❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖
✣❸■ ❍➴❈ ✣⑨ ◆➂◆●
❑❍❖❆ ❚❖⑩◆

✖✖✖✖✖
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P

Ù◆● ❉Ư◆● P❍❺◆ ▼➋▼ ▼❆❚❍❊▼❆❚■❈❆
❚➐▼ ◆●❍■➏▼ ●❺◆ ✣Ĩ◆● ❈Õ❆ P❍×❒◆● ❚❘➐◆❍
❇➀◆● P❍×❒◆● P❍⑩P ◆❊❲❚❖◆

❙✐♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥✿ ✣❖⑨◆
ữợ

◆➤♥❣✱ ✵✺✴✷✵✶✻


▼ö❝ ❧ö❝
▲❮■ ❈❷▼ ❒◆
▼Ð ✣❺❯
✶ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚
✶✳✶

▲Þ ❚❍❯❨➌❚ ❙❆■ ❙➮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✳✶ ❙❛✐ sè t✉②➺t ✤è✐✱ s❛✐ sè t÷ì♥❣ ✤è✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✳✷ ❙❛✐ sè t❤✉ ❣å♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✳✸ ❙❛✐ sè t➼♥❤ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷ ◆●❍■➏▼ ❱⑨ ❑❍❖❷◆● P❍❹◆ ▲■ ◆●❍■➏▼ ✳ ✳ ✳
✶✳✷✳✶ ◆❣❤✐➺♠ t❤ü❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♠ët ➞♥ ✳ ✳ ✳ ✳
✶✳✷✳✷ Þ ♥❣❤➽❛ ❤➻♥❤ ❤å❝ ❝õ❛ ♥❣❤✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
ỹ tỗ t tỹ ừ ữỡ tr ởt

❑❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳ ✳
✳ ✳
✳ ✳
✳ ✳
✳ ✳
✳ ✳
✳ ✳
➞♥
✳ ✳

✳
✳
✳
✳
✳
✳
✳
✳
✳

✳
✳
✳
✳
✳
✳
✳
✳

✳

✳
✳
✳
✳
✳
✳
✳
✳
✳

✹
✺
✼

✼
✼
✼
✽
✾
✾
✾
✶✶
✶✷

✷ ❚➐▼ ◆●❍■➏▼ ●❺◆ ✣Ĩ◆● ❈Õ❆ P❍×❒◆● ❚❘➐◆❍ ❇➀◆●
P❍×❒◆● P❍⑩P ◆❊❲❚❖◆
✶✻
✷✳✶

●■❰■ ❚❍■➏❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✶✳✶ ✣➦t ✈➜♥ ✤➲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✶✳✷ ❈→❝❤ ❣✐↔✐ q✉②➳t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷ P❍×❒◆● P❍⑩P ◆❊❲❚❖◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷✳✶ ▼ỉ t↔ ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷✳✷ ❙ü ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✈➔ ✤→♥❤ ❣✐→ s❛✐ sè
✷✳✸ ▼❐❚ ❙➮ ❇⑨■ ❚❖⑩◆ ❚➐▼ ◆●❍■➏▼ ●❺◆ ✣Ĩ◆●
P❍×❒◆● P❍⑩P ◆❊❲❚❖◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷

✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳
✳ ✳ ✳ ✳ ✳
❇➀◆●
✳ ✳ ✳ ✳ ✳

✶✻
✶✻
✶✼
✶✾
✶✾
✷✸
✷✺

✸ Ù◆● ❉Ư◆● P❍❺◆ ▼➋▼ ▼❆❚❍❊▼❆❚■❈❆ ❚➐▼ ◆●❍■➏▼
●❺◆ ✣Ĩ◆● ❈Õ❆ P❍×❒◆● ❚❘➐◆❍ ❇➀◆● P❍×❒◆●
P❍⑩P ◆❊❲❚❖◆
✷✼
✸✳✶

▼❐❚ ❱⑨■ ◆➆❚ ❱➋ P❍❺◆ ▼➋▼ ▼❆❚❍❊▼❆❚■❈❆ ✳ ✳ ✳ ✳ ✳ ✷✼
✸✳✶✳✶ ●✐ỵ✐ t❤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✸✳✶✳✷ ●✐❛♦ ❞✐➺♥ t÷ì♥❣ t→❝ ❝õ❛ ▼❛t❤❡♠❛t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✸✳✶✳✸ ❈→❝ t➼♥❤ ♥➠♥❣ ❝õ❛ ▼❛t❤❡♠❛t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✸✳✶✳✹ ▼ët sè ❤➔♠ t❤ỉ♥❣ t❤÷í♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✸✳✷ Ù◆● ❉Ư◆● P❍❺◆ ▼➋▼ ▼❆❚❍❊▼❆❚■❈❆ ❱⑨❖ ●■❷■ P❍×❒◆●
❚❘➐◆❍ ❇➀◆● P❍×❒◆● P❍⑩P ◆❊❲❚❖◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

❑➌❚ ▲❯❾◆
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✸✾
✹✵

✸


▲❮■ ❈❷▼ ❒◆
❊♠ ①✐♥ ❜➔② tä sü ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❇❛♥ ●✐→♠ ❍✐➺✉ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷
♣❤↕♠ ✲ ✣↕✐ ❍å❝ ✣➔ ♥➤♥❣✱ ❜❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥✱ ✤➣ t↕♦ ❝ì ❤ë✐ ❝❤♦ ❝❤ó♥❣
❡♠ ✤÷đ❝ ❧➔♠ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳ ❈❤ó♥❣ ❡♠ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥✱ ❧í✐ tr✐ ➙♥
s➙✉ s➢❝ ✤➳♥ t➜t ❝↔ ❝→❝ t❤➛② ❝æ ❣✐→♦ tr♦♥❣ tr÷í♥❣✱ ✤➦❝ ❜✐➺t ❧➔ ❝→❝ t❤➛② ❝ỉ ❣✐→♦
tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✤➣ t➟♥ t➻♥❤ ❝❤➾ ❞↕②✱ tr✉②➲♥ ✤↕t ❝❤♦ ❝❤ó♥❣ ❡♠ ỳ
tự ờ qỵ tr sốt tớ ❣✐❛♥ ✈ø❛ q✉❛✳ ❊♠ ❝↔♠ ì♥ sü ❣✐ó♣ ✤ï✱

❝❤✐❛ s➫ ❝õ❛ t➜t ❝↔ ❝→❝ ❜↕♥ tr♦♥❣ ❧ỵ♣✱ ❝→❝ ❛♥❤ ❝❤à ❦❤â❛ tr➯♥ tr♦♥❣ t❤í✐ ❣✐❛♥
❝❤ó♥❣ ❡♠ ❧➔♠ ♥❣❤✐➯♥ ❝ù✉✳
❈✉è✐ ❝ị♥❣✱ ❡♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ t❤➛② ▲➯ ❍↔✐ ❚r✉♥❣ ữớ trỹ
t ữợ ú ổ q t➙♠✱ ✤ë♥❣ ✈✐➯♥ ❝❤➾ ❞➝♥ t➟♥ t➻♥❤ ✤➸
❝❤ó♥❣ ❡♠ ❤♦➔♥ t❤➔♥❤ tèt ❧✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣ ♥➔②✳
❚✉② ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ s♦♥❣ ❧✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣ ✈➝♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐
♥❤ú♥❣ t❤✐➳✉ sât ✈➲ ♥ë✐ ❞✉♥❣ ❧➝♥ ❤➻♥❤ t❤ù❝ tr➻♥❤ ❜➔②✱ ❝❤ó♥❣ ❡♠ r➜t ♠♦♥❣
♥❤➟♥ ✤÷đ❝ sü ✤â♥❣ ❣â♣ ❝õ❛ qỵ t ổ ồ
t ì♥✦

✹


é
ỵ ỹ ồ t

t t❤ü❝ t➳ ✭tr♦♥❣ ✈➟t ❧➼✱ ❝ì ❤å❝✱ t❤✐➯♥ ✈➠♥ ❤å❝✱ ❦❤♦❛ ❤å❝ ❦ÿ
t❤✉➟t✱ ✤♦ ✤↕❝ r✉ë♥❣ ✤➜t✳✳✳✮ ❞➝♥ ✤➳♥ ✈✐➺❝ ❝➛♥ ♣❤↔✐ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ♣❤ù❝
t↕♣✳ ❈→❝ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❦❤â ❝â t❤➸ ❣✐↔✐ ✤÷đ❝ ✭✤÷❛ ✈➲ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤
❝ì ❜↔♥✮ ❜➡♥❣ ❝→❝ ❜✐➳♥ ✤ê✐ ✤↕✐ sè✱ t❤➟♠ ❝❤➼ tr♦♥❣ ♠ët sè tr÷í♥❣ ❤đ♣ ❝ơ♥❣ ❝â
t❤➸ ❦❤ỉ♥❣ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ t÷í♥❣ ♠✐♥❤✳ ❍ì♥ ♥ú❛✱ ✈➻ ❝→❝ ❝ỉ♥❣ t❤ù❝ ♥❣❤✐➺♠
t❤÷í♥❣ ♣❤ù❝ t↕♣✱ ❝ỉ♥❣ ❦➲♥❤ ♥➯♥ ❝❤♦ ❞ị ❝â ❝ỉ♥❣ t❤ù❝ ♥❣❤✐➺♠✱ ✈✐➺❝ ❦❤↔♦ s→t
❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♥â ❝ô♥❣ ❣➦♣ ♣❤↔✐ r➜t ♥❤✐➲✉ ❦❤â ❦❤➠♥✳ ❱➻ ✈➟②✱ ♥❣❛② tø t❤í✐
❆r❝❤✐♠❡❞❡s✱ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣➛♥ ✤ó♥❣ ✤➣ ✤÷đ❝ ①➙② ❞ü♥❣✳ ◆❤✐➲✉ ♣❤÷ì♥❣
♣❤→♣ ✤➣ trð t❤➔♥❤ ❦✐♥❤ ✤✐➸♥ ✈➔ ✤÷đ❝ sû ❞ư♥❣ rë♥❣ r➣✐ tr♦♥❣ t❤ü❝ t➳✳ ❇ð✐ ✈➟②✱
✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣➛♥ ✤ó♥❣ ✤➸ t➻♠ ♥❣❤✐➺♠ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤
♥➔② trð ♥➯♥ ❝➜♣ t❤✐➳t ✈➔ tü ♥❤✐➯♥✳
❈ị♥❣ ✈ỵ✐ sü ♣❤→t tr✐➸♥ ❝õ❛ t✐♥ ❤å❝✱ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❣➛♥ ✤ó♥❣
õ ỵ tỹ t ỡ ởt ♣❤÷ì♥❣ tr➻♥❤ ❜➡♥❣ t❛② tr➯♥ ❣✐➜②✱
❝â ❦❤✐ ♣❤↔✐ ♠➜t r➜t ♥❤✐➲✉ t❤í✐ ❣✐❛♥ ✈ỵ✐ ♥❤ú♥❣ s❛✐ sât ❞➵ ①↔② r❛✱ t❤➻ ✈ỵ✐ sü

❤é trå ❝õ❛ ❝→❝ ♣❤➛♥ ♠➲♠ ❝❤✉②➯♥ ❞ư♥❣ ❝❤ó♥❣ t❛ ❝❤➾ ❝➛♥ ✈➔✐ ♣❤ót✱ t❤➟♠ ❝❤➼
✈➔✐ ❣✐➙②✳ ▼➦t ỵ tt sỹ ở tử tố ✤ë ❤ë✐ tö✱ ✤ë ❝❤➼♥❤
①→❝✱ ✤ë ♣❤ù❝ t↕♣ t➼♥❤ t♦→♥✳✳✳✮ s➩ ✤÷đ❝ ♥❤➻♥ t❤➜② rã ❤ì♥ ❦❤✐ sû ❞ư♥❣ ❝→❝ ♣❤➛♥
♠➲♠ ♥➔②✳ ❱➻ ✈➟②✱ ✈✐➺❝ sû ❞ö♥❣ t❤➔♥❤ t❤↕♦ ❝→❝ ❝ỉ♥❣ ❝ư t➼♥❤ t♦→♥ ❧➔ ❝➛♥ t❤✐➳t
❝❤♦ ❝ỉ♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉✱ ♥❤➜t ❧➔ ✤è✐ ✈ỵ✐ ❤å❝ s✐♥❤ s✐♥❤ ✈✐➯♥✳
❱ỵ✐ ♠♦♥❣ ♠✉è♥ ❝â t❤➸ ❤✐➸✉ rã ❤ì♥ ✈➲ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤
♣❤✐ t✉②➳♥ ♥❤➡♠ ✤→♣ ù♥❣ ♥❣✉②➺♥ ✈å♥❣ ♥❣❤✐➯♥ ự ồ ừ t
ỗ tớ ữủ sỹ ủ ỵ ở ừ ữợ ❚❙✳ ▲➯ ❍↔✐
❚r✉♥❣ ♥➯♥ tỉ✐ ❝❤å♥ ✤➲ t➔✐✿ ✧Ù♥❣ ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ t➻♠ ♥❣❤✐➺♠
❣➛♥ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥✧ ❧➔♠ ❧✉➟♥ ✈➠♥ tèt
♥❣❤✐➺♣ ❝õ❛ ♠➻♥❤✳

✺


✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉

▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧➔ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ✭t✐➳♣ t✉②➳♥✮ ✤➸ ①❡♠
①➨t ✈➔ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✱ tø ✤â s♦ s s số ợ
ừ ữỡ tr õ ỗ tớ ự ự ử
tt t ❝❤÷ì♥❣ tr➻♥❤ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣
◆❡✇t♦♥ ✈➔ ổ t ừ ữỡ tr ỗ t❤à t❤ỉ♥❣ q✉❛
❝→❝ ❣â✐ ❧➺♥❤ ✤➣ ✤÷đ❝ ❧➟♣ tr➻♥❤✳

✸✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉

❚➻♠ ✤å❝ t➔✐ ❧✐➺✉ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ✈➔ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ t➼♥❤ ①➜♣ ①➾
❦❤→❝❀ ♣❤➙♥ t➼❝❤ t➔✐ ❧✐➺✉ ✈➔ ❤➺ t❤è♥❣ ❤â❛ ❝→❝ ❦✐➳♥ t❤ù❝❀ tr❛♦ ✤ê✐✱ t ợ
ữợ t tữớ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣
✤➲ t➔✐✳

✹✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉

✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉✿ ◆❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ✤➸ t➻♠ ♥❣❤✐➺♠ ❣➛♥
✤ó♥❣ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❧➟♣ tr➻♥❤ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ tr♦♥❣ ▼❛t❤❡♠❛t✐❝❛✳
P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉✿ ◆❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ❝❤♦ ❝→❝ ♣❤÷ì♥❣
tr➻♥❤ ♣❤✐ t✉②➳♥✳

✺✳ ❈➜✉ tró❝ ❧✉➟♥ ✈➠♥

◆❣♦➔✐ ♣❤➛♥ ▼ð ✤➛✉ ✈➔ ❑➳t ỗ ữỡ
ữỡ ởt số t❤ù❝ ❝❤✉➞♥ ❜à
❈❤÷ì♥❣ ✷✿ P❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣✳
❈❤÷ì♥❣ ✸✿ Ù♥❣ ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛
♣❤÷ì♥❣ tr➻♥❤ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥✳

✻


ữỡ


ị ❙➮
✶✳✶✳✶ ❙❛✐ sè t✉②➺t ✤è✐✱ s❛✐ sè t÷ì♥❣ ✤è✐
❚r♦♥❣ t➼♥❤ t t tữớ ợ tr ❣➛♥ ✤ó♥❣ ❝õ❛ ❝→❝
✤↕✐ ❧÷đ♥❣✳ ❚❛ ♥â✐ a ❧➔ sè ❣➛♥ ✤ó♥❣ ❝õ❛ a∗ ♥➳✉ a ❦❤ỉ♥❣ s❛✐ ❦❤→❝ a∗ ♥❤✐➲✉✳ ✣↕✐
❧÷đ♥❣ ∆ ✿❂ ⑤a−a∗ | ❣å✐ ❧➔ s❛✐ sè t❤➟t sü ❝õ❛ a✳ ❉♦ ❦❤æ♥❣ ❜✐➳t a∗ ♥➯♥ t❛ ❝ơ♥❣
❦❤ỉ♥❣ ❜✐➳t ∆✳ ❚✉② ♥❤✐➯♥✱ t❛ ❝â t❤➸ t➻♠ ✤÷đ❝ ∆a ≥ 0 ❣å✐ ❧➔ s❛✐ sè t✉②➺t ✤è✐
❝õ❛ a✱ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥✿
|a − a∗ | ≤ ∆a

✭✶✳✶✮

❤❛② a − ∆a ≤ a∗ ≤ a + ∆a✳ ✣÷ì♥❣ ♥❤✐➯♥ ∆a t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✶✳✶✮ ❝➔♥❣
♥❤ä ❝➔♥❣ tèt✳ ❙❛✐ sè t÷ì♥❣ ✤è✐ ❝õ❛ a ❧➔
δa :=

∆a
|a|

✶✳✶✳✷ ❙❛✐ sè t❤✉ ❣å♥
▼ët sè t❤➟♣ ♣❤➙♥ a ❝â ❞↕♥❣ tê♥❣ q✉→t ♥❤÷ s❛✉✿
a = ±(βp 10p + βp−1 10p−1 + ... + βp−s 10p−s )

❚r♦♥❣ ✤â 0 ≤ βi ≤ 9(i = p − 1, 1 − s); βp > 0 ❧➔ ♥❤ú♥❣ sè ♥❣✉②➯♥✳ ◆➳✉
p − s ≥ 0 t❤➻ a ❧➔ sè ♥❣✉②➯♥❀ p − s = −m (m > 0) t a õ ỗ m
ỳ số s = +∞✱ a ❧➔ sè t❤➟♣ ♣❤➙♥ ✈æ ❤↕♥✳ ❚❤✉ ❣å♥ ♠ët sè a ❧➔ ✧✈ùt

✼


❜ä✧ ♠ët sè ❝→❝ ❝❤ú sè ❜➯♥ ♣❤↔✐ a ✤➸ ✤÷đ❝ ♠ët sè a ♥❣➢♥ ❣å♥ ❤ì♥ ✈➔ ❣➛♥
✤ó♥❣ ♥❤➜t ✈ỵ✐ a✳
◗✉② t➢❝ t❤✉ ❣å♥✿ ●✐↔ sû
a = (βp 10p + ... + βj 10j + ... + βp−s 10p−s )

✈➔ t❛ ❣✐ú ❧↕✐ ✤➳♥ sè ❤↕♥❣ t❤ù ❥✳ ●å✐ ♣❤➛♥ ✧✈ùt ❜ä✧ ❧➔ ϕ ✱ t❛ ✤➦t
a = βp 10p + ... + βj+1 10j+1 + βj 10j

tr♦♥❣ ✤â✿


β + 1 ♥➳✉ 0.5 × 10j < ϕ < 10j ,
j
βj :=
β
♥➳✉ 0 < ϕ < 0.5 × 10j ,
j

✭✶✳✷✮

♥➳✉ ϕ = 0.5 × 10j t❤➻ βj = βj ♥➳✉ βj ❝❤➤♥ ✈➔ βj = βj + 1 ♥➳✉ βj ❧➫ ✈➻ t➼♥❤
t♦→♥ ✈ỵ✐ sè ❝❤➤♥ t❤✉➟♥ t✐➺♥ ❤ì♥✳

✶✳✶✳✸ ❙❛✐ sè t➼♥❤ t♦→♥
❚r♦♥❣ t➼♥❤ t♦→♥ t❛ t❤÷í♥❣ ❣➦♣ ✹ ❧♦↕✐ s❛✐ sè s❛✉✿
❛✮ ❙❛✐ sè ❣✐↔ t❤✐➳t ✲ ❉♦ ♠æ õ ỵ tữ õ t tỹ t sè
♥➔② ❦❤ỉ♥❣ ❧♦↕✐ trø ✤÷đ❝✳
❜✮ ❙❛✐ sè ♣❤÷ì♥❣ ♣❤→♣ ✲ ❈→❝ ❜➔✐ t♦→♥ t❤÷í♥❣ ❣➦♣ r➜t ♣❤ù❝ t↕♣✱ ❦❤ỉ♥❣ t❤➳
❣✐↔✐ ✤ó♥❣ ✤÷đ❝ ♠➔ ♣❤↔✐ sû ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣➛♥ ✤ó♥❣✳ ❙❛✐ sè ♥➔② s➩
✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❝❤♦ tø♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝ö t❤➸✳
❝✮ ❙❛✐ sè ❝→❝ sè ❧✐➺✉ ✲ ❈→❝ sè ❧✐➺✉ t❤÷í♥❣ t❤✉ ✤÷đ❝ ❜➡♥❣ t❤ü❝ ♥❣❤✐➺♠ ❞♦
✤â ❝â s❛✐ sè✳
❞✮ ❙❛✐ sè t➼♥❤ t♦→♥ ✲ ❈→❝ sè ✈è♥ ✤➣ ❝â s❛✐ sè✱ ❝á♥ t❤➯♠ s❛✐ sè t❤✉ ❣å♥ ♥➯♥
❦❤✐ t➼♥❤ t♦→♥ s➩ ①✉➜t ❤✐➺♥ s❛✐ sè t➼♥❤ t♦→♥✳
●✐↔ sû ♣❤↔✐ t➻♠ ✤↕✐ ❧÷đ♥❣ y t❤❡♦ ❝ỉ♥❣ t❤ù❝✿
y = f (x1 , x2 , ..., xn )

●å✐ x∗i , y ∗ (i = 1, n) ✈➔ xi , y (i = 1, n) ❧➔ ❝→❝ ❣✐→ trà ✤ó♥❣ ✈➔ ❣➛♥ ✤ó♥❣ ❝õ❛
❝→❝ ✤è✐ sè ✈➔ ❤➔♠ sè✳ ◆➳✉ f ❞÷ì♥❣ ❦❤↔ ✈✐ ❧✐➯♥ tư❝ t❤➻

n
∗

|y − y | = |f (x1 , x2 , ..., xn ) −

f (x∗1 , ..., x∗n )|

|fi ||xi − x∗i |.

=
i=1

✽


df
tr♦♥❣ ✤â fi ❧➔ ✤↕♦ ❤➔♠ dx
t➼♥❤ t↕✐ t❤í✐ ✤✐➸♠ tr✉♥❣ ❣✐❛♥✳ ❉♦
i
∆xi ❦❤→ ❜➨✱ t❛ ❝â t❤➸ ❝♦✐

df
dxi

❧✐➯♥ tö❝ ✈➔

n

|fi (x1 , ..., xn )|∆xi .

∆y =
i=1

❉♦ ✤â

∆y
=
δ=
|y|

n

|
i=1

d
ln f |∆xi .
dxi

✶✳✷ ◆●❍■➏▼ ❱⑨ ❑❍❖❷◆● P❍❹◆ ▲■ ◆●❍■➏▼
✶✳✷✳✶ ◆❣❤✐➺♠ t❤ü❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♠ët ➞♥
❳➨t ♣❤÷ì♥❣ tr➻♥❤ ♠ët ➞♥✿
f (x) = 0



tr õ f ởt số trữợ ❝õ❛ ✤è✐ sè x✳
◆❣❤✐➺♠ t❤ü❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮ ❧➔ sè t❤ü❝ α t❤ä❛ ♠➣♥ ✭✶✳✸✮ tù❝ ❧➔
❦❤✐ t❤❛② α ✈➔♦ x ð ✈➳ tr→✐ t❛ ✤÷đ❝✿
f (α) = 0.

✭✶✳✹✮

✶✳✷✳✷ ị ồ ừ
ỗ t ừ ❤➔♠ sè✿
y = f (x)

✭✶✳✺✮

tr♦♥❣ ♠ët ❤➺ tå❛ ✤ë ✈✉æ♥❣ õ sỷ ỗ t t trử
t↕✐ ♠ët ✤✐➸♠ ▼ t❤➻ ✤✐➸♠ ▼ ♥➔② ❝â t✉♥❣ ✤ë y = 0 ✈➔ ❤♦➔♥❤ ✤ë x = α✳
❚❤❛② ❝❤ó♥❣ ✈➔♦ ✭✶✳✺✮ t❛ ✤÷đ❝✿
0 = f (α)

❱➟② ❤♦➔♥❤ ✤ë α ❝õ❛ ❣✐❛♦ ✤✐➸♠ ▼ ❝❤➼♥❤ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✸✮✳

✾

✭✶✳✻✮


ị ồ ừ
rữợ ỗ t t ụ õ t t ữỡ tr ữỡ
tr tữỡ ữỡ
g(x) = h(x)

ỗ ỗ t ừ ❤❛✐ ❤➔♠ sè ✭❍➻♥❤ ✶✳✷✮
y = g(x), y = h(x)

✭✶✳✽✮

●✐↔ sỷ ỗ t t t ❝â ❤♦➔♥❤ ✤ë s = α t❤➻ t❛ ❝â✿
g(α) = h()

ỗ t số g(x), h(x)





❱➟② ❤♦➔♥❤ ✤ë α ❝õ❛ ❣✐❛♦ ✤✐➸♠ ▼ ❝õ❛ ❤❛✐ ỗ t ởt
ừ tự ừ

ỹ tỗ t tỹ ừ ữỡ tr ởt
rữợ t t ú tỹ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮ t❛
♣❤↔✐ tü ❤ä✐ ①❡♠ ♥❣❤✐➺♠ t❤ü❝ õ tỗ t ổ tr ớ t õ t
ũ ữỡ ỗ t ử tr ❚❛ ❝ơ♥❣ ❝â t❤➸ ❞ị♥❣ ✤à♥❤ ❧➼ s❛✉✿

✣à♥❤ ❧➼ ✶✳✶✳ ◆➳✉ ❝â ❤❛✐ sè t❤ü❝ a ✈➔ b (a < b) s❛♦ ❝❤♦ f (a) ✈➔ f (b) tr→✐
❞➜✉✱ tù❝



f (a).f (b) < 0

ỗ tớ f (x) tử tr➯♥ [a, b] t❤➻ ð tr♦♥❣ ❦❤♦↔♥❣ (a, b) ❝â ➼t ♥❤➜t ♠ët
♥❣❤✐➺♠ t❤ü❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮✳

❈❤ù♥❣ ♠✐♥❤✿

❈❤✐❛ ✤ỉ✐ ✤♦↕♥ [a, b] t❤➔♥❤ ❤❛✐ ♣❤➛♥ ❜➡♥❣ ♥❤❛✉ ❜ð✐ ✤✐➸♠
• ◆➳✉ f ( a+b
2 ).f (a) > 0 t❤➻ ✤➦t a1 =

a+b
2 ; b1

• ◆➳✉ f ( a+b
2 ).f (a) < 0 t❤➻ ✤➦t a1 = a; b1 =

= b✳
a+b
2 ✳

❈❤✐❛ ✤♦↕♥ [a1 , b1 ] t❤➔♥❤ ❤❛✐ ♣❤➛♥ ❜➡♥❣ ♥❤❛✉ ❜ð✐ ✤✐➸♠
1
• ◆➳✉ f ( a1 +b
2 ).f (a1 ) > 0 t❤➻ ✤➦t a2 =

a+b
2 ✳

a1 +b1
2 ; b2

= b1 ✳

1
• ◆➳✉ f ( a1 +b
2 ).f (a1 ) > 0 t❤➻ ✤➦t a2 = a1 ; b2 =

a1 +b1
2 ✳

a1 +b1
2 ✳

❈ù t✐➳♣ tö❝ q✉→ tr➻♥❤ tr➯♥ t❛ ①→❝ ✤à♥❤ ✤÷đ❝ ❤❛✐ ❞➣② an ✱ bn ♠➔ an ❧➔ ❞➣② t➠♥❣✱
bn ❧➔ ❞➣② ❣✐↔♠ ✈➔ f (an ) ❝ị♥❣ ❞➜✉ ✈ỵ✐ f (a)✱f (bn ) ❝ị♥❣ ❞➜✉ ✈ỵ✐ f (b)✳
⇒ f (an ).f (bn ) < 0, ∀n ∈ N✳
❉➣② an t➠♥❣✱ ❜à ❝❤➦♥ tr➯♥ ❜ð✐ b lim an tỗ t
x

bn ữợ a lim bn tỗ t
t = lim an ✱ β = lim bn ✳
x→∞
x→∞
⇒ lim (bn − an ) = β − α✳

x→∞

x→∞

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

x→∞

✶✶


⇒ lim f (bn ) = f (α) = lim f (an )✳
x→∞

x→∞

⇒ 0 ≥ f (α) ≥ 0 ⇒ f (α) = 0 ✈➔ α ∈ (a, b)✳
❱➟② ✤à♥❤ ❧➼ ✶✳✶ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳
✣✐➲✉ ✤â ❝â t❤➸ ✤÷đ❝ ồ tr ỗ t ỗ t ừ ❤➔♠ sè
y = f (x) t↕✐ a ≤ x ≤ b ❧➔ ♠ët ✤÷í♥❣ ❧✐➲♥ ♥è✐ ❤❛✐ ✤✐➸♠ ❆ ✈➔ ữợ
tr trử ❝➢t trö❝ ❤♦➔♥❤ t↕✐ ➼t ♥❤➜t ♠ët ✤✐➸♠ ð tr♦♥❣
❦❤♦↔♥❣ tø a ✤➳♥ b✳ ❱➟② ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮ ❝â ➼t t ởt tr
(a, b)

ỗ t ❤➔♠ sè y(x)t↕✐ a ≤ x ≤ b

✶✳✷✳✹ ❑❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❑❤♦↔♥❣ (a, b) ♥➔♦ ✤â ❣å✐ ❧➔ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ ❝õ❛
♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮ ♥➳✉ ♥â ❝❤ù❛ ♠ët ✈➔ ❝❤➾ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤â✳
✣➸ t➻♠ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ t❛ ❝â ✤à♥❤ ❧➼✿

✣à♥❤ ❧➼ ✶✳✷✳ ◆➳✉ (a, b) ❧➔ ♠ët ❦❤♦↔♥❣ tr♦♥❣ ✤â ❤➔♠ số f (x) tử

ỡ ỗ tớ f (a) ✈➔ f (b) tr→✐ ❞➜✉✱ tù❝ ❧➔ ❝â ✭✶✳✶✵✮ t❤➻ (a, b) ❧➔ ♠ët
❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✶✳✸✳

❈❤ù♥❣ ♠✐♥❤✿ ❚ø ❣✐↔ t❤✐➳t✱ ✈➻ f (x) ❧✐➯♥ tư❝ ✈➔ ✤ì♥ ✤✐➺✉ ♥➯♥ tr➯♥ (a, b), f (x)

t➠♥❣ ❤♦➦❝ ❣✐↔♠✳
❍ì♥ ♥ú❛✱ tø ✤✐➲✉ ❦✐➺♥ f (a).f (b) < 0 ự tọ út ừ ỗ t
số f (x) ♥➡♠ ✈➲ ❤❛✐ ♣❤➼❛ ❝õ❛ trö❝ ❤♦➔♥❤✳

✶✷


❑➳t ❤đ♣ ✈ỵ✐ t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❤➔♠ sè t❛ s✉② r❛ (a, b) ❧➔ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐
♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ f (x) = 0✳
❱➟② ✤à♥❤ ❧➼ ✶✳✷ ✤➣ ✤÷đ❝ ự
õ t ồ ỗ t ỗ t ừ số
y = f (x) ❝➢t trö❝ ❤♦➔♥❤ t↕✐ ♠ët ✈➔ ❝❤➾ ♠ët ✤✐➸♠ ð tr♦♥❣ (a, b)✳ ❱➟② (a, b)
❝❤ù❛ ♠ët ✈➔ ❝❤➾ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮✳

❍➻♥❤ ✶✳✹✿ ❑❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ f (x) = 0
◆➳✉ f (x) ❝â ✤↕♦ ❤➔♠ t❤➻ ✤✐➲✉ ❦✐➺♥ ✤ì♥ ✤✐➺✉ ❝â t❤➸ t❤❛② ❜➡♥❣ ✤✐➲✉ ❦✐➺♥
❦❤æ♥❣ ✤ê✐ ❞➜✉ ❝õ❛ ✤↕♦ ❤➔♠ ✈➻ ✤↕♦ ❤➔♠ ❦❤ỉ♥❣ ✤ê✐ ❞➜✉ t❤➻ ❤➔♠ sè ✤ì♥ ✤✐➺✉✳
❚❛ ❝â✿

✣à♥❤ ❧➼ ✶✳✸✳ ◆➳✉ (a, b) ❧➔ ♠ët ❦❤♦↔♥❣ tr♦♥❣ ✤â ❤➔♠ f (x) ❧✐➯♥ tư❝✱ ✤↕♦ ❤➔♠

f (x) ❦❤ỉ♥❣ ✤ê✐ ❞➜✉ ✈➔ f (a), f (b) tr→✐ ❞➜✉ t❤➻ (a, b) ❧➔ ♠ët ❦❤♦↔♥❣ ♣❤➙♥ ❧✐
♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮✳

▼✉è♥ t➻♠ ❝→❝ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮ t❤÷í♥❣ ♥❣÷í✐
t❛ ♥❣❤✐➯♥ ❝ù✉ sü ❜✐➳♥ t❤✐➯♥ ❝õ❛ ❤➔♠ sè y = f (x) rỗ ử

❞ư ✶✳✶✳ ❈❤♦ ♣❤÷ì♥❣ tr➻♥❤✿
f (x) = 2x3 − 3x + 5

✭✶✳✶✶✮

❍➣② ❝❤ù♥❣ tä ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✶✮ ❝â ♥❣❤✐➺♠ t❤ü❝ t

rữợ t t t sỹ t❤✐➯♥ ❝õ❛ ❤➔♠ sè f (x)✳ ◆â ①→❝ ✤à♥❤ ✈➔ ❧✐➯♥ tö❝

✶✸


t ồ x ỗ tớ
1
f (x) = 6x2 3 = 0 ⇔ x = ± √ .
2

❚❛ ❝â ❜↔♥❣ ❜✐➳♥ t❤✐➯♥ ♥❤÷ tr♦♥❣ ❍➻♥❤ ✶✳✺✿
tr♦♥❣ ✤â✿
√
1
1
1
f (m) = f (− √ ) = 2.(− √ ) + 3. √ + 5 = 5 + 2 > 0.
2
2 2
2
√
1
1

1
f (M ) = f ( √ ) = 2.( √ ) − 3. √ + 5 = 5 − 2 > 0.
2
2 2
2

❍➻♥❤ ✶✳✺✿ ❇↔♥❣ ❜✐➳♥ t❤✐➯♥ ❝õ❛ ❤➔♠ sè f (x) = 2x3 3x + 5
ỗ t t trö❝ ❤♦➔♥❤ t↕✐ ✶ ✤✐➸♠ ❞✉② ♥❤➜t ✭❍➻♥❤ ✶✳✻✮✱ ❞♦ ✤â ♣❤÷ì♥❣
tr➻♥❤ ✭✶✳✶✶✮ ❝â ✶ ♥❣❤✐➺♠ t❤ü❝ ❞✉② ♥❤➜t✱ ❦➼ ❤✐➺✉ ♥â ❧➔ α.

✶✹


ỗ t số f (x) = 2x3 − 3x + 5 tr♦♥❣ ❦❤♦↔♥❣ ❬✲✷✱✷❪
❚❛ t➼♥❤ t❤➯♠ f (−2) = 2.(−2)3 − 3.(−2) + 5 = −5 < 0; f (0) = 2.(0)3 −
3(0) + 5 = 5 > 0✳ ◆❤÷ ✈➟②✱ f (−2).f (0) < 0✳
❱➟② ❦❤♦↔♥❣ (−2, 0) ❝❤ù❛ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✶✮✳

✶✺


ữỡ

ể ế
Pì
Pì PP
●■❰■ ❚❍■➏❯
✷✳✶✳✶ ✣➦t ✈➜♥ ✤➲
❈❤ó♥❣ t❛ t❤÷í♥❣ t➻♠ ❤✐➸✉ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ trü❝ t✐➳♣✳ ◆➳✉
♠å✐ t➼♥❤ t♦→♥ ❝õ❛ t❛ ❧➔ ❝❤➼♥❤ ①→❝ t❤➻ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✤â ❝❤♦ ❦➳t q✉↔ ❤♦➔♥

t♦➔♥ ❝❤➼♥❤ ①→❝✳ ❚✉② ♥❤✐➯♥ tr♦♥❣ t❤ü❝ t➳ ❦❤✐ t➼♥❤ t♦→♥ ❝❤ó♥❣ t❛ t❤÷í♥❣ ①✉②➯♥
♣❤↔✐ ❧➔♠ trá♥ ❝→❝ sè✱ ♥❣❤➽❛ ❧➔ t❛ ❝❤➾ t➼♥❤ t♦→♥ tr➯♥ ❝→❝ sè ú tổ
õ ữ rt ợ t q ố ũ ố ợ ữỡ tr
số t✉②➳♥ t➼♥❤ ✧♥❤↕② ❝↔♠✧ ✈ỵ✐ s❛✐ sè✳ ❱➻ ✈➟②✱ ❝❤ó♥❣ t❛ ♥➯♥ ♥❣❤✐➯♥ ❝ù✉ ❝→❝
♣❤÷ì♥❣ ♣❤→♣ ❣➛♥ ✤ó♥❣ ✤➸ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ♠ët ♣❤÷ì♥❣ tr➻♥❤✳
❈❤♦ ♣❤÷ì♥❣ tr➻♥❤✿
f (x) = 0
✭✷✳✶✮
tr♦♥❣ ✤â f ❧➔ ♠ët ❤➔♠ ✤↕✐ sè ❤♦➦❝ s✐➯✉ ✈✐➺t✳ ❚➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
✭✷✳✶✮ ❧➔ ♠ët ❜➔✐ t♦→♥ t❤÷í♥❣ ❣➦♣ tr♦♥❣ ❦ÿ t❤✉➟t✳ ◆➳✉ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ ❧➔
♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè ❜➟❝ ♥ ❝â ❞↕♥❣✿
a0 xn + a1 xn−1 + ... + an−1 x + an = 0 (a0 = 0)

✭✷✳✷✮

t❤➻ ✈ỵ✐ n = 1, n = 2✱ t❛ ❝â ❝ỉ♥❣ t❤ù❝ t➼♥❤ ♥❣❤✐➺♠ ♠ët ❝→❝❤ ✤ì♥ ❣✐↔♥✳ ◆❣÷í✐
t❛ ❝ơ♥❣ t➻♠ r❛ ♥❤ú♥❣ ❝ỉ♥❣ t❤ù❝ t➼♥❤ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✷✮ ❦❤✐ n = 3 ✈➔ n = 4✱

✶✻


ữ sỷ ử rt ự t ỏ ợ ỳ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè tø ❜➟❝
✺ trð ❧➯♥ ❤♦➦❝ ♣❤÷ì♥❣ tr➻♥❤ s✐➯✉ ✈✐➺t t❤➻ ❦❤æ♥❣ ❝â ❝æ♥❣ t❤ù❝ t➼♥❤ ♥❣❤✐➺♠✳
❱➻ ✈➟②✱ ✈✐➺❝ t➻♠ ♥❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❣➛♥ ✤ó♥❣ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè
✈➔ s✐➯✉ ✈✐➺t ❝ơ♥❣ ♥❤÷ ✈✐➺❝ ✤→♥❤ ❣✐→ ♠ù❝ ✤ë ❝❤➼♥❤ ①→❝ ❝õ❛ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣
t➻♠ ✤÷đ❝ ❝â ♠ët ✈❛✐ trá r➜t q✉❛♥ trå♥❣✳

✷✳✶✳✷ ❈→❝❤ ❣✐↔✐ q✉②➳t
❚❤æ♥❣ tữớ q tr ữỡ tr ỗ ữợ s
ã ữợ sỡ ở é t❛ t➻♠ ♠ët ❦❤♦↔♥❣ ✤õ ❜➨ ❝❤ù❛ ♥❣❤✐➺♠

❝õ❛ f (x)✳
• ữợ t ợ ở ❝➛♥ t❤✐➳t✳

✣➸ ❣✐↔✐ sì ❜ë ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶ ✮t❛ sû ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✤ì♥ ❣✐↔♥ ♥❤÷
♣❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ✤ỉ✐ ữỡ ỗ t
Pữỡ ổ
sỷ ❤➔♠ f (x) ❧✐➯♥ tö❝ tr➯♥ [a, b] ✈➔ f (a).f (b) < 0✳
●å✐ ∆0 := [a, b]✱ t❛ ❝❤✐❛ ✤æ✐ ∆0 ✈➔ ❝❤å♥ ∆1 := [a1 , b1 ] ❧➔ ♠ët tr♦♥❣ ❤❛✐
♥û❛ ❝õ❛ ∆0 s❛♦ ❝❤♦ f (a1 ).f (b1 ) 0
õ ữợ tự n t❛ ❝â✿
∆n = [an , bn ] ⊂ ∆n−1 ⊂ ...∆0 .

✭✷✳✸✮

◆❣♦➔✐ r❛ bn − an = (b−a)
2n −→ 0 ✭❦❤✐ n −→ ∞✮✳ ❉➵ t❤➜② ❞➣② an ✤ì♥ ✤✐➺✉
t➠♥❣✱ ❜à ❝❤➦♥ tr➯♥ ❜ð✐ b ❝á♥ ❞➣② bn ✤ì♥ ✤✐➺✉ ữợ a ỡ
ỳ bn an −→ 0 s✉② r❛ an , bn −→ α (n −→ ∞)✳
❱➻ f (an ).f (bn ) ≤ 0 ♥➯♥ ❝❤å♥ n −→ ∞✱ t❛ ❝â [f (α)]2 ≤ 0✱ s✉② r❛ f (α) = 0✳
◆❣♦➔✐ r❛✱ t❛ ❝â ÷ỵ❝ ❧÷đ♥❣ s❛✐ sè s❛✉✿
b−a
0 ≤ α − an ≤ b n an = n .
2
ì ừ ữỡ ♣❤→♣ ❝❤✐❛ ✤ỉ✐ ❧➔ t❤✉➟t t♦→♥ r➜t ✤ì♥ ❣✐↔♥✱ ❞♦ ✤â ❞➵
❧➟♣ tr➻♥❤ tr➯♥ ♠→② t➼♥❤✳ ▼➦t ❦❤→❝✱ ✈➻ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ✤ỉ✐ sû ❞ư♥❣ r➜t ➼t
t❤ỉ♥❣ t✐♥ ✈➲ ❤➔♠ f ♥➯♥ tè❝ ✤ë ❤ë✐ tư ❦❤→ ❝❤➟♠✳
❜✳ P❤÷ì♥❣ ♣❤→♣ ỗ t
ỗ t số y = f (x) tr ổ ổ ở ừ
ỗ t❤à ♥â✐ tr➯♥ ✈ỵ✐ trư❝ ❤♦➔♥❤ ❝❤➼♥❤ ❧➔ ♥❣❤✐➺♠ ❝➛♥ t➻♠✳ ◆❤✐➲✉ ❦❤✐ t❛ ❜✐➳♥

✶✼


✤ê✐ ♣❤÷ì♥❣ tr➻♥❤ f (x) = 0 ✈➲ ❞↕♥❣ t÷ì♥❣ ✤÷ì♥❣ ϕ(x) = ψ(x)✳ ◆❣❤✐➺♠ ❝➛♥
t➻♠ ❧➔ ❤♦➔♥❤ ✤ë ❣✐❛♦ ừ ỗ t y = (x) y = ψ(x)✳

❱➼ ❞ö ✷✳✶✳ ❚➻♠ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✉✿
x3 −

√
3

✭✷✳✹✮

x−5=0

√
✣➦t y = f (x) = x3 − 3 x − 5✳ ❚❛ ❝â t❤➸ ❞➵ ❞➔♥❣ t➼♥❤ ✤÷đ❝ f (0) = −5 < 0
√
✈➔ f (3) = 33 − 3 3 − 5 20, 5577504 > 0✳ ❉♦ ✤â✱ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✹✮ ❝â ➼t
♥❤➜t ✶ ♥❣❤✐➺♠ tr♦♥❣ ❦❤♦↔♥❣ (0, 3)✳ ✣➸ ①❡♠ ❞→♥❣ ✤✐➺✉ ❤➻♥❤ ❤å❝ ừ ỗ t
t õ t ớ tt ỗ t tr (0, 3)

ỗ t sè f (x) = x3 − √x − 5
3

◆❤➻♥ ✈➔♦ ỗ t t t ỗ t t trử t↕✐ ♠ët ✤✐➸♠ tr♦♥❣
❦❤♦↔♥❣ (0, 3)✳ ❚✉② ♥❤✐➯♥✱ ✤➸ ❝❤➼♥❤ ①→❝ ❤ì♥ t❛ ❝➛♥ t➼♥❤ ❣✐→ trà ❝õ❛ ❤➔♠ sè t↕✐
❝→❝ ✤✐➸♠ x = 1 ✈➔ x = 2✳ ❚❛ ❝â✿ f (1) = −5 < 0, f (2) = 1, 740079 > 0✳
❱➟② ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✹✮ ❝â ♠ët ♥❣❤✐➺♠ ❞✉② ♥❤➜t tr♦♥❣ ❦❤♦↔♥❣ (1, 2)✳

❙❛✉ ❦❤✐ ✤➣ t→❝❤ ✤÷đ❝ ♥❣❤✐➺♠ t❤➻ ❝æ♥❣ ✈✐➺❝ t✐➳♣ t❤❡♦ ❧➔ ❝❤➼♥❤ ①→❝ ❤â❛
♥❣❤✐➺♠ ở tt tỹ ữợ ♥➔②✱ t❛ ❝â t❤➸ sû ❞ư♥❣
♠ët sè ♣❤÷ì♥❣ ♣❤→♣ s❛✉✿ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣✱ ♣❤÷ì♥❣ ♣❤→♣ ❞➙② ❝✉♥❣✱ ♣❤÷ì♥❣
♣❤→♣ ◆❡✇t♦♥ ✭t✐➳♣ t✉②➳♥✮ ✳✳✳ ◆❤÷♥❣ ❞♦ t❤í✐ ❣✐❛♥ ❝â ❤↕♥ ♥➯♥ tr♦♥❣ ♣❤↕♠ ✈✐
❜➔✐ ❧✉➟♥ ✈➠♥ ♥➔②✱ tỉ✐ s➩ tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ✭t✐➳♣ t✉②➳♥✮ ❣✐↔✐ ❣➛♥
✤ó♥❣ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✳

✶✽


Pì PP
ổ t ữỡ
sỷ r t❛ t➻♠ ✤÷đ❝ ♠ët ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮
❧➔ (a, b)✳ ❚❛ ❧✉æ♥ ❣✐↔ t❤✐➳t ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t❤ä❛ ♠➣♥✿
❛✳ P❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ ❝â ♥❣❤✐➺♠ α ❞✉② ♥❤➜t tr➯♥ (a, b)✳
❜✳ f ∈ C 2 [a, b] ✈➔ f (x), f (x) ❦❤æ♥❣ ✤ê✐ ❞➜✉ tr➯♥ (a, b)
ị ừ ừ ữỡ t t t ữỡ tr
t ố ợ x ởt ữỡ tr ú t t ố ợ x
rữợ t t ỵ tr ❝õ❛ ♠ët ❤➔♠ ♥❤÷ s❛✉✿
❈❤♦ ❤➔♠ sè f (x) ①→❝ ✤à♥❤ ✈➔ ✤↕♦ ❤➔♠ ✤➳♥ ❝➜♣ n + 1 t↕✐ x0 ✈➔ ❧➙♥ ❝➟♥ ❝õ❛
x0 ✳ ❚❛ ❝â ❝æ♥❣ t❤ù❝ s❛✉ ✤➙② ✤÷đ❝ ❣å✐ ❧➔ ❦❤❛✐ tr✐➸♥ ❚❛②❧♦ ❜➟❝ n ❝õ❛ f (x) t↕✐
x0 ✿
(x − x0 )2
f (x0 ) + ...+
f (x) = f (x0 ) + (x − x0 )f (x0 ) +
2!
(x − x0 )n+1 (n+1)
(x − x0 )n (
f n)(x0 ) +
.f

(c)
✭✷✳✺✮
+
n!
(n + 1)!
c = x0 + θ(x − x0 ), 0 < θ < 1
✭✷✳✻✮
❈æ♥❣ t❤ù❝ ♥➔② ❝â ❣✐→ trà t↕✐ x ð ❧➙♥ ❝➟♥ x0 ✳ ❈æ♥❣ t❤ù❝ ✭ ✷✳✻✮ ♠✉è♥ ♥â✐ r➡♥❣
❝ ❧➔ ♠ët sè tr✉♥❣ ❣✐❛♥ ❣✐ú❛ x0 ✈➔ x✳
❇➙② ❣✐í ①➨t ữỡ tr ợ tt õ õ tỹ α ♣❤➙♥ ❧✐
ð tr♦♥❣ ❦❤♦↔♥❣ [a, b]✳ ●✐↔ sû ❤➔♠ f ❝â ✤↕♦ ❤➔♠ f (x) = 0 t↕✐ x ∈ [a, b] ✈➔
✤↕♦ ❤➔♠ ❝➜♣ ❤❛✐ f (x) t↕✐ x ∈ (a, b)✳ ❚❛ ❝❤å♥ x0 ∈ [a, b] rỗ t tr
t ừ f t x0 ✿
1
f (x) = f (x0 ) + (x − x0 )f (x0 ) + (x − x0 )2 f (c)
2
x ∈ [a, b], c = x0 + θ(x − x0 ) ∈ (a, b).

◆❤÷ ✈➟② ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ ✈✐➳t✿
1
f (x0 ) + (x − x0 )f (x0 ) + (x − x0 )2 f (c) = 0
2

❚❛ ❜ä q✉❛ sè ❤↕♥❣ ❝✉è✐ ❝ị♥❣ ✈➔ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤
f (x0 ) + (x − x0 )f (x) = 0

✶✾

✭✷✳✼✮

◆❤÷ ✈➟②✱ t❛ ✤➣ t❤❛② ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ ❜➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✼✮ ✤ì♥ ❣✐↔♥
❤ì♥ ♥❤✐➲✉ ✈➻ ✭✷✳✼✮ t✉②➳♥ t➼♥❤ ✤è✐ ợ x
ữỡ t t õ ✤ó♥❣✳ ●å✐ x1 ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✼✮
t❛ ❝â✿
f (x0 )
✭✷✳✽✮
x1 = x0 −
f (x0 )
❚ø x1 t❛ t➼♥❤ ♠ët ❝→❝❤ t÷ì♥❣ tü r❛ x2 ✈✳✈✳✳✳ ✈➔ ♠ët ❝→❝❤ tê♥❣ q✉→t✱ ❦❤✐ ✤➣
❜✐➳t xn t❛ t➼♥❤ xn+1 t❤❡♦ ❝æ♥❣ t❤ù❝✿
xn+1 = xn

f (xn )
f (x)

x0 ồ trữợ [a, b]




①❡♠ xn ❧➔ ❣✐→ trà ❣➛♥ ✤ó♥❣ ❝õ❛ ♥❣❤✐➺♠ α✳
P❤÷ì♥❣ ♣❤→♣ t➼♥❤ xn t❤❡♦ ✭ ✷✳✾✮✱ ✭ ✷✳✶✵✮ ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥✳
❱➻ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✼✮ ❞ị♥❣ ✤➸ t❤❛② ❝❤♦ ữỡ tr
t t ố ợ x ữỡ ♣❤→♣ ◆❡✇t♦♥ ❝ơ♥❣ ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ t✉②➳♥
t➼♥❤ ❤â❛ ✳
◆❤➻♥ ✭✷✳✾✮✱ ✭✷✳✶✵✮ t❛ t❤➜② ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ t❤✉ë❝ ❧♦↕✐
♣❤÷ì♥❣ ♣❤→♣ ợ

ú ỵ

ú ỵ

(x) = x

ú ỵ t ồ

f (x)
f (x)



t f (x) số õ ừ t t ừ ỗ
t ❤➔♠ sè y = f (x) t↕✐ x0 ✳ ❳➨t ởt trữớ ủ ử t
ỗ t tr ỗ t t trử t ❝â ❤♦➔♥❤
✤ë ❝❤➼♥❤ ❧➔ ♥❣❤✐➺♠ α✳ ✣➸ t➼♥❤ ❣➛♥ ✤ó♥❣ α t❛ t❤❛② ♠ët ❝→❝❤ ❣➛♥ ✤ó♥❣ ❝✉♥❣
❆❇ ❜ð✐ t✐➳♣ t✉②➳♥ t↕✐ ❇ ✱ ❇ ❝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↕✐ ❇✱ ✈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➳♥ ✳
√
❚➼♥❤ 3 ❜➡♥❣ ❝→❝❤ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ s❛✉✿

❱➼ ❞ư ✷✳✷✳

f (x) = x2 − 3 = 0

✭✷✳✶✷✮

❇➔✐ ❣✐↔✐✿

❚❛ ❝â✿
f (1) = −2 < 0
f (2) = 1 > 0

❉♦ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✷✮ ❝â ♥❣❤✐➺♠ t❤ü❝ α tr♦♥❣ ❦❤♦↔♥❣ [1, 2]✳ ❚r♦♥❣
❦❤♦↔♥❣ ✤â✿
f (x) = 2x > 0
f (x) = 2 > 0

✷✶


❱➻ f (2) = 1 > 0✱ ♥❤÷ ✈➟② t❤➻✿ f (2).f (x) = 1.2 = 2 > 0✳ ◆➯♥ ❝❤å♥ x0 = 2✳
❱➟② t❛ ❝â t❤➸ →♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ✤➸ t➼♥❤ ♥❣❤✐➺♠ ①➜♣ ①➾ ❝õ❛
♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✷✮✳ ❚❛ ❝â ❜↔♥❣ s❛✉✿

♥
✵

✶
✷
✸

x0 = 2; xn+1 = xn −

f (x)
f (xn )

= xn −

x2n −3
2nn

✷
✶✱✼✺
✶✱✼✸✷✶✹✷✽✺✼
✶✱✼✸✷✵✺✵✽✶

❇↔♥❣ ✷✳✶✿ ❈→❝ ♥❣❤✐➺♠ ①➜♣ ①➾ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✷✮
√
❚❛ ❝â t❤➸ ❧➜② ♥❣❤✐➺♠ ①➜♣ ①➾ ❧➔ ✶✱✼✸✷✵✺✶✳ ❚❛ ❜✐➳t r➡♥❣ 3 = 1, 732050808...✳
◆❤÷ ✈➟② ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ◆❡✇t♦♥ ❤ë✐ tư r➜t ♥❤❛♥❤✳
◆❣♦➔✐ r❛✱ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ❝ơ♥❣ ❝â t❤➸ t❤ü❝ ❤✐➺♥ t❤❡♦ ❝→❝ ữợ s
t
x0 = b f (a).f (a) < 0
x0 = a ❦❤✐ f (a).f (a) > 0

✭✷✳✶✸✮

✈➔ t➼♥❤ xk+1 t❤❡♦ ❝æ♥❣ t❤ù❝ ✤➺ q✉②✿
xk+1 = xk −

f (xk )
f (xk )
= xk
tan(k )
f (xk )



ú ỵ f ❦❤ỉ♥❣ ✤ê✐ ❞➜✉✱ t❤➻ ❞➣② ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ t❤❡♦ ữỡ
ỡ õ ợ s số s ữợ tự
t ①➨t ❞➜✉ f (xk ).f (xk + s.∆)✱ tr♦♥❣ ✤â ♥➳✉ ❞➣② ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ t➠♥❣
✭tù❝ ❧➔ ❦❤✐ x0 = a✮ t❤➻ ❧➜② s = 1❀ ❣✐↔♠ ✭x0 = b✮ t❤➻ ❧➜② s = −1✳ ◆➳✉
f (xk ).f (xk + s.∆) < 0 t❤➻ ❝â ✤→♣ sè✿
x = xk ± ∆

❈❤➼♥❤ ①→❝ ❤ì♥✿
x = (xk +

❱➼ ❞ư ✷✳✸✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤✿

s.∆
∆
)±
2
2

x3 + 3x − 5 = 0

✷✷

✭✷✳✶✺✮


ợ s số = 106
rữợ t✐➯♥✱ t❛ ✤✐ t➻♠ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠✿
❚❛ ♥❤➟♥ ①➨t ❤➔♠ f (x) = x3 + 3x − 5 ❝â ✤↕♦ ❤➔♠ ❧➔ f (x) = 3x2 + 3 ❧✉æ♥
❧✉æ♥ ❞÷ì♥❣✱ ♥➯♥ ✤ì♥ ✤✐➺✉ t➠♥❣✱ f (1) = −1, f (2) = 9✱ ❞♦ ✤â ♣❤÷ì♥❣ tr➻♥❤ ❝â
♥❣❤✐➺♠ ❞✉② ♥❤➜t tr♦♥❣ ❦❤♦↔♥❣ (1, 2)✳
✯ ❱ỵ✐ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ ❧➔ (1, 2)✿
❚r➯♥ ❦❤♦↔♥❣ (1, 2) ❝â f (x) = 6x > 0, f (1) = 1 õ ợ ữỡ ♣❤→♣
◆❡✇t♦♥ ❝❤å♥ x0 = b = 2, s = −1 ✈➔ t➼♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝ ❧➦♣ ✭✷✳✶✹✮✳
❙û ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ✺✳✷ ✈ỵ✐ s❛✐ sè ❝❤♦ ♣❤➨♣ ❧➔ ∆ = 106 t
s ữợ t ữủ tr ữ tr

ố ữợ ●✐→ trà ♥❣❤✐➺♠ ①➜♣ ①➾ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✺✮
✶
✶✱✹
✷
✶✱✶✽✶✵✽
✸
✶✱✶✺✹✺✸
✹
✶✱✶✺✹✶✼
❇↔♥❣ ✷✳✷✿ ❈→❝ ♥❣❤✐➺♠ ①➜♣ ①➾ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✺✮

✷✳✷✳✷ ❙ü ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✈➔ ✤→♥❤ ❣✐→ s❛✐ sè

❛✮ ❚➼♥❤ ❤ë✐ tư

▼ư❝ ✤➼❝❤ ❝õ❛ t❛ ❧➔ t➼♥❤ ❣➛♥ ✤ó♥❣ ❝õ❛ α✳ ✣✐➲✉ ✤â ❝❤➾ ❝â t❤➸ t❤ü❝ ❤✐➺♥ ❜➡♥❣
♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ♥➳✉ xn → α✳ ❑❤✐ n → ∞✳ ❚❛ ❝â ❦➳t q✉↔ s❛✉✿

✣à♥❤ ❧➼ ✷✳✶✳ ●✐↔ sû [a, b] ❧➔ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ α ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

✭✷✳✶✮✱ f ❝â ✤↕♦ ❤➔♠ f , f ✈ỵ✐ f ✈➔ f ❧✐➯♥ tư❝ tr➯♥ [a, b], f ✈➔ f ❦❤ỉ♥❣ ✤ê✐
❞➜✉ tr♦♥❣ (a, b)✳ ❳➜♣ ①➾ ✤➛✉ x0 ❝❤å♥ ❧➔ a ❤❛② b s❛♦ ❝❤♦ f (x0 ) ❝ị♥❣ ❞➜✉ ✈ỵ✐
f ✳ ❑❤✐ ✤â xn t➼♥❤ ❜ð✐ ✭✷✳✾✮ ✭✷✳✶✵✮ ❤ë✐ tü ✈➲ 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 n t❛ ✤÷đ❝ xn ✈➔ ①❡♠ xn ❧➔ ❣✐→ trà ❣➛♥
✤ó♥❣ ❝õ❛ α✳

❜✮ ✣→♥❤ ❣✐→ s❛✐ sè

✷✸


●✐↔ sû f (x) ❧✐➯♥ tư❝ ✈➔ ❦❤ỉ♥❣ ✤ê✐ ❞➜✉ tr➯♥ [a, b] ✈➔ t❤ä❛ ♠➣♥✿
∃m1 , M2 ❞÷ì♥❣ s❛♦ ❝❤♦ m1 ≤ |f (x)|; f (x) ≤ M2 ✈ỵ✐ ∀x ∈ [a, b].

❑❤✐ ✤â t❛ ❝â✿
|xn − α| ≤

M2
|xn − xn−1 |2
2m1

✭✷✳✶✻✮

✭✷✳✶✼✮

❈❤ù♥❣ ♠✐♥❤✿

❉ị♥❣ ❝ỉ♥❣ t❤ù❝ ❦❤❛✐ tr✐➸♥ ❚❛②❧♦ ❝❤♦ f (xn ) t↕✐ xn−1 t❛ ❝â✿
xn − xn−1
(xn − xn−1 )2
f (xn ) = f (xn−1 ) +
.f (xn−1 ) +
.f (c)
1!
2!

✭✷✳✶✽✮

tr♦♥❣ ✤â c ∈ (xn−1 , xn )✳
❚❤❡♦ ✭✷✳✾✮
xn = xn−1 −

f (xn−1 )
f (xn−1 )

❚ø ✤➙②
f (xn−1 ) + (xn − xn−1 ).f (xn−1 ) = 0

❚❤❛② ✈➔♦ ✭✷✳✶✽✮ t❛ ❝â✿
(xn − xn−1 )2
f (xn ) =
.f (c)
2!

◆❤÷ ✈➟②
|f (xn )| (xn − xn−1 )2
M2
|xn − α| ≤
=
.f (c) ≤
.|xn − xn−1 |2
m1
2m1
2m1

▲➔ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳

❱➼ ử t ữỡ tr
x3


3

x5=0



ợ ✤➣ ❜✐➳t ð ❱➼ ❞ö ✷✳✶ ❧➔ (1, 2)✳
2 −5
❚r➯♥ ❦❤♦↔♥❣ (1, 2) ❝â f (x) = 6x + .x 3 > 0; f (1) = −5 < 0✳ ❉♦ õ ợ
9
ữỡ t ồ x0 = b = 2 ✈➔ t➼♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝ ❧➦♣ ✭✷✳✶✹✮✳
❙û ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ✺✳✷ ✈ỵ✐ x0 = 2, s = −1 ✈➔ t t ữủ

x3 = 1, 83959 ợ = 105 ✳

✷✹


❍➻♥❤ ✷✳✸✿ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ x3 − √x − 5 = 0 ❜➡♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛
3

❚ø ✤â s✉② r❛✿ f (x3 + ∆) tr→✐ ❞➜✉ ✈ỵ✐ f (x3 )✳
❱➟② t❛ ❝â ♥❣❤✐➺♠✿ x = x3 ± ∆ = 1, 83959 ± 10−5 ✳

✷✳✸ ▼❐❚ ❙➮ ❇⑨■ ❚❖⑩◆ ❚➐▼ ◆●❍■➏▼ ●❺◆ ể
Pì PP
ử ữỡ tr sè ❜➟❝ ❝❛♦✿
x7 + 10x5 + 15x + 5 = 0

✭✷✳✷✶✮

❚✉② ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✶✮ ❝❤➾ ❧➔ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ✤❛ t❤ù❝✱ t✉② ♥❤✐➯♥ ❜➟❝
❝õ❛ ♥â ❦❤→ ❝❛♦ ♥➯♥ ❦❤â ❝â t❤➸ ❣✐↔✐ ✤÷đ❝ ❜➡♥❣ ❝→❝ ❦➽ t❤✉➟t ❝õ❛ ✤↕✐ sè ✭✤➦t
➞♥ ♣❤ư✱ ♥❤â♠ ❤↕♥❣ tû✱✳✳✳✮ ✤➸ ✤÷❛ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ t❤➜♣ ❤ì♥✳
✣➦t f (x) = x7 + 10x5 + 15x + 5✳
❉♦ f (x) = 7x6 + 50x4 + 15 > 0, x số ỗ tr t♦➔♥ trư❝
sè✳ ❚❛ ❞➵ ❞➔♥❣ t➼♥❤ ✤÷đ❝ f (−1) = −21 < 0, f (0) = 5 > 0✳ ❉♦ ✤â ♣❤÷ì♥❣
tr➻♥❤ ✭✷✳✷✶✮ ❝â ❞✉② ♥❤➜t ♥❣❤✐➺♠ tr♦♥❣ ❦❤♦↔♥❣ (−1, 0)✳ ❚✐➳♣ t❤❡♦ t❛ ❞ị♥❣
♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ✤➸ t➻♠ ♥❣❤✐➺♠ ú ừ ữỡ tr
ợ ❧➔ (−1, 0) t❛ ❝â f (x) = 42x5 + 200x3 <
0, f (−1) = −21 < 0✳ ❉♦ ✤â ❝❤å♥ x0 = a = −1, s = 1 ✈➔ t➼♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝
❧➦♣ ✭✷✳✶✹✮✳
❙û ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ✺✳✷ ợ x0 = 1, s = 1 t ữủ

x4 = −0, 330676 ✈ỵ✐ s❛✐ sè ∆ = 10−5 ✳

✷✺