✳
✰ ◆➳✉ g(a) = 0 ✈➔ g(b) = 0 t s r g(a).g(b) < 0
ỵ s r tỗ t c (a, b) s❛♦ ❝❤♦
g(c) = 0✳
⇒ f (c) − λ = 0 f (c) =

tỗ t tr

q ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳

❈❤➼♥❤ ✈➻ ♥ë✐ ❞✉♥❣ ❝õ❛ ❍➺ q✉↔ ♥➔② ỵ tr ỏ t

ỵ ❣✐→ trà tr✉♥❣ ❣✐❛♥ ❝õ❛ ❤➔♠ ❧✐➯♥ tö❝
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳
(a, b)
❦❤♦↔♥❣ ♣❤➙♥ ❧✐
♥❣❤✐➺♠
✳

❑❤♦↔♥❣

♥➔♦ ✤â ✤÷đ❝ ❣å✐ ❧➔

✭❝á♥ ❣å✐ ❧➔ ❦❤♦↔♥❣ ❝→❝❤ ❧✐ ♥❣❤✐➺♠ ❤❛② ❦❤♦↔♥❣ t→❝❤ ♥❣❤✐➺♠✮

❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ ♥➳✉ ♥â ❝❤ù❛

♠ët ✈➔ ❝❤➾ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣

tr➻♥❤ ✤â✳

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠


✶✸

❈❤÷ì♥❣ ✷
P❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ✤➸ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤✱ ❝❤ó♥❣ t❛ ❧✉ỉ♥ ❣✐↔
t❤✐➳t r➡♥❣

f (x)

❧➔ ♠ët ❤➔♠ ①→❝ ✤à♥❤ ✈➔ ❧✐➯♥ tö❝ tr➯♥ ♠ët ✤♦↕♥ ♥➔♦ ✤â

❝õ❛ ✤÷í♥❣ t❤➥♥❣ t❤ü❝✳ ❚❛ ❝ơ♥❣ ❣✐↔ t❤✐➳t r➡♥❣ ❝→❝ ổ tự
tỗ t ởt ❝õ❛ ✤✐➸♠

α

♣❤÷ì♥❣ tr➻♥❤✳ ❑❤♦↔♥❣ ❧➙♥ ❝➟♥ ✭❝❤ù❛
❝õ❛ ♥❣❤✐➺♠

❦❤ỉ♥❣ ❝❤ù❛ ❝→❝ ♥❣❤✐➺♠ ❦❤→❝ ❝õ❛

α

✮ ♥➔② ✤÷đ❝ ❣å✐ ❧➔

❦❤♦↔♥❣ ❝→❝❤ ❧✐

α✳

✷✳✶ ❈→❝ ữợ ú ữỡ tr
ú ữỡ tr



f (x) = 0

ữủ t t ữợ

ữợ sỡ ❜ë

❈â ✸ ♥❤✐➺♠ ✈ö✿
✲ ❱➙② ♥❣❤✐➺♠✿ ❧➔ t➻♠ ①❡♠ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❝â t❤➸ ♥➡♠
tr➯♥ ♥❤ú♥❣ ✤♦↕♥ ♥➔♦ ❝õ❛ trö❝

x✳

✲ ❚→❝❤ ♥❣❤✐➺♠✿ ❧➔ t➻♠ ❝→❝ ❦❤♦↔♥❣ ❝❤ù❛ ♥❣❤✐➺♠ s❛♦ ❝❤♦ tr♦♥❣ ♠é✐
❦❤♦↔♥❣ ❝❤➾ ❝â ✤ó♥❣ ♠ët ♥❣❤✐➺♠✳
✲ ❚❤✉ ❤➭♣ ❦❤♦↔♥❣ ❝❤ù❛ ♥❣❤✐➺♠✿ ❧➔ ❧➔♠ ❝❤♦ ❦❤♦↔♥❣ ❝❤ù❛ ♥❣❤✐➺♠
ọ tốt

ữợ sỡ ở t õ ❝❤ù❛ ♥❣❤✐➺♠ ✤õ ♥❤ä✳
✣➸ t➻♠ ❦❤♦↔♥❣ ❝❤ù❛ ♥❣❤✐➺♠✱ t❛ ❝â t❤➸ ❞ò♥❣ ♠ët tr♦♥❣ ❝→❝ t✐➯✉ ❝❤✉➞♥
s❛✉✿

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠




ỵ


q ừ ỵ

sỷ f (x) ♠ët ❤➔♠ ❧✐➯♥ tư❝ ✈➔ ✤ì♥ ✤✐➺✉ ❝❤➦t tr➯♥
✤♦↕♥ [a, b]✳ ❑❤✐ ➜②✱ ♥➳✉ f (a).f (b) < 0 t❤➻ ♣❤÷ì♥❣ tr➻♥❤ f (x) = 0 ❝â ❞✉②
♥❤➜t ♠ët ♥❣❤✐➺♠ tr (a, b)
ỵ



ị ồ ừ ỵ ỗ t ừ ởt số ❧✐➯♥
tư❝ t➠♥❣ ❝❤➦t ✭❣✐↔♠ ❝❤➦t✮ ❧➔ ♠ët ✤÷í♥❣ ❝♦♥❣ ❧✐➯♥ tư❝ ✭❧✐➲♥ ♥➨t✮ ❧✉ỉ♥ ✤✐
❧➯♥ ✭✤✐ ①✉è♥❣✮✳ ❑❤✐ ❞✐ ❝❤✉②➸♥ tø ✤✐➸♠

❆ ✭❛✱ ❢✭❛✮✮ s❛♥❣ ✤✐➸♠ ❇ ✭❜✱ ❢✭❜✮✮

♥➡♠ ð ừ trử t ỗ t ♣❤↔✐ ❝➢t ✈➔ ❝❤➾ ❝➢t

trö❝ ❤♦➔♥❤ ♠ët ❧➛♥✳

❈❤ù♥❣ ♠✐♥❤✳

[a, b]✱ f (x)

❚ø ❣✐↔ t❤✐➳t✱ ✈➻

f (x)

❧➔ ❧✐➯♥ tư❝ ✈➔ ✤ì♥ ✤✐➺✉ ♥➯♥ tr➯♥

t➠♥❣ ❤♦➦❝ ❣✐↔♠✳ ❍ì♥ ♥ú❛✱ tø ✤✐➲✉ ❦✐➺♥

f (a).f (b) < 0

ự

tọ út ừ ỗ t ❤➔♠ sè ♥➡♠ ✈➲ ❤❛✐ ♣❤➼❛ ❝õ❛ trư❝ ❤♦➔♥❤✳ ❑➳t
❤đ♣ ợ t ỡ ừ số s r
ừ ữỡ tr➻♥❤

f (x) = 0

✳

(a, b) ❧➔ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠

✣à♥❤ ỵ ữủ ự

ồ ỵ ỗ t



ỵ ỵ ✷✳✶ ❝❤➾ ✤á✐ ❤ä✐ t➼♥❤ ❧✐➯♥ tư❝ ♠➔ ❦❤ỉ♥❣ ✤á✐ ọ
t tỗ t ừ

f (x)



f (x)

õ ✤↕♦ ❤➔♠ t❤➻ ✤✐➲✉

❦✐➺♥ ✤ì♥ ✤✐➺✉ ❝â t❤➸ t❤❛② ❜➡♥❣ ✤✐➲✉ ❦✐➺♥ ❦❤æ♥❣ ✤ê✐ ❞➜✉ ❝õ❛ ✤↕♦ ❤➔♠✱
✈➻ ✤↕♦ ❤➔♠ ❦❤ỉ♥❣ ✤ê✐ ❞➜✉ t❤➻ ❤➔♠ sè ✤ì♥ ✤✐➺✉✳

✯ ❍➺ q✉↔ ừ ỵ
õ tốt

❚r➙♠


✶✺

●✐↔ sû ❤➔♠ sè f (x) ❝â ✤↕♦ ❤➔♠ f (x) ✈➔ ✤↕♦ ❤➔♠ f (x)
❝õ❛ ♥â ❦❤æ♥❣ ✤ê✐ ❞➜✉ ✭❧✉ỉ♥ ❞÷ì♥❣ ❤♦➦❝ ❧✉ỉ♥ ➙♠✮ tr➯♥ ✤♦↕♥ [a, b]✳ ❑❤✐
➜②✱ ♥➳✉ f (a).f (b) < 0 t❤➻ ♣❤÷ì♥❣ tr➻♥❤ f (x) = 0 ❝â ❞✉② ♥❤➜t ♠ët ♥❣❤✐➺♠
tr♦♥❣ ❦❤♦↔♥❣ (a, b)

ỵ

ự

tỗ t
t ừ
ừ

f (x)✳

α

α

s❛♦ ❝❤♦

f (α) = 0

❧➔ ❤➺ q✉↔ ✶✳✶✱ ✈➔ t➼♥❤

q ừ t ỡ

ỵ ữủ ự

f (x) ổ ờ

ứ ỵ tr t❛ ✤✐ ✤➳♥ ✷ ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ❦❤♦↔♥❣ ❝→❝❤ ❧✐
♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

f (x) = 0 ✭❦❤♦↔♥❣ ❝❤ù❛ ❞✉② ♥❤➜t ởt

ữỡ ồ ữỡ t

ã

Pữỡ ♣❤→♣ ❣✐↔✐ t➼❝❤✿

f (x) = 0 tr♦♥❣ ❦❤♦↔♥❣
(a, b)✳ ❚❛ ✤✐ t➼♥❤ ❣✐→ trà f (a), f (b) ✈➔ ❝→❝ ❣✐→ trà f (xi ) ❝õ❛ ❤➔♠ sè t↕✐
♠ët sè ✤✐➸♠ xi ∈ (a, b)✱ i = 1, 2, ..., n✳ ◆➳✉ ❤➔♠ f (x) ✤ì♥ ✤✐➺✉ ❝❤➦t
tr➯♥ ❦❤♦↔♥❣ (xi , xi+1 ) ✈➔ ✤✐➲✉ ❦✐➺♥ f (xi ).f (xi+1 ) < 0 ✤÷đ❝ t❤ä❛ ♠➣♥
t❤➻ (xi , xi+1 ) ❧➔ ♠ët ❦❤♦↔♥❣ ❝→❝❤ ❧✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ f (x) = 0✳
◆➳✉ t❤æ♥❣ t✐♥ ✈➲ ❤➔♠ f (x) q✉→ ➼t t❤➻ t❛ t❤÷í♥❣ ❞ị♥❣ q✉② tr➻♥❤ ❝❤✐❛
✤♦↕♥ t❤➥♥❣ ✭❝❤✐❛ ❦❤♦↔♥❣ (a, b) t❤➔♥❤ ✷✱ ✹✱ ✽✱ ✳✳✳ ♣❤➛♥✮ ✈➔ t❤û ✤✐➲✉ ❦✐➺♥
f (xi ).f (xi+1 ) < 0 ✤➸ t➻♠ ❦❤♦↔♥❣ ❝→❝❤ ❧✐ ♥❣❤✐➺♠✳
▼ët ✤❛ t❤ù❝ ❜➟❝ n ❝â ❦❤ỉ♥❣ q✉→ n ♥❣❤✐➺♠✳ ❱➻ ✈➟②✱ ♣❤÷ì♥❣ tr➻♥❤ ✤❛
t❤ù❝ ❝â ❦❤æ♥❣ q✉→ n ❦❤♦↔♥❣ ❝→❝❤ ❧✐ ♥❣❤✐➺♠✳
❑❤✐ ❤➔♠ f (x) ✤õ tèt ✭❝â ✤↕♦ ❤➔♠✱ ❝â ❞↕♥❣ ❝ö t❤➸✱ ✳✳✳✮✱ t❛ ❝â t❤➸
●✐↔ sû t❛ ♣❤↔✐ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr

st ỗ t trử số t ờ ừ
ỗ ừ số


ã

Pữỡ ồ

r trữớ ủ ỗ t số tữỡ ố t õ t

ỗ t t➻♠ ❦❤♦↔♥❣ ❝→❝❤ ❧✐ ♥❣❤✐➺♠ ❤♦➦❝ ❣✐→ trà t❤æ ❝õ❛ ữ
ú ừ ỗ t ợ trư❝ ❤♦➔♥❤✳ ❙❛✉ ✤â✱ ♥❤í t➼♥❤ t♦→♥✱
t❛ ✏t✐♥❤ ❝❤➾♥❤✑ ✤➸ ✤✐ ✤➳♥ ❦❤♦↔♥❣ ❝→❝❤ ❧✐ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❤ì♥✳

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠





ữợ t

ú t ②➯✉ ❝➛✉ ✤➦t r❛✳ ❈â ✹ ♣❤÷ì♥❣ ♣❤→♣ ❝ì
❜↔♥ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❧➔✿
✰ P❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ✤ỉ✐✳
✰ P❤÷ì♥❣ ♣❤→♣ ❧➦♣✳
✰ P❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣✳
✰ P❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥ ✭❍❛② ❝á♥ ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥✱
♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ✲ s

Pữỡ t t


ở ữỡ

ị ừ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥ ❧➔ t➻♠ ❝→❝❤ t❤❛② ♣❤÷ì♥❣ tr
t ố ợ
ố ợ

x

ởt ữỡ tr ✤ó♥❣✱ t✉②➳♥ t➼♥❤

x✳

❚❛ t❤❛② ❝✉♥❣ ❝õ❛ ✤÷í♥❣ ❝♦♥❣
✤✐➸♠

y = f (x)

tr➯♥

[a, b]

❜➡♥❣ t✐➳♣ t✉②➳♥ t↕✐

❆ ✭❛✱ ❢✭❛✮✮ ❤♦➦❝ ✤✐➸♠ ❇ ✭❜✱ ❢✭❜✮✮ ✈➔ ❝♦✐ ❣✐❛♦ ✤✐➸♠ ❝õ❛ t✐➳♣ t✉②➳♥

✈ỵ✐ trư❝ ừ ữỡ tr
rữợ t t❛ ♥❤➢❝ ❧↕✐ ❝æ♥❣ t❤ù❝ ❚❛②❧♦r✿

❈æ♥❣ t❤ù❝ ❚❛②❧♦r✳ ❈❤♦ ❤➔♠

n+1

tr➯♥

(a, b)✳

❑❤✐ ✤â✱

f (x) ①→❝ ✤à♥❤ ✈➔ ❝â ✤↕♦ ❤➔♠ ✤➳♥
x0 , x (a, b) tỗ t c ♥➡♠ ❣✐ú❛ x0 ✈➔ x s❛♦

❝❤♦✿

x − x0
(x − x0 )2
(x − x0 )n (n)
f (x) = f (x0 )+
.f (x0 )+
.f (x0 )+...+
.f (x0 )+
1!
2!
n!
(x − x0 )n+1 (n+1)
+
.f
(c). ✭✷✳✶✮
(n + 1)!
❈ỉ♥❣ t❤ù❝ ✭✷✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ ❦❤❛✐ tr✐➸♥ r

n

ừ

f (x)

ớ t ữỡ tr ợ t❤✐➳t ♥â ❝â ♥❣❤✐➺♠
♣❤➙♥ ❧✐ ð tr♦♥❣ ❦❤♦↔♥❣

x0 ✳
t❤ü❝ α
t↕✐

(a, b)✳

f ❝â ✤↕♦ ❤➔♠ f (x) = 0 t↕✐ x ∈ [a, b] ✈➔ ✤↕♦ ❤➔♠ ❝➜♣ ❤❛✐
x ∈ (a, b) ỗ tớ f (x), f (x) tử ❦❤æ♥❣ ✤ê✐ ❞➜✉

●✐↔ sû ❤➔♠

f (x) t↕✐
tr➯♥ [a, b]✳

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠


✶✼
✣à♥❤ ♥❣❤➽❛

✷✳✶✳ ✣✐➸♠

f (x).f (x) > 0

✭

x ∈ [a, b]

✣✐➲✉ ❦✐➺♥ ❋♦✉r✐❡r ✮✳

❑❤æ♥❣ ❣✐↔♠ tê♥❣ q✉→t✱ ❤➔♠
❧➔ ❝â ✤↕♦ ❤➔♠

f (x) > 0

✱ ♥➳✉

✤÷đ❝ ❣å✐ ❧➔

✤✐➸♠ ❋♦✉r✐❡r

♥➳✉

f (x) tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ ❝â t❤➸ ❝♦✐
❦❤ỉ♥❣ t❛ ①➨t ♣❤÷ì♥❣ tr➻♥❤ g(x) = 0 ✈ỵ✐

g := −f ✳
❈❤å♥ ①➜♣ ①➾ ❜❛♥ ✤➛✉
❞ü♥❣ ❞➣②

x0

❧➔ ✤✐➸♠ ❋♦✉r✐❡r✿

f (x0 ).f (x0 ) > 0✳

❚❛ ①➙②

{xn }n=0,∞ ✳

❑❤❛✐ tr✐➸♥ ❚❛②❧♦r ❜➟❝ ♥❤➜t ❝õ❛

f

t↕✐

x0

❧➔

f (x) = f (x0 ) + (x − x0 ).f (x0 ) +
x ∈ [a, b]
x0 < c < x✳

✈ỵ✐

✱

c = x0 + θ.(x − x0 ) ∈ (a, b)

(x − x0 )2
.f (c)
2
ợ

0< <1

õ

ữ ữỡ tr ữủ t t❤➔♥❤✿

f (x0 ) + (x − x0 ).f (x0 ) +
●✐↔ sû r➡♥❣

x0

❣➛♥ ✈ỵ✐

x✱

(x − x0 )2
.f (c) = 0.
2

✭✷✳✷✮

t❛ ❝â t❤➸ ❜ä q✉❛ sè ❤↕♥❣ ❝✉è✐ tr♦♥❣ ✭✷✳✷✮ ✈➔

✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤✿

f (x0 ) + (x − x0 ).f (x0 ) = 0.

✭✷✳✸✮

◆❤÷ ✈➟②✱ t❛ ✤➣ t❤❛② ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ ❜➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✸✮ ✤ì♥
❣✐↔♥ ❤ì♥ ♥❤✐➲✉ ✈➻ ✭✷✳✸✮ t✉②➳♥ t ố ợ

x

ữỡ t t õ ❣➛♥ ✤ó♥❣✳ ●å✐

x1

❧➔ ♥❣❤✐➺♠ ❝õ❛

✭✷✳✸✮ t❛ ❝â✿

f (x0 ) + (x1 − x0 ).f (x0 ) = 0
f (x0 )
⇔ x1 = x0 −
.
f (x0 )
f (x1 )
❚ø x1 ✱ t❛ t➼♥❤ ♠ët ❝→❝❤ t÷ì♥❣ tü r❛ x2 = x1 −
✱ ✳ ✳
f (x1 )
f (xn−1 )
xn = xn−1 −
✳
f (xn−1 )
❚ê♥❣ q✉→t✱ ❦❤✐ ✤➣ ❜✐➳t xn t❛ t➼♥❤ xn+1 t❤❡♦ ❝æ♥❣ t❤ù❝✿
xn+1 = xn −
❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

f (xn )
(n ≥ 0)
f (xn )

✭✷✳✹✮

✳ ✱

✭✷✳✺✮

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠



x0
ú s ởt số




[a, b]

ữợ t

ừ ữỡ tr

Pữỡ t

ú ỵ

n

ồ trữợ

xn

t ồ

xn

❣✐→ trà ❣➛♥ ✤ó♥❣ ❝õ❛

♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥

✳

✷✳✶✳ ❱➻ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✸✮ ❞ị♥❣ ✤➸ t❤❛② ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤

✭✶✳✶✮✱ ✭✷✳✸✮ ❧➔ t✉②➳♥ t ố ợ

x

ữỡ t t õ

ữỡ t ụ ồ



ú ỵ ứ t t ữỡ t tở

ữỡ

ợ

(x) = x
ú ỵ
ỗ t

f (x)
.
f (x)



t ồ t f (x0) ❧➔ ❤➺ sè ❣â❝ ❝õ❛ t✐➳♣ t✉②➳♥ ❝õ❛

y = f (x)

t↕✐

x0

✳

❳➨t ♠ët tr÷í♥❣ ❤đ♣ ❝ư t❤➸ ♥❤÷ s❛✉✿ t ỗ t tr

õ tốt

◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠





ỗ t



AB

t trử t

t ú



M

õ ❤♦➔♥❤ ✤ë ❝❤➼♥❤ ❧➔ ♥❣❤✐➺♠

t❛ t❤❛② ♠ët ❝→❝❤ ❣➛♥ ✤ó♥❣ ❝✉♥❣

AB

❜ð✐ t✐➳♣

t✉②➳♥ t↕✐ B ✱ B ❝â ❤♦➔♥❤ ✤ë x0✱ t✐➳♣ t✉②➳♥ ♥➔② ❝➢t trö❝ ❤♦➔♥❤ t↕✐ P ✱ P
❝â ❤♦➔♥❤ ✤ë x1 ✈➔ t❛ ①❡♠ x1 ❧➔ ❣✐→ trà ❣➛♥ ✤ó♥❣ ❝õ❛ α✳

✣➸ t➼♥❤

x1

t❛ ✈✐➳t ♣❤÷ì♥❣ tr➻♥❤ t✐➳♣ t✉②➳♥ t↕✐

B

✿ ✈ỵ✐

x0 = b

t❛ ❝â✿

Y − f (x0 ) = f (x0 ).(X − x0 ).
❚↕✐

P

t❛ ❝â

X = x1 , Y = 0✱

♥➯♥ ❝â✿

−f (x0 ) = f (x0 ).(x1 − x0 ).
❚ø ✤â✱ t❛ s✉② r❛ ✭✷✳✹✮✳ ❈❤♦ ♥➯♥ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ❝á♥ ❝â t➯♥ ❧➔

♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥
✷✳✷✳✷

✳

❙ü ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥

α✳ ✣✐➲✉ ✤â ❝❤➾ ❝â t❤➸ t❤ü❝ ❤✐➺♥
✤÷đ❝ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ♥➳✉ xn → α ❦❤✐ n → ∞✳ ❚❛ ❝â ❦➳t q✉↔
▼ö❝ ừ t t ú

s

ỵ ❦✐➺♥ ✤õ ✤➸ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥ ❤ë✐ tư✮✳

♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ✤÷đ❝ t❤ä❛ ♠➣♥✿
✰ ✣✐➲✉ ❦✐➺♥ ✶✿ (a, b) ❧➔ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠

α

●✐↔ sû

❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

✭✶✳✶✮ ✳
❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠


✷✵

❍➔♠ f (x) ❝â ✤↕♦ ❤➔♠ ❜➟❝ ♥❤➜t f (x) ✈➔ ❜➟❝ ❤❛✐ f (x)✱
✈ỵ✐ f (x) ✈➔ f (x) ❧✐➯♥ tư❝ tr➯♥ [a, b]✳ f ✈➔ f ❦❤ỉ♥❣ ✤ê✐ ❞➜✉ tr♦♥❣ (a, b)
✭♥❣❤➽❛ ❧➔ ❤➔♠ f (x) ✤ì♥ ✤✐➺✉✱ ỗ ó tr [a, b]
✸✿ ❳➜♣ ①➾ ✤➛✉ ✤✐➸♠ ❋♦✉r✐❡r x0 ✤÷đ❝ ❝❤å♥ ❧➔ ♠ët tr♦♥❣
❤❛✐ ✤➛✉ ♠ót a ❤♦➦❝ b ✭✈✐➺❝ ❝❤å♥ ✤✐➸♠ ❜❛♥ ✤➛✉ x0 r➜t q✉❛♥ trå♥❣ ✮ s❛♦
❝❤♦ f (x0) ❝ị♥❣ ✤ê✐ ❞➜✉ ✈ỵ✐ f (x)✱ tù❝ ❧➔ f (x0).f (x) > 0 ỗ t
ồ tr ➙♠✱ ❤➔♠ ❧ã♠ t❤➻ ❝❤å♥ ♣❤➼❛ ❣✐→ trà ❞÷ì♥❣ ✮✳
❑❤✐ ✤â✱ xn t➼♥❤ ❜ð✐ ✭✷✳✺✮ ❤ë✐ tö ✈➲ α ❦❤✐ n → ∞✳ ❈ư t❤➸ ❤ì♥ t❛ ❝â
xn ✤ì♥ ✤✐➺✉ t➠♥❣ tỵ✐ α ♥➳✉ f .f < 0 ✈➔ xn ✤ì♥ ✤✐➺✉ ❣✐↔♠ tỵ✐ α ♥➳✉
✰ ✣✐➲✉ ❦✐➺♥ ✷✿

f .f > 0



ứ ữợ t tự

ú ừ ✳

❈❤ù♥❣ ♠✐♥❤✳

n

①→❝ ✤à♥❤✱ t❛ ✤÷đ❝

xn

✈➔

①❡♠ xn ❧➔ ❣✐→ trà

f (x) > 0✳
f (x) > 0 ❤♦➔♥

◆❤÷ ✤➣ ♥â✐ ð ♠ư❝ ❬✷✳✷✳✶❪✱ t❛ ❧✉æ♥ ❝â t❤➸ ❝♦✐

❙❛✉ ✤➙②✱ t❛ ❝❤➾ ①➨t tr÷í♥❣ ❤đ♣

f (x) < 0✳

❚r÷í♥❣ ❤đ♣

t♦➔♥ t÷ì♥❣ tü✳
❑❤❛✐ tr✐➸♥ ❚❛②❧♦r ❜➟❝ ✶ ❝õ❛

f (xn )

t↕✐ ✤✐➸♠

xn−1 ✱

t❛ ❝â✿

(xn − xn−1 )2
.f (αn−1 ).
f (xn ) = f (xn−1 ) + (xn − xn−1 ).f (xn−1 ) +
2
❚ø ✭✷✳✼✮ s✉② r❛

(xn − xn−1 )2
.f (αn−1 ) ≥ 0

f (xn ) =
2

✭✷✳✼✮

✳

▼➦t ❦❤→❝✱

f (xn )
(xn − xn−1 )2 .f (αn−1 )
xn+1 − xn = −
=
≥ 0,
f (xn )
2.f (xn )
{xn } ✤ì♥
f (xn ) < f (α) = 0 ✳
❞♦ ✤â ❞➣②

✤✐➺✉ t➠♥❣✳ ◆➳✉ ❝â

xn > α

✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❜➜t ✤➥♥❣ t❤ù❝

t❤➻ ❞♦

f (xn ) ≥ 0

f (x) < 0

♥➯♥

✳ ◆❤÷ ✈➟②✱

a ≤ xn ≤ xn+1 ≤ ... ≤ α b,
s r tỗ t ợ

lim xn = x

n



õ ❧↕✐✱ ❞➣② ❝→❝ ①➜♣ ①➾ ❧✐➯♥ t✐➳♣ ❧➔ ♠ët ❞➣② ✤ì♥ ✤✐➺✉ t➠♥❣ ✈➔ ❜à ❝❤➦♥

f (x).f (x) < 0 ✮ ❤♦➦❝ ✤ì♥ ✤✐➺✉ ❣✐↔♠ ✈➔ ❜à ❝❤➦♥
f (x).f (x) > 0 tỗ t ợ lim xn = x

tr trữớ ủ
trữớ ủ

õ tốt

ữợ

n

❚❤à ▼✐♥❤ ❚r➙♠

✷✶
❉➵ t❤➜② r➡♥❣

x

❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

f (x) = 0

❚❤➟t ✈➟②✱ ❝❤✉②➸♥ q✉❛ ❣✐ỵ✐ ❤↕♥ tr♦♥❣ ❜✐➸✉ t❤ù❝
❝â

α

x = x−

f (x)
f (x)

✳ ❙✉② r❛

f (x) = 0

❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ ♥➯♥

✳ ❉♦

x=α

(a, b)

✳

xn+1 = xn −

f (xn )
f (xn )

t❛

❧➔



ồ ỵ ❤➻♥❤ ✈➩✳ ✭❍➻♥❤ ✷✳✸ ✈➔ ❍➻♥❤ ✷✳✹✮✳

❍➻♥❤ ✷✳✸✿

❍➻♥❤ ✷✳✹✿

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠


✷✷
✷✳✷✳✸

✣→♥❤ ❣✐→ s❛✐ sè ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥

●✐↔ sû
✤→♥❤ s số

ỵ



| xn |

0 < m ≤| f (x) |

| f (x) |≤ M ✳

❑❤✐ ➜②✱ t❛ ❝â

| f (xn ) | max{| f (xn ) |, x ∈ [a, b]}
≤
m
m

✈➔
| xn − α |≤

❈❤ù♥❣ ♠✐♥❤✳

✈➔

M

| xn xn1 |2 .
2m

ử ỵ tr tr✉♥❣ ❜➻♥❤ ▲❛❣r❛♥❣❡ ✭❝æ♥❣

t❤ù❝ sè ❣✐❛ ❤ú✉ ❤↕♥✮✱ t❛ ❝â✿

f (xn ) − f (α) = (xn − α).f (c)
✈ỵ✐
❱➻

c ∈ (xn , α) ⊂ (a, b).
f (α) = 0 ✈➔ 0 < m ≤| f (x) |

♥➯♥

| f (xn ) − f (α) |=| (xn − α).f (c) |≥ m. | xn − α | .
❙✉② r❛

| f (xn ) |
.
m
❚❛②❧♦r ❝õ❛ f (x)

| xn − α |≤

❉ò♥❣ ❦❤❛✐ tr✐➸♥

t↕✐

xn−1

✿

f (xn ) = f (xn−1 ) + (xn − xn−1 ).f (xn−1 )+
1
+ .(xn − xn−1 )2 .f (c).
2
xn ✈➔ xn−1 ✳
f (xn−1 )
xn = xn−1 −
♥➯♥
f (xn−1 )

tr♦♥❣ ✤â
❉♦

c

♥➡♠ ❣✐ú❛

f (xn−1 ) + (xn − xn−1 ).f (xn−1 ) = 0.
❚❤❛② ✈➔♦ ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ✤÷đ❝✿

❚ø ❝ỉ♥❣ t❤ù❝

1
| f (xn ) |=| .(xn − xn−1 )2 .f (c) | .
2
| f (xn ) |

tr➯♥ ✈➔ ❝æ♥❣ t❤ù❝ | xn − α |≤
t❛ s✉②
m
| f (xn ) |
M
| xn − α |≤
≤
| xn − xn−1 |2 .
m
2m

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

r❛✿

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠


✷✸
◆❤÷ ✈➟②✱ tè❝ ✤ë ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥ ❧➔ ❜➟❝ ❤❛✐✳
P❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ❤ë✐ tư r➜t ♥❤❛♥❤ õ tữớ ữủ sỷ ử
tr ữợ t♦➔♥ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮✳

✷✳✷✳✹

❱➼ ❞ư →♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ t✉②➳♥ ◆❡✇t♦♥

❱➼ ❞ư ✷✳✷✳✶✳ ❚➼♥❤

√

2

❜➡♥❣ ❝→❝❤ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ s❛✉✿

f (x) = x2 − 2 = 0.

✭✷✳✽✮

●✐↔✐✳

f (1) = −1, f (2) = 2 ⇒ f (1).f (2) < 0 ♥➯♥ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐
♥❣❤✐➺♠ ❧➔ [1, 2]✳ ◆❤÷ ✈➟②✱ ✤✐➲✉ ❦✐➺♥ ✶ ✤÷đ❝ t❤ä❛ ♠➣♥✳
f (x) = 2x > 2 ✈ỵ✐ ♠å✐ x ∈ [1, 2]✳
f (x) = 2 > 1 ✈ỵ✐ ♠å✐ x ∈ [1, 2]✳ ❱➟② ✤✐➲✉ ❦✐➺♥ ✷ ✤÷đ❝ t❤ä❛ ♠➣♥✳
❱➻ f (2) = 2 ♥➯♥ t❛ ❝❤å♥ x0 = 2✱ ♥❤÷ ✈➟② t❤➻ f (2).f (x) = 2.2 = 4 > 0
❚❛ t❤➜②

✈➔ ✤✐➲✉ ❦✐➺♥ ✸ ✤÷đ❝ t❤ä❛ ♠➣♥✳
❱➟② t❛ ❝â t❤➸ →♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ◆❡✇t♦♥ ✤➸ t➼♥❤ ♥❣❤✐➺♠ ①➜♣ ①➾
❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✽✮✳
❚❛ ❝â ❜↔♥❣ s❛✉✿

n

x0 = 2

n)
✈ỵ✐ xn+1 = xn − ff (x
(x )

n

0

2

1

1, 5

2

1, 417

3

1, 41421

❇↔♥❣ ✷✳✶✿
❚❛ ❝â t❤➸ ❧➜② ♥❣❤✐➺♠ ①➜♣ ①➾ ❧➔

1, 41421✳ ❚❛ ❜✐➳t r➡♥❣

√

2 = 1, 414213562...✱

♥❤÷ ✈➟② ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ◆❡✇t♦♥ ❤ë✐ tư r➜t ♥❤❛♥❤✳

❱➼ ❞ư ✷✳✷✳✷✳ ❉ị♥❣ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤

x3 − 2x − 10 = 0
[2, 3] ✳

✈ỵ✐ ✤ë ❝❤➼♥❤ ①→❝

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

10−3 ✱

❜✐➳t ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ ❧➔

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠


✷✹
●✐↔✐✳
✣➦t

f (x) = x3 − 2x − 10✳

❑❤✐ ✤â t❛ ❝â✿

f (x) = 3x2 − 2.
f (x) = 6x.
❉➵ t❤➜② r➡♥❣

{xn }n=1,∞

f (3).f (3) > 0

♥➯♥ t❛ ❝❤å♥

x0 = 3✳

❚❛ ①➙② ❞ü♥❣ ❞➣②

♥❤÷ s❛✉✿


3

xn+1 = xn − f (xn ) = xn − xn − 2xn − 10 ,
f (xn )
3x2n 2

n 0
ợ

x0 = 3

t t ữủ

x1 = 2, 5600

f (x1 ) = 1, 6572

x2 = 2, 4662

f (x2 ) = 0, 0668

x3 = 2, 4621

|f (x3 )| = 1, 2501.10−4

❇↔♥❣ ✷✳✷✿
❈❤å♥

m = 10, M = 18
|x3 − x∗ | ≤

❦❤✐ ✤â✿

M
18
.|x3 − x2 |2 ≤ .|0, 0041|2 < 10−3 .
2m
20

❱➻ t❤➳ t❛ ❝â t❤➸ ❝❤å♥ ♥❣❤✐➺♠

x∗ ≈ x3



ì ừ ữỡ t t


ì

ã số ữỡ t õ ❤❛✐ ♥➯♥ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥
❝â tè❝ ✤ë ❤ë✐ tư ❜➟❝ ỗ tớ số ữợ ọ t ♣❤÷ì♥❣
♣❤→♣ ◆❡✇t♦♥ ❧➔♠ ✈✐➺❝ t❤➻ ♥â ❤ë✐ tư ✤➳♥ ♥❣❤✐➺♠ ỡ t ữỡ


ã

ớ sỷ ử ✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ sè

f (x)

♥➯♥ ♥â✐ ❝❤✉♥❣ ♣❤÷ì♥❣

♣❤→♣ ◆❡✇t♦♥ ❤ë✐ tư ♥❤❛♥❤ ❤ì♥ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ✤ỉ✐ ✈➔ ♣❤÷ì♥❣ ♣❤→♣
❞➙② ❝✉♥❣✳
❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠



ã

Pữỡ t s ởt ữỡ t sự ỡ

Pữỡ tỹ sỹ õ ỵ t trữợ ừ
số r trữớ ủ ổ t trữợ t ụ õ t
ử ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ❜➡♥❣ ❝→❝❤ t➼♥❤ ①➜♣ ①➾ ❣✐→ trà ✤↕♦ ❤➔♠ t↕✐ tø♥❣
✤✐➸♠ ✭♠ët tr♦♥❣ ♥❤ú♥❣ ❝→❝❤ ①➜♣ ①➾ ❧➔ t➼♥❤ ❤✐➺✉
❣➛♥✑ ✈ỵ✐

x1 ✮✳

f (x2 ) − f (x1 ) ợ x2

ừ

Pữỡ t õ ỵ

t tr t





❦✐➸♠ tr❛ ✤✐➲✉ ❦✐➺♥ ✤➸ →♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ♣❤ù❝ t↕♣
❤ì♥ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ✤ỉ✐✱ ♣❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣✳ ◆❤ú♥❣ ✤✐➲✉ ❦✐➺♥ ✤➸
♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ❤ë✐ tư ❧➔ q✉❛♥ trå♥❣ ✈➔ ❝➛♥ t❤✐➳t ♣❤↔✐ ❦✐➸♠ tr❛
❦❤✐ →♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♥➔②✳ ❈â tr÷í♥❣ ❤đ♣ ♥➳✉ →♣ ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣
♣❤→♣ ❝❤✐❛ ✤æ✐ ❤♦➦❝ ❞➙② ❝✉♥❣ t❤➻ q✉→ tr➻♥❤ ❧➦♣ s➩ ❤ë✐ tư✱ ❝á♥ ♥➳✉ t❛
→♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ♥❤÷♥❣ ❝❤å♥ ✤✐➸♠ ①✉➜t ♣❤→t ❜❛♥ ✤➛✉

x0

❦❤ỉ♥❣ t❤➼❝❤ ❤đ♣ t❤➻ ❦❤ỉ♥❣ ✤↕t ✤÷đ❝ ❦➳t q✉↔ ♥❤÷ ♠♦♥❣ ♠✉è♥✳

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❙❱❚❍✿ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚r➙♠