✣❸■ ❍➴❈ ✣⑨ ◆➂◆●
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

✖✖✖✖✖

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P

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

❙✐♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥✿ ◆❣✉②➵♥ ❚❤à Pữỡ
ữợ r

✵✺✴✷✵✶✻


ử ử

é
é








ỵ tt s❛✐ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✶✳✶

❙❛✐ sè t✉②➺t ✤è✐✱ s❛✐ sè t÷ì♥❣ ✤è✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✶✳✷

❙❛✐ sè t❤✉ ❣å♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✶✳✸

❙❛✐ sè t➼♥❤ t♦→♥

✾

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

◆❣❤✐➺♠ ✈➔ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

✶✳✷✳✶

◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

✶✳✷✳✷

Þ ♥❣❤➽❛ ❤➻♥❤ ❤å❝ ❝õ❛ ♥❣❤✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳





ỹ tỗ t tỹ ừ ữỡ tr

✳ ✳ ✳ ✳

✶✷

✶✳✷✳✹

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

✶✹

✷ ❚➐▼ ◆●❍■➏▼ ●❺◆ ✣Ĩ◆● ❈Õ❆ P❍×❒◆●
Pì PP

ợ t

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

✶✽

✷✳✶✳✶

✣➦t ✈➜♥ ✤➲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✽

✷✳✶✳✷

❈→❝❤ ❣✐↔✐ q✉②➳t

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

✶✾

P❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ✤ỉ✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✾

✷✳✷✳✶

▼ỉ t↔ ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✾

✷✳✷✳✷

❙ü ❤ë✐ tö ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✶

✷✳✷✳✸

✣→♥❤ ❣✐→ s❛✐ sè

✷✷

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

ởt số t t ú ợ ữỡ ♣❤→♣ ❝❤✐❛
✤æ✐

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

✷

✷✷


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

✷✺
✸✳✶

✸✳✷

▼ët ✈➔✐ ♥➨t ✈➲ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✺

✸✳✶✳✶

●✐ỵ✐ t❤✐➺✉

✷✺

✸✳✶✳✷

▼ët sè ❤➔♠ t❤ỉ♥❣ ❞ư♥❣

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

✷✻

Ù♥❣ ❞ö♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❜➡♥❣
♣❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ✤ỉ✐✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❑➌❚ ▲❯❾◆
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✷✽

✸✼
✸✽

✸


▲❮■ ❈❷▼ ❒◆
❊♠ ①✐♥ ❜➔② tä sü ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❇❛♥ ●✐→♠ ❍✐➺✉ tr÷í♥❣ ✣↕✐ ❤å❝
❙÷ ♣❤↕♠ ✲ ✣↕✐ ❍å❝ ✣➔ ◆➤♥❣✱ ❜❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥✱ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥
✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ t❤ü❝ ❤✐➺♥ ✈➔ ❤♦➔♥ t❤➔♥❤✳ ❈❤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♥❣ ❡♠
♥❤ú♥❣ 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 tr♦♥❣ ✈➟t ❧➼✱ ❝ì ❤å❝✱ t❤✐➯♥ ✈➠♥ ❤å❝ ✈➔ ♠ët sè ❧➽♥❤ ✈ü❝ ❦❤→❝
t❤æ♥❣ q✉❛ ♠æ ❤➻♥❤ ❤â❛ t♦→♥ ❤å❝ ❞➝♥ ✤➳♥ ✈✐➺❝ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ♣❤✐
t✉②➳♥✳ ❚✉② ♥❤✐➯♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② t❤÷í♥❣ ♣❤ù❝ t↕♣✱ ❞♦ ✤â ♥â✐ ❝❤✉♥❣
❦❤â ❝â t❤➸ ❣✐↔✐ ✤÷đ❝ ✭✤÷❛ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❝ì ❜↔♥✮ ❜➡♥❣ ❝→❝ ❜✐➳♥ ✤ê✐ ✤↕✐ sè
t❤➟♠ ❝❤➼ tr♦♥❣ ♠ët sè tr÷í♥❣ ❤đ♣ ❝ơ♥❣ ❦❤ỉ♥❣ t❤➸ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ t÷í♥❣
♠✐♥❤✳ ❍ì♥ ♥ú❛✱ ✈➻ ❝→❝ ❝ỉ♥❣ t❤ù❝ ♥❣❤✐➺♠ t❤÷í♥❣ ♣❤ù❝ t↕♣✱ ỗ
ũ õ ổ tự 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➻♥❤ ♣❤✐ t✉②➳♥ ♥➔② trð ♥➯♥ ❝➜♣
t❤✐➳t ✈➔ tü ♥❤✐➯♥✳
❇➯♥ ❝↕♥❤ ✤â✱ ✈ỵ✐ ✈✐➺❝ ♣❤→t tr✐➸♥ ❝õ❛ ❝ỉ♥❣ ♥❣❤➺✱ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❣➛♥
✤ó♥❣ ỵ ỡ ự ử ổ ❝ư ♠→② t➼♥❤ ✤✐➺♥ tû ❤❛②
♣❤➛♥ ♠➲♠ t♦→♥ ❤å❝ ♥❤÷ ▼❛♣❧❡✱ ▼❛t❤❡♠❛t✐❝❛✳✳✳ ✣➸ ❣✐↔✐ ♠ët ♣❤÷ì♥❣ tr➻♥❤
❜➡♥❣ t❛② tr➯♥ ❣✐➜②✱ ♣❤↔✐ ♠➜t r➜t ♥❤✐➲✉ t❤í✐ ❣✐❛♥ ✈ỵ✐ ♥❤ú♥❣ s❛✐ sât ❞➵ ①↔②
r❛✱ t❤➻ ✈ỵ✐ sü ❤é trđ ❝õ❛ ❝→❝ ♣❤➛♥ ♠➲♠ ❝❤✉②➯♥ ❞ư♥❣ ❝❤ó♥❣ t❛ ❝❤➾ ❝➛♥ ♠➜t
✈➔✐ ♣❤ót t❤➟♠ ❝❤➼ ✈➔✐ ❣✐➙②✳ ▼➦t ❦❤→❝✱ ♥❤✐➲✉ ✈➜♥ ✤➲ ❧➼ t❤✉②➳t ✭sü ❤ë✐ tö✱ tè❝
✤ë ❤ë✐ tö✱ ✤ë ❝❤➼♥❤ ①→❝✱ ✤ë ♣❤ù❝ t↕♣ t➼♥❤ t♦→♥✱✳✳✳✮ s➩ ✤÷đ❝ ♥❤➻♥ t❤➜② rã ❤ì♥
❦❤✐ sû ❞ö♥❣ ❝→❝ ♣❤➛♥ ♠➲♠ ♥➔②✳ ❱➻ ✈➟②✱ ✈✐➺❝ sû ❞ư♥❣ t❤➔♥❤ t❤↕♦ ❝→❝ ❝ỉ♥❣ ❝ư
t♦→♥ ❤å❝ ❧➔ ❝➛♥ t❤✐➳t ❝❤♦ ❝ỉ♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉✱ ♥❤➜t ❧➔ ✤è✐ ✈ỵ✐ ❤å❝ s✐♥❤✱ s✐♥❤
✈✐➯♥✳
❱ỵ✐ ♠♦♥❣ ♠✉è♥ ❝â t❤➸ ❤✐➸✉ ❦➽ ❤ì♥ ✈➲ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤
♣❤✐ t✉②➳♥ ♥❤➡♠ ✤→♣ ù♥❣ ồ ự ồ ừ t
ỗ tớ ữủ sỹ ủ ỵ ở ừ ữợ




❍↔✐ ❚r✉♥❣ ♥➯♥ tỉ✐ ❧ü❛ ❝❤å♥ ✤➲ t➔✐✿ ✧Ù♥❣ ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ t➻♠

♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ✤ỉ✐ ✧ ❝❤♦ ❧✉➟♥
✈➠♥ tèt ♥❣❤✐➺♣ ❝õ❛ ♠➻♥❤✳

✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉
▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧➔ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ✤ỉ✐ ✤➸ ①❡♠ ①➨t ✈➔

t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✱ tø ✤â s♦ s→♥❤ s số ợ
ừ ữỡ tr õ ỗ t❤í✐✱ ♥❣❤✐➯♥ ❝ù✉ ù♥❣ ❞ư♥❣ ♣❤➛♥ ♠➲♠
▼❛t❤❡♠❛t✐❝❛ ✤➸ ✈✐➳t ❝❤÷ì♥❣ tr➻♥❤ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣
❝❤✐❛ ✤ỉ✐ ✈➔ ổ t ữỡ tr ỗ t t❤ỉ♥❣ q✉❛ ❝→❝
❣â✐ ❧➺♥❤ ✤➣ ✤÷đ❝ ❧➟♣ tr➻♥❤✳

✸✳ ✣è✐ t÷đ♥❣✱ ♣❤↕♠ ✈✐ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉
✣➲ t➔✐ t➟♣ tr✉♥❣ ♥❣❤✐➯♥ ❝ù✉ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ♣❤✐ t✉②➳♥ ❜➡♥❣ ♣❤÷ì♥❣

♣❤→♣ ❝❤✐❛ ổ ỵ tt ỡ ú t➼♥❤ ❤ë✐ tö✱ ✤→♥❤
❣✐→ s❛✐ sè✳✳✳ ✈➔ sû ❞ö♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ✤➸ ❣✐↔✐ ♠ët sè ♣❤÷ì♥❣ tr➻♥❤
♣❤✐ t✉②➳♥ ♣❤ù❝ t↕♣ ♠➔ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ t➼♥❤ t❛② ❦❤â ❝â t❤➸ ❧➔♠ ✤÷đ❝✳
P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉✿ ◆❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ✤ỉ✐ ❝❤♦ ❝→❝ ♣❤÷ì♥❣
tr➻♥❤ ♣❤✐ t✉②➳♥ ♠ët ❜✐➳♥ t❤ü❝✳
P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉✿ ❚❤❛♠ ❦❤↔♦ ❝→❝ t➔✐ ❧✐➺✉ ✈➔ ❤➺ t❤è♥❣ ❤â❛ ❝→❝ ❦✐➳♥
t❤ù❝❀ tr❛♦ ✤ê✐✱ t❤↔♦ ❧✉➟♥ ✈ỵ✐ ❣✐→♦ ✈✐➯♥ ữợ t tữớ
t q ự tr♦♥❣ ✤➲ t➔✐✳

✹✳ Þ ♥❣❤➽❛ ❦❤♦❛ ❤å❝ ✈➔ t❤ü❝ t✐➵♥ t
t õ ỵ t ỵ tt ❝â t❤➸ sû ❞ư♥❣ ♥❤÷ ❧➔ t➔✐ ❧✐➺✉ t❤❛♠

❦❤↔♦ ❞➔♥❤ ❝❤♦ s✐♥❤ ✈✐➯♥ ✈➔ ❝→❝ ✤è✐ t÷đ♥❣ ❝â ♠è✐ q✉❛♥ t➙♠ ✤➳♥ ♣❤÷ì♥❣ ♣❤→♣
❝❤✐❛ ✤ỉ✐ ✈➔ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛✳

✺✳ ❈➜✉ tró❝ ❧✉➟♥ ✈➠♥

◆❣♦➔✐ ♣❤➛♥ ▼ð ✤➛✉ ✈➔ ❑➳t ❧✉➟♥✱ ❧✉➟♥ ữủ tr ỗ ữỡ

ữỡ tự ♠ð ✤➛✉
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ tê♥❣ qt ỵ tt s
sổ t t ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤

f (x) = 0✳

❈❤÷ì♥❣ ✷✿ ❚➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐❛
✤ỉ✐✳
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❝→❝❤ ①➙② ❞ü♥❣ ❝æ♥❣ t❤ù❝ ❧➦♣✱ t➼♥❤ ❤ë✐ tư✱
✤→♥❤ ❣✐→ s❛✐ sè ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ✤ỉ✐✳

✻


❈❤÷ì♥❣ ✸✿ Ù♥❣ ❞ư♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤♠❛t✐❝❛ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛
♣❤÷ì♥❣ tr➻♥❤ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ✤ỉ✐✳
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② s➩ tr➻♥❤ ❜➔② ❝→❝ ✈➼ ❞ö✱ ❝→❝ ❝➙✉ ❧➺♥❤ ✈➔ þ ♥❣❤➽❛ ❝→❝ ❝➙✉
❧➺♥❤ ✤➸ ❣✐↔✐ ♠ët sè ♣❤÷ì♥❣ tr➻♥❤ ♣❤✐ t✉②➳♥ ❜➡♥❣ ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛✳

✼


ữỡ

é

ỵ tt s số
sè t✉②➺t ✤è✐✱ s❛✐ sè t÷ì♥❣ ✤è✐
❚r♦♥❣ t➼♥❤ t♦→♥✱ t❛ tữớ ợ tr ú ❝õ❛ ❝→❝

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❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥✿

✤↕✐ ❧÷đ♥❣✳ ❚❛ ♥â✐

|a − a∗ | ≤ ∆a

✭✶✳✶✮

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

❤❛②

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 )
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 sè✳ ◆➳✉ s = +∞✱ a ❧➔ sè t❤➟♣ ♣❤➙♥ ✈æ ❤↕♥✳ ❚❤✉ ❣å♥ ♠ët sè a ❧➔ ✧✈ùt
❚r♦♥❣ ✤â

✽


❜ä✧ ♠ët sè ❝→❝ ❝❤ú sè ❜➯♥ ♣❤↔✐
✤ó♥❣ ♥❤➜t ✈ỵ✐

a

✤➸ ✤÷đ❝ ♠ët sè

a

♥❣➢♥ ❣å♥ ❤ì♥ ✈➔ ❣➛♥

a✳

◗✉② t➢❝ t❤✉ ❣å♥✿
●✐↔ sû

a = (βp 10p + ... + βj 10j + ... + βp−s 10p−s )

ϕ✱

✈➔ t❛ ❣✐ú ❧↕✐ ✤➳♥ sè ❤↕♥❣ t❤ù ❥✳ ●å✐ ♣❤➛♥ ✧✈ùt ❜ä✧ ❧➔

t❛ ✤➦t

a = βp 10p + ... + βj+1 10j+1 + βj 10j
tr♦♥❣ ✤â✿


β + 1
j
βj :=
β

ϕ = 0.5 × 10j

t❤➻

βj = βj

♥➳✉

✭✶✳✷✮

j

0 < ϕ < 0.5 × 10 ,

♥➳✉

j

◆➳✉

0.5 × 10j < ϕ < 10j ,

♥➳✉

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 ❤➻♥❤ ❤â❛✱ ỵ tữ õ t tỹ t số
ổ ❧♦↕✐ 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♥❣ ♣❤→♣ ❝ư 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)

❝→❝ ✤è✐ sè ✈➔ ❤➔♠ sè✳

xi , y(i = 1, n)
◆➳✉ f ❞÷ì♥❣ ❦❤↔
✈➔

❧➔ ❝→❝ ❣✐→ trà ✤ó♥❣ ✈➔ ❣➛♥ ✤ó♥❣ ❝õ❛
✈✐ ❧✐➯♥ tư❝ t❤➻

n
∗

|y − y | = |f (x1 , x2 , ..., xn ) −

f (x∗1 , ..., x∗n )|

|fi ||xi − x∗i |.

=
i=1

✾


tr♦♥❣ ✤â

∆xi

fi

❧➔ ✤↕♦ ❤➔♠

df
df
dxi t➼♥❤ t↕✐ t❤í✐ ✤✐➸♠ tr✉♥❣ ❣✐❛♥✳ ❉♦ dxi ❧✐➯♥ tö❝ ✈➔

❦❤→ ❜➨✱ t❛ ❝â t❤➸ ❝♦✐

n

|fi (x1 , ..., xn )|∆xi .

∆y =
i=1
❉♦ ✤â

∆y
=
δ=
|y|

n

|
i=1

d
lnf |∆xi .
dxi

✶✳✷ ◆❣❤✐➺♠ ✈➔ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠
✶✳✷✳✶ ◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
❳➨t ♣❤÷ì♥❣ tr➻♥❤ ♠ët ➞♥✿

f (x) = 0.
❚r♦♥❣ ✤â

f

✭✶✳✸✮

❧➔ ♠ët ❤➔♠ số trữợ ừ ố số

tỹ ừ ữỡ tr ✭✶✳✸✮ ❧➔ sè t❤ü❝
❦❤✐ t❤❛②

α

✈➔♦

x

x✳
α t❤ä❛

♠➣♥ ✭✶✳✸✮ tù❝ ❧➔

ð ✈➳ tr t ữủ

f () = 0.



ị ồ ừ
ỗ t ừ số

y = f (x)



Oxy

sỷ ỗ t t trử t

tr ởt ❤➺ trư❝ tå❛ ✤ë ✈✉ỉ♥❣ ❣â❝

♠ët ✤✐➸♠

M

♥➔② ❝â t✉♥❣ ✤ë

y = 0 ✈➔ ❤♦➔♥❤ ✤ë x = α✳ ❚❤❛② ❝❤ó♥❣ ✈➔♦ ✭✶✳✸✮

t❛ ✤÷đ❝✿

0 = f (α)
❱➟② ❤♦➔♥❤ ✤ë

α

❝õ❛ ❣✐❛♦ ✤✐➸♠

M

❝❤➼♥❤ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✸✮✳

✶✵

✭✶✳✻✮


ị ồ ừ
rữợ ỗ t t ụ õ t t ữỡ tr ữỡ
tr tữỡ ữỡ

g(x) = h(x),



rỗ ỗ t ❤➔♠ sè ✭❍➻♥❤ ✶✳✷✮✿

y = g(x), y = h(x).
●✐↔ sû ỗ t t t

M

õ ở

g() = h(α)

✶✶

✭✶✳✽✮

x=α

t❤➻ t❛ ❝â✿
✭✶✳✾✮


ỗ t số g(x), h(x)
ở

ừ M

ừ ỗ t ♠ët ♥❣❤✐➺♠

❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✼✮✱ tù❝ ❧➔ ❝õ❛ ✭✶✳✸✮✳

✶✳✷✳✸ ❙ü tỗ t tỹ ừ ữỡ tr
rữợ t t➼♥❤ ❣➛♥ ✤ó♥❣ ♥❣❤✐➺♠ t❤ü❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮ t❛
♣❤↔✐ tỹ ọ tỹ õ tỗ t ổ tr ớ t õ t
ũ ữỡ ỗ t ụ õ t ũ ỵ s

ỵ ✶✳✶✳ ◆➳✉ ❝â ❤❛✐ sè t❤ü❝ ❛ ✈➔ ❜ (a < b) s❛♦ ❝❤♦ f (a) ✈➔ f (b) tr→✐
❞➜✉✱ tự

f (a).f (b) < 0
ỗ tớ

f (x)

tử tr

[a, b]

✭✶✳✶✵✮

t❤➻ ð tr♦♥❣ ❦❤♦↔♥❣

(a, b)

♥❣❤✐➺♠ t❤ü❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮✳

❈❤ù♥❣ ♠✐♥❤✿

❈❤✐❛ ✤♦↕♥

[a, b]

t❤➔♥❤ ❤❛✐ ♣❤➛♥ ❜➡♥❣ ♥❤❛✉ ❜ð✐ ✤✐➸♠

•

◆➳✉

f ( a+b
2 ).f (a) > 0

t❤➻ ✤➦t

a1 =

•

◆➳✉

f ( a+b
2 ).f (a) < 0

t❤➻ ✤➦t

a1 = a; b1 =

✶✷

a+b
2 ; b1

= b✳
a+b
2 ✳

a+b
2

❝â ➼t ♥❤➜t ♠ët


❈❤✐❛ ✤♦↕♥

[a1 , b1 ]

a1 +b1
2

t❤➔♥❤ ❤❛✐ ♣❤➛♥ ❜➡♥❣ ♥❤❛✉ ❜ð✐ ✤✐➸♠

•

◆➳✉

1
f ( a1 +b
2 ).f (a1 ) > 0

t❤➻ ✤➦t

a2 =

a1 +b1
2 ; b2

= b1 ✳

•

◆➳✉

1
f ( a1 +b
2 ).f (a1 ) < 0

t❤➻ ✤➦t

a2 = a1 ; b 2 =

a1 +b1
2 ✳

❈ù t✐➳♣ tö❝ q✉→ tr➻♥❤ tr➯♥ t❛ ①→❝ ✤à♥❤ ✤÷đ❝ ❤❛✐ ❞➣②

bn ❧➔ ❞➣② ❣✐↔♠ ✈➔ f (an ) ❝ị♥❣ ❞➜✉ ✈ỵ✐ f (a), f (bn )
⇒ f (an ).f (bn ) < 0✱ ∀n ∈ N ✳
❉➣② an t➠♥❣✱ ❜à ❝❤➦♥ tr➯♥ ❜ð✐ ❜ ⇒ lim an tỗ t

t

an , bn



an



ũ ợ

f (b)

n



bn

ữợ

lim bn
n

tỗ t

= lim an ✱ β = lim bn ✱
n→∞
n→∞

⇒ lim (bn − an ) = β − α✳
✣➦t

n→∞

bn − an = b−a
2n ; ∀n ∈ N ✳
♥➯♥ 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 )✳
▼➔

n→∞

n→∞

⇒ 0 ≥ f (α) ⇒ f (α) = 0

✈➔

α ∈ (a, b)✳

❱➟② ✤à♥❤ ❧➼ ✶✳✶ ✤➣ ✤÷đ❝ ự
õ õ t ồ tr ỗ t ỗ t ừ số

y = f (x)

a x ≤ b ❧➔ ♠ët ✤÷í♥❣ ❧✐➲♥ ♥è✐ ❤❛✐ ✤✐➸♠ A B A ữợ B
ữỡ tr õ t t ởt tr

ỗ t ❤➔♠ sè y = f (x) t↕✐ a ≤ x ≤ b
✶✸

❧➔ trö❝ ❤♦➔♥❤

a
(a, b)✳

♥➯♥ ♣❤↔✐ ❝➢t trö❝ ❤♦➔♥❤ t↕✐ ➼t ♥❤➜t ♠ët ✤✐➸♠ tr♦♥❣ ❦❤♦↔♥❣ tø

t↕✐

✤➳♥

b✳

❱➟②


✶✳✷✳✹ ❑❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠

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

ỵ (a, b) ❧➔ ♠ët ❦❤♦↔♥❣ tr♦♥❣ ✤â ❤➔♠ sè f (x) ❧✐➯♥ tử
ỡ ỗ tớ

f (a)



f (b)

tr tự ❝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 ✤✐➲✉ ❦✐➺♥
sè

f (x)

f (a).f (b) < 0

❝❤ù♥❣ tä ❤❛✐ ✤➛✉ ♠ót ừ ỗ t

ừ trử

t ❤đ♣ ✈ỵ✐ t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❤➔♠ sè t❛ s✉② r❛
♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

(a, b)

❧➔ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐

f (x) = 0✳

❱➟② ✤à♥❤ ❧➼ ✶✳✷ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳
✣✐➲✉ ♥➔② ❝â t❤➸ ồ ỗ t ỗ t ừ ❤➔♠ sè

y = f (x)

❝➢t trö❝ ❤♦➔♥❤ t↕✐ ♠ët ✈➔ ❝❤➾ ♠ët ✤✐➸♠ ð tr♦♥❣

(a, b)✳

❝❤ù❛ ♠ët ✈➔ ❝❤➾ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮✳

❍➻♥❤ ✶✳✹✿ ❑❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ f (x) = 0
✶✹

❱➟②

(a, b)


◆➳✉

f (x)

❝â ✤↕♦ ❤➔♠ t❤➻ ✤✐➲✉ ❦✐➺♥ ✤ì♥ ✤✐➺✉ ❝â t❤➸ t❤❛② ❜➡♥❣ ✤✐➲✉ ❦✐➺♥

❦❤æ♥❣ ✤ê✐ ❞➜✉ ❝õ❛ ✤↕♦ ❤➔♠ ✈➻ ✤↕♦ ❤➔♠ ❦❤ỉ♥❣ ✤ê✐ ❞➜✉ t❤➻ ❤➔♠ sè ✤ì♥
t õ

ỵ [a, b] ởt tr♦♥❣ ✤â ❤➔♠ f (x) ❧✐➯♥ tư❝✱ ✤↕♦ ❤➔♠
f (x)

❦❤ỉ♥❣ ✤ê✐ ❞➜✉ ✈➔

f (a), f (b)

tr→✐ ❞➜✉ t❤➻

(a, b)

❧➔ ♠ët ❦❤♦↔♥❣ ♣❤➙♥ ❧✐

♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮✳
▼✉è♥ t➻♠ ❝→❝ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮ t❤÷í♥❣ ♥❣÷í✐

t❛ ♥❣❤✐➯♥ ❝ù✉ sü ❜✐➳♥ t❤✐➯♥ ❝õ❛ ❤➔♠ sè

❱➼ ❞ư

y = f (x)

rỗ ử

ữỡ tr➻♥❤✿

f (x) = x3 − x − 1.

✭✶✳✶✶✮

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

rữợ t t t sỹ t ừ số
ồ

x

f (x)

õ tử t

ỗ tớ

1
f (x) = 3x2 − 1 = 0 ⇐⇒ x = ± √ .
3

❚❛ ❝â ❜↔♥❣ ❜✐➳♥ t❤✐➯♥ ♥❤÷ tr♦♥❣ ❤➻♥❤ ✶✳✺✿
❚r♦♥❣ ✤â✿

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

1
1
1
2
f (m) = f ( √ ) = − √ − √ − 1 = − √ − 1 < 0.
3
3 3
3
3 3

✶✺


❍➻♥❤ ✶✳✺✿ ❇↔♥❣ ❜✐➳♥ t❤✐➯♥ ❝õ❛ ❤➔♠ sè f (x) = x3 x 1
ỗ t t trử ❤♦➔♥❤ t↕✐ ♠ët ✤✐➸♠ ❞✉② ♥❤➜t ✭❍➻♥❤ ✶✳✻✮✱ ❞♦ ✤â ♣❤÷ì♥❣
tr➻♥❤ ✭✶✳✶✶✮ ❝â ♠ët ♥❣❤✐➺♠ t❤ü❝ ❞✉② ♥❤➜t✱ ❦➼ ❤✐➺✉ õ

ỗ t số f (x) = x3 − x − 1 t↕✐ ❦❤♦↔♥❣ (1, 2)
f (1) = 13 − 1 − 1 = −1 < 0; f (2) = 23 − 2 − 1 = 5 > 0✳
◆❤÷ ✈➟②✱ f (1).f (2) < 0✳
❱➟② ❦❤♦↔♥❣ (1, 2) ❝❤ù❛ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✶✮✳
❚❛ t➼♥❤ t❤➯♠✿

✶✻


❱➼ ❞ư ✶✳✷✳

❚➻♠ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✉✿

√
πx 2
27
− 2(x + 1) = 0
3( sin ) + e 2x −
2
11
✣➦t

2
y = f (x) = 3( sin πx
2 ) + e

√

2x

−

27
11

− 2(x + 1)✳

✭✶✳✶✷✮

✣➸ t➻♠ ❦❤♦↔♥❣ ♣❤➙♥

❧✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✷✮✱ t❛ ♥❤í ♣❤➛♥ ♠➲♠ ▼❛t❤❡♠❛t✐❝❛ ✈➩ ỗ t
số

y = f (x) = 3(

2
sin x
2 )



+ e

2x

−

27
11

− 2(x + 1)

tr♦♥❣ ❦❤♦↔♥❣ ✭✷✱ ✸✮

♥❤÷ tr♦♥❣ ❤➻♥❤ ✭✶✳✼✮✳

❍➻♥❤ ỗ t số f (x) = 3( sin πx2 )2 +

√

e

2x

−

27
11

− 2(x + 1)

tr♦♥❣ ❦❤♦↔♥❣ (0, 5)

◆❤➻♥ ✈➔♦ ỗ t t t ỗ t t trử ❤♦➔♥❤ t↕✐ ♠ët ✤✐➸♠
tr♦♥❣ ❦❤♦↔♥❣ ✭✷✱ ✸✮✳ ❚✉② ♥❤✐➯♥ ✤➸ ❝❤➼♥❤ ①→❝ ❤ì♥ t❛ ❝➛♥ t➼♥❤ ❣✐→ trà ❤➔♠ sè

x = 2 ✈➔ x = 3✳

❚❛ ❝â✿ f (2) = −1.0565 < 0; f (3) = 1.14814 > 0✳ ◆❤÷ ✈➟②✱ f (2).f (3) < 0✳
❱➟② ❦❤♦↔♥❣ (2, 3) ❝❤ù❛ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✷✮✳
t↕✐ ❝→❝ ✤✐➸♠

✶✼


ữỡ

ể ế
Pì
Pì PP
ợ t
t
ú t tữớ t ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ trü❝ t✐➳♣✳ ◆➳✉
♠å✐ t➼♥❤ t♦→♥ ❝õ❛ ❝❤ó♥❣ ❧➔ ❝❤➼♥❤ 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➯♥ ❝→❝ số ú
tổ õ ữ rt ợ t q ố ũ ố ợ ữỡ
tr sè 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♥❣ ❝æ♥❣ 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 ởt trỏ q trồ

qt
ị tữ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ ❧➔
①➙② ❞ü♥❣ ♠ët ❞➣② ❝→❝ sè

x0 , x1 , x2 , ..., xn , ....

✈ì✐

x0

❧➔ ❣✐→ trà ①✉➜t ♣❤→t s❛♦

❝❤♦✿

lim xn = .

n

ữ ợ ợ t õ t ①❡♠

xn

❧➔ ①➜♣ ①➾ ❝õ❛ ♥❣❤✐➺♠

α✳

❚❛ ❝â t❤➸ ✤÷❛ r❛ ♠ët ✤→♥❤ ❣✐→ ✈➲ s❛✐ sè tê♥❣ q✉→t ❝❤♦ ❤➛✉ ❤➳t
ữ s

ỵ

ợ

f (x)

tử ❦❤↔ ✈✐ tr➯♥ ✤♦↕♥

[a, b]✱

♥❣♦➔✐ r❛✿

∃m1 : 0 < m1 < |f (x)|; ∀x ∈ [a, b]

✭✷✳✸✮

❦❤✐ ✤â t❛ ❝â ✤→♥❤ ❣✐→✿

|x0 − α| ≤

|f (xn )|
m1

✭✷✳✹✮

✷✳✷ P❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ✤ỉ✐
✷✳✷✳✶ ▼ỉ t↔ ♣❤÷ì♥❣ ♣❤→♣
f (x) ❧✐➯♥ tư❝ tr➯♥ [a, b] ✈➔ f (a), f (b) tr→✐ ❞➜✉✳ ◆❤÷ ✈➟② tr♦♥❣ ❦❤♦↔♥❣
♥➔② ❝â ➼t ♥❤➜t ♠ët ♥❣❤✐➺♠ α✳ ❚❛ s➩ t➻♠ ♥❣❤✐➺♠ ♥➔② ❜➡♥❣ ❝→❝❤ ❝❤✐❛ ✤æ✐
❦❤♦↔♥❣ (a, b)✱ ❝❤å♥ ự rỗ ổ t

sû

❝❤ù❛ ♥❣❤✐➺♠ ♥➔② ❝❤♦ ✤➳♥ ❦❤✐ t➻♠ t❤➜② ♥❣❤✐➺♠ ❤♦➦❝ ❦❤♦↔♥❣ ❝♦♥ ✤➣ ✤õ ♥❤ä ✤➸
✤↔♠ ❜↔♦ r➡♥❣ ♠å✐ ❣✐→ trà tr♦♥❣ ❦❤♦↔♥❣ ✤â ✤➲✉ ❝â t❤➸ ①❡♠ ❧➔ ①➜♣
ử t trữợ t t t

a0 = a, b0 = b trữợ ởt tr > 0 ✤õ ♥❤ä

✤➸ ❞ò♥❣ ❧➔♠ ✤✐➲✉ ❦✐➺♥ ①➜♣ ①➾ ♥❣❤✐➺♠ ✈➔ ❞ø♥❣ q✉→ tr➻♥❤ t➼♥❤ t♦→♥✳
❙❛✉ ✤â t❛ t❤ü❝ ❤✐➺♥ ữợ s




ữợ


0
x0 = a0 +b
2
f (a0 )f (b0 ) < 0✱ ❞♦ ✤â ♠ët

✣➦t

✶✳

f (x0 ) = 0✳

✷✳

f (x0 ) = 0✳ ◆➳✉ f (a)f (x0 ) < 0 t❤➻ ♥❣❤✐➺♠ s➩ ð tr♦♥❣ ❦❤♦↔♥❣ (a, x0 ) ❞♦ ✤â

❚❛ ❝â

x0

tr♦♥❣ ✷ tr÷í♥❣ ❤đ♣ s❛✉ ①↔② r❛✿

❧➔ ♥❣❤✐➺♠ ✈➔ ❦➳t t❤ó❝✳

t❛ ✤➦t

a1 = a0 , b1 = x0
◆➳✉

f (x0 )f (b) < 0

t❤➻ ♥❣❤✐➺♠ s➩ ð tr♦♥❣ ❦❤♦↔♥❣

(x0 , b)

❞♦ ✤â t❛ ✤➦t

a1 = x0 , b1 = b.

❱➻ ♥❣❤✐➺♠

α ∈ (a1 , b1 )✱

t❛ ❝â

|x0 − α| ≤ |b1 a1 | =

ba
2

s ữợ

ữợ



1
x1 = a1 +b
2
f (a1 )f (b1 ) < 0✱ ❞♦ ✤â ♠ët

✣➦t

❛✳

f (x1 ) = 0✳

❜✳

f (x1 ) = 0✳

❚❛ ❝â

f (a1 )f (x1 ) < 0

◆➳✉

x1

tr♦♥❣ ✷ tr÷í♥❣ ❤đ♣ s❛✉ ①↔② r❛✿

❧➔ ♥❣❤✐➺♠ ✈➔ ❦➳t t❤ó❝✳

t❤➻ ♥❣❤✐➺♠ s➩ ð tr♦♥❣ ❦❤♦↔♥❣

(a1 , x1 )

❞♦ ✤â t❛ ✤➦t

a2 = a1 , b2 = x1 .
f (x1 )f (b1 ) < 0

◆➳✉

t❤➻ ♥❣❤✐➺♠ s➩ ð tr♦♥❣ ❦❤♦↔♥❣

(x1 , b1 )

❞♦ ✤â t❛ ✤➦t

a2 = x1 , b2 = b1 .

❱➻ ♥❣❤✐➺♠

α ∈ (a2 , b2 )✱

t❛ ❝â

|x1 − α| ≤ |b2 a2 | =

ba
22

s ữợ


ữợ


n
xn = an +b
2
f (an )f (bn ) < 0✱ ❞♦ ✤â ♠ët

✣➦t

❛✳

f (xn ) = 0✳

❜✳

f (xn ) = 0✳

❚❛ ❝â

xn

tr♦♥❣ ✷ tr÷í♥❣ ❤đ♣ s❛✉ ①↔② r❛✿

❧➔ ♥❣❤✐➺♠ ✈➔ ❦➳t t❤ó❝✳

✷✵


◆➳✉

f (an )f (xn ) < 0

t❤➻ ♥❣❤✐➺♠ s➩ ð tr♦♥❣ ❦❤♦↔♥❣

(an , xn )

❞♦ ✤â t❛ ✤➦t

an+1 = an , bn+1 = xn
◆➳✉

f (xn )f (bn ) < 0

t❤➻ ♥❣❤✐➺♠ s➩ ð tr♦♥❣ ❦❤♦↔♥❣

(xn , bn )

❞♦ ✤â t❛ ✤➦t

an+1 = xn , bn+1 = bn
α ∈ (an+1 , bn+1 )✱ t❛ ❝â |xn − α| ≤ |bn+1 − an+1 | = 2b−a
n+1 ✳
b−a
tr❛ ①❡♠ ♥➳✉ n+1 ≤ ε t t tú ổ t s
2




ữợ

n + 1.

❱➼ ❞ư ✷✳✶✳

❳➨t ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✶✮✿

x3 − x − 1 = 0,
✈ỵ✐ s❛✐ sè

ε = 10−1 .

❚❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❝❤➾ ❝â ♠ët ♥❣❤✐➺♠ t❤ü❝

α

✤➣

(1, 2)✳ ❱➟②✿
α ∈ (1, 2) ✈➔ f (1) = 1 − 1 − 1 < 0 ❀ f (2) = 23 − 2 − 1 > 0
3
❚❛ ❝❤✐❛ ✤æ✐ ❦❤♦↔♥❣ (1, 2) ✤✐➸♠ ❝❤✐❛ ❧➔ ✳
2
3 2
3
3
3
2 = ( 2 ) − 2 − 1 > 0 tr→✐ ❞➜✉ f (1)✳ ❱➟② α ∈ (1, 2 )✳
3
5
5
❚❛ ❝❤✐❛ ✤æ✐ ❦❤♦↔♥❣ (1, )✱ ✤✐➸♠ ❝❤✐❛ ❧➔ ✳ ❚❛ ❝â f ( ) < 0✱ ❝ị♥❣ ❞➜✉ ✈ỵ✐ f (1)✳
2
4
4
5 3
❱➟② α ∈ ( , )✳
4 2
5 3
11
11
5
❚❛ ❝❤✐❛ ✤æ✐ ❦❤♦↔♥❣ ( , )✱ ✤✐➸♠ ❝❤✐❛ ❧➔
4 2
8 ✳ ❚❛ ❝â f ( 8 ) > 0✱ tr→✐ ❞➜✉ f ( 4 )✳
5 11
❱➟② α ∈ ( ,

4 8 )✳
5 11
21
❚❛ ❝❤✐❛ ✤æ✐ ❦❤♦↔♥❣ ( ,
4 8 )✱ ✤✐➸♠ ❝❤✐❛ ❧➔ 16 ✳
1
1
−1
❱➻ s❛✐ sè 4 =
2
16 = 0.0625 < ε = 10 . ◆➯♥ t❛ ❞ø♥❣ q✉→ tr➻♥❤ ❝❤✐❛ ✤æ✐
21
t↕✐ ✤➙② ✈➔ ❧➜②
16 = 1.3125 ❧➔ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✳
♣❤➙♥ ❧✐ tr♦♥❣ ❦❤♦↔♥❣

✷✳✷✳✷ ❙ü ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣
❉➣②

an

❧➔ ❞➣② ✤ì♥ ✤✐➺✉ t➠♥❣✱ ❜à ❝❤➦♥ tr➯♥ ❜ð✐

a ♥➯♥
bn − an = ba
2n

ữợ

r



bn



❤❛✐ ❞➣② ✤➲✉ ❝â ❣✐ỵ✐ ❤↕♥✳
♥➯♥

lim (bn − an ) = 0

n→∞

t❛ ✤÷đ❝

❤❛②

lim (an ) = lim (bn ) = x✳

n→∞

y = f (x)✱ ❧➜② ❣✐ỵ✐ ❤↕♥
f 2 (x) = lim f (an ).f (bn ) ≤ 0✳

❉♦ t➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ ❤➔♠ sè

f (an ).f (bn ) ≤ 0

b✱

n→∞

tr♦♥❣ ❜✐➸✉ t❤ù❝

n→∞

f (x) = 0 ❤❛② x ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ f (x) = 0 tr♦♥❣ [a, b]✳

✷✶


s số
ữợ tự

n

an < x < bn ✈➔ bn − an = b−a
an .
˜| ≤ bn − an = b−a
✤ó♥❣ ❧➔ x = an t❤➻ |x − x
2n .
b−a
✤ó♥❣ ❧➔ x
˜ = bn t❤➻ |˜
x − x| ≤ bn − an = 2n .
n
t❤➻ t❛ ❝â ✤→♥❤ ❣✐→✿ ✿
✤ó♥❣ ❧➔ x

˜ = an +b
2

t❛ ❝â

◆➳✉ ❝❤å♥ ♥❣❤✐➺♠ ❣➛♥
◆➳✉ ❝❤å♥ ♥❣❤✐➺♠ ❣➛♥
◆➳✉ ❝❤å♥ ♥❣❤✐➺♠ ❣➛♥

|˜
x − x|
ữ s ữợ tự

n

bn an
ba
= n+1 .
2
2

ồ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❧➔

x˜ = cn =

an +bn
2 ✱

t❛ s➩ ✤÷đ❝ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❤ì♥✳

an +bn
t❤➻
2

bn −an
2

= 2b−a
n+1 . ❉♦ õ ợ ộ > 0
trữợ ở trữợ t õ |x x
˜| < ε ✈ỵ✐ ♠å✐ n > log2 ( b−a
ε )
n
t ộ ữợ t ồ x
= an +b
t❤➻ t❛ ❝ô♥❣ ❝â✿
2
◆➳✉ ❝❤å♥

x˜ =

|x − x˜| ≤

|xn+1 − xn | = |(xn+1 − x) + (x − xn )| ≤

b−a
ba
b−a
+
≤

,
2n+2
2n+1
2n

❞♦ ✤â ❦❤✐ t➼♥❤ t♦→♥✱ t❛ ❝â t❤➸ ❞ø♥❣ t➼♥❤ t♦→♥ ❦❤✐

xn−1 = xn = xn+1 = ....

✤ó♥❣ ✤➳♥ ❝❤ú sè t❤➟♣ ♣❤➙♥ ❝➛♥ t❤✐➳t✳

✷✳✸ ▼ët sè ❜➔✐ t t ú ợ ữỡ
ổ
ử

ữỡ tr

x99 + x 10 = 0,
ợ



= 10−6 .

❚➻♠ ❦❤♦↔♥❣ t→❝❤ ♥❣❤✐➺♠✿

f (x) = 99x98 + 1
❧✉ỉ♥ ❧✉ỉ♥ ❞÷ì♥❣✱ ♥➯♥ ✤ì♥ ✤✐➺✉ t➠♥❣✱ f (1) = −8 ✈➔ f (1.03) = 9.688 ❞♦ ✤â
♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t tr♦♥❣ ❦❤♦↔♥❣ (1, 1.03).

❚ø ♥❤➟♥ ①➨t ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✺✮ ❧➔ ♠ët ✤❛ t❤ù❝ ❝â ✤↕♦ ❤➔♠

x0 = 1+1.03
= 1.015❀ ∆x0 = 0.015❀ f (x0 ) = −4.61845 tr→✐ ❞➜✉ ✈ỵ✐ f (1.03).
2
⇒①1 = 1.015+1.03
= 1.0225❀ ∆x1 = 0.0075❀ f (x1 ) = 0.07291 tr→✐ ❞➜✉ ✈ỵ✐ f (1).
2
⇒①2 = 1..0225+1
= 1.01125❀ ∆x2 = 0.00375❀ f (x2 ) = −5.9619 tr→✐ ❞➜✉ ✈ỵ✐
2

✷✷


f (1.103).
...........
ữỡ ổ q ữợ ❧➦♣ t❛ s➩ ❝â
sè

1.03−1
212

x12 = 1.02242

✈➔ s❛✐

= 0.00000732 < 10−6 .

❱➼ ❞ư ✷✳✸✳

●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè s✐➯✉ ✈✐➺t✿

√
x2016 + 21 5 x − 5 cosx − 14 = 0.
❚ø ♥❤➟♥ ①➨t ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✻✮ ❧➔ ♠ët ✤❛ t❤ù❝ ❝â ✤↕♦ ❤➔♠

√
21 5

✭✷✳✻✮

f (x) = 2016x2015 +

x4 + 5 sinx, f (0) = −19 ✈➔ f (1) = 3 ❞♦ ✤â ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ ❞✉②
♥❤➜t tr♦♥❣ ❦❤♦↔♥❣ (0, 1)✳
◆❣❤✐➺♠ ừ ữỡ tr ữ ừ ỗ t❤à ❤➔♠ sè y =
√
x2016 + 21 5 x − 14 = 0 ✈➔ y = 5 cosx✳ ❚❛ ❝â t ớ tt
ỗ t tr (0, 1) tr➯♥ ❝ị♥❣ ♠ët ❤➺ trư❝ tå❛ ✤ë ♥❤÷ ❤➻♥❤ ✭✷✳✶✮✳
5

✷✸


ỗ t số y = x2016 + 21 √x − 14 ✈➔ y = 5 cosx
5

◆❤➻♥ ỗ t t t số t ♥❤❛✉ t↕✐ ♠ët ✤✐➸♠ tr♦♥❣

(0.4, 0.7)✳ ❚✉② ♥❤✐➯♥✱ ✤➸ ❝❤➼♥❤ ①→❝ ❤ì♥ t❛ ❝➛♥ t➼♥❤ ❣✐→ trà ❝õ❛ ❤➔♠
sè t↕✐ x = 0.4; x = 0.7✳ ❚❛ ❝â✿ f (0.4) = −1.51626; f (0.7) = 0.55452✳
❱➟② ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✻✮ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t tr♦♥❣ ❦❤♦↔♥❣ (0.4, 0.7)✳
❱ỵ✐ ❦❤♦↔♥❣ ♣❤➙♥ ❧✐ (0.4, 0.7) ợ ữỡ ổ ũ ợ sỹ ộ
trủ ừ tt s ữợ t❛ ❝â ♥❣❤✐➺♠ x20 = 0.510998
0.7−0.4
✈➔ s❛✐ sè ♥❤ä ❤ì♥
= 2.861022949.10−7 ✳
220
❦❤♦↔♥❣

✷✹


ữỡ

ệ P

ể ế Pì
❇➀◆● P❍×❒◆● P❍⑩P
❈❍■❆ ✣➷■
✸✳✶ ▼ët ✈➔✐ ♥➨t ✈➲ ♣❤➛♥ ♠➲♠ tt
ợ t
tt t ữủ r sr ♣❤→t ❤➔♥❤ ✈➔♦
♥➠♠ ✶✾✽✽ ❧➔ ♠ët ❤➺ t❤è♥❣ ♥❤➡♠ t❤ü❝ ❤✐➺♥ ❝→❝ t➼♥❤ t♦→♥ t♦→♥ ❤å❝ tr➯♥ ♠→②
t➼♥❤ ✤✐➺♥ tû✳ õ ởt tờ ủ t t ỵ t t số
ỗ t ổ ỳ ❧➟♣ tr➻♥❤ t✐♥❤ ✈✐✳ ▲➛♥ ✤➛✉ t✐➯♥ ❦❤✐ ✈❡rs✐♦♥ ✶ ❝õ❛
▼❛t❤❡♠❛t✐❝❛ ✤÷đ❝ ♣❤→t ❤➔♥❤✱ ♠ư❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ♣❤➛♥ ♠➲♠ ♥➔② ❧➔ ✤÷❛ ✈➔♦
sû ❞ư♥❣ ❝❤♦ ❝→❝ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ t ỵ ổ t ồ ũ ợ
tớ ▼❛t❤❡♠❛t✐❝❛ trð t❤➔♥❤ ♣❤➛♥ ♠➲♠ q✉❛♥ trå♥❣ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝

❦❤♦❛ ❤å❝ ❦❤→❝✳
▼❛t❤❡♠❛t✐❝❛ ❧➔ ♥❣ỉ♥ ♥❣ú t➼❝❤ ❤đ♣ ✤➛② ✤õ ♥❤➜t ❝→❝ t➼♥❤ t♦→♥ ❦ÿ t❤✉➟t✳
▲➔ ❞↕♥❣ ♥❣æ♥ ♥❣ú ❞ü❛ tr ỵ ỷ ỵ ỳ tữủ trữ ◆❣ỉ♥
♥❣ú ▼❛t❤❡♠❛t✐❝❛ ❝â ÷✉ ✤✐➸♠ ✈÷đt trë✐ ❤ì♥ ✈➲ ❣✐❛♦ ❞✐➺♥ t❤➙♥ t❤✐➺♥✱ ✈➲ ❦❤↔

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