Một số phương pháp cơ bản giải phương trình và hệ phương trình

✣❸■ ❍➴❈ ✣⑨ ◆➂◆● ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❱➹ ◆●➴❈ ❚❘➑ ▼❐❚ ❙➮ P❍×❒◆● P❍⑩P ❈❒ ❇❷◆ ●■❷■ P❍×❒◆● ❚❘➐◆❍ ❱⑨ ❍➏ P❍×❒◆● ❚❘➐◆❍ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣➔ ◆➤♥❣ ✲ ✷✵✷✵ ✣❸■ ❍➴❈ ✣⑨ ◆➂◆● ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❱➹ ◆●➴❈ ❚❘➑ ▼❐❚ ❙➮ P❍×❒◆● P❍⑩P ❈❒ ❇❷◆ ●■❷■ P❍×❒◆● ❚❘➐◆❍ ❱⑨ ❍➏ P❍×❒◆● ❚❘➐◆❍ ❈❤✉②➯♥ ♥❣➔♥❤✿ ữ P ữợ ❞➝♥ ❚❙✳ ▲➊ ❱❿◆ ❉Ô◆● ✣➔ ◆➤♥❣ ✲ ✷✵✷✵ ✶ ▼Ư❈ ▲Ư❈ ▲í✐ ❝↔♠ ì♥ ✸ ▼ð ✤➛✉ ✹ ❈❤÷ì♥❣ ỡ s ỵ tt ữỡ ♣❤÷ì♥❣ tr➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✶ P❤÷ì♥❣ tr➻♥❤ ♠ët ➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✷ P❤÷ì♥❣ tr➻♥❤ t÷ì♥❣ ✤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✸ P❤➨♣ ❜✐➳♥ ✤ê✐ t÷ì♥❣ ✤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✹ P❤÷ì♥❣ tr➻♥❤ ❤➺ q✉↔ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✺ P❤÷ì♥❣ tr➻♥❤ ♥❤✐➲✉ ➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ P❤÷ì♥❣ tr➻♥❤ ❜➟❝ ♥❤➜t ✈➔ ❜➟❝ ❤❛✐ ♠ët ➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✶ P❤÷ì♥❣ tr➻♥❤ ❜➟❝ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✷ P❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✸ ỵ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ P❤÷ì♥❣ tr➻♥❤ q✉② ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ♥❤➜t✱ ❜➟❝ ❤❛✐ ✳ ✳ ✶✳✸✳✶ P❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ➞♥ tr♦♥❣ ❞➜✉ ❣✐→ trà tt ố Pữỡ tr ự ữợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ♥❤➜t ♥❤✐➲✉ ➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹✳✶ ❍➺ ❤❛✐ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ♥❤➜t ❤❛✐ ➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹✳✷ ❍➺ ❜❛ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ♥❤➜t ❜❛ ➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺ ❙ü ỗ số ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺✳✶ ❚➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺✳✷ ◗✉② t➢❝ ①➨t t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤÷ì♥❣ ✷✳ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❝ì ❜↔♥ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✻ ✼ ✼ ✼ ✽ ✽ ✽ ✾ ✾ ✶✵ ✶✵ ✶✵ ✶✶ ✶✶ ✶✶ ✶✷ ✶✷ ✶✸ ✶✹ ✷✳✶ ▲ô② t❤ø❛ ❤❛✐ ✈➳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶✳✶ ▼ư❝ ✤➼❝❤✱ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✉♥❣✱ ❞➜✉ ❤✐➺✉ ♥❤➟♥ ❜✐➳t ✳ ✶✹ ✷ ✷✳✶✳✷ ❱➼ ❞ö ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✸ ❇➔✐ t➟♣ ✈➟♥ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ✣➦t ➞♥ ♣❤ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✶ ▼ư❝ ✤➼❝❤✱ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✉♥❣✱ ❞➜✉ ❤✐➺✉ ♥❤➟♥ ❜✐➳t ✷✳✷✳✷ ❱➼ ❞ö ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✸ ❇➔✐ t➟♣ ✈➟♥ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ ❙û ❞ư♥❣ t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✶ ▼ư❝ ✤➼❝❤✱ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✉♥❣✱ ❞➜✉ ❤✐➺✉ ♥❤➟♥ ❜✐➳t ✷✳✸✳✷ ❱➼ ❞ö ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✸ ❇➔✐ t➟♣ ✈➟♥ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹ ▼ët sè ❜➔✐ t♦→♥ ❝❤å♥ ❧å❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✾ ✷✵ ✷✵ ✷✶ ✷✻ ✷✼ ✷✼ ✷✽ ✸✶ ✸✶ ❈❤÷ì♥❣ ✸✳ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❝ì ❜↔♥ ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✸✻ ✸✳✶ ✣➦t ➞♥ ♣❤ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✳✶ ❱➼ ❞ö ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✳✷ ❇➔✐ t➟♣ ✈➟♥ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✸✳✷ ❙û ❞ư♥❣ t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❤➔♠ sè ✸✳✷✳✶ ❱➼ ❞ö ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷✳✷ ❇➔✐ t➟♣ ✈➟♥ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✸✳✸ ▼ët sè ❜➔✐ t♦→♥ ❝❤å♥ ❧å❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✻ ✹✷ ✹✸ ✹✸ ✹✼ ✹✽ ❑➳t ❧✉➟♥ ✺✸ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✺✸ ✸ ▲❮■ ❈❷▼ ❒◆ ▲í✐ ✤➛✉ t✐➯♥ ❡♠ ①✐♥ ❝❤➙♥ t ỡ qỵ ổ trữớ ữ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ✣➔ ◆➤♥❣✱ ✤➦❝ ❜✐➺t ❧➔ t❤➛② ❝ỉ ❑❤♦❛ ❚♦→♥ ✤➣ q✉❛♥ t➙♠✱ ❣✐ó♣ ✤ï ✈➔ tr✉②➲♥ t ố tự qỵ ú ❡♠ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳ ❊♠ ỡ ỳ ữớ ợ ✤➣ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï ♠➻♥❤ ✈➲ ♠➦t t✐♥❤ t❤➛♥ tr♦♥❣ q✉→ tr➻♥❤ ❧➔♠ ❦❤â❛ ❧✉➟♥ ✈➔ ♥❤ú♥❣ ✤✐➲✉ ✤â ✤➣ t↕♦ t✐➲♥ ✤➲ ❝❤♦ ❡♠ ❤♦➔♥ t❤➔♥❤ ❜➔✐ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ♥➔②✳ ❱ỵ✐ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ♥❤➜t✱ ❡♠ ①✐♥ ❣û✐ ✤➳♥ t❤➛② ▲➯ ❱➠♥ ❉ơ♥❣✱ ♥❣÷í✐ t t ữợ ú ù t ❜➔✐ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳ ❚r♦♥❣ ❦❤✉æ♥ ❦❤ê ♠ët ❧✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝✱ ❝ơ♥❣ ♥❤÷ sü ❤↕♥ ❝❤➳ ✈➲ t❤í✐ ❣✐❛♥✱ ✈➲ ❦❤↔ ♥➠♥❣ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ s❛✐ sât✳ ❘➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ ❝→❝ þ ❦✐➳♥ ✤â♥❣ ❣â♣ ❝õ❛ ❝→❝ ❚❤➛②✱ ❈æ ✈➔ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❈✉è✐ ❝ị♥❣ ❡♠ ①✐♥ ❦➼♥❤ ❝❤ó❝ ❝→❝ t❤➛② ❝ỉ ♠↕♥❤ ❦❤ä❡✱ ❤↕♥❤ ♣❤ó❝ ✈➔ t❤➔♥❤ ❝ỉ♥❣ tr♦♥❣ ❝ỉ♥❣ ✈✐➺❝ ❣✐↔♥❣ ❞↕②✳ ❙✐♥❤ ✈✐➯♥✿ ❱➹ ◆●➴❈ ❚❘➑ ✹ ▼Ð ✣❺❯ ✶✳ ỵ t Pữỡ tr ữỡ tr ❧➔ ♠ët ♥ë✐ ❞✉♥❣ ❦❤→ q✉❡♥ t❤✉ë❝ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ ❚♦→♥ ♣❤ê t❤ỉ♥❣ ❝ơ♥❣ ♥❤÷ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ ❚♦→♥ ❝❛♦ ❝➜♣✳ Ð ❜➟❝ ❚r✉♥❣ ❤å❝ ♣❤ê t❤æ♥❣✱ ♥ë✐ ❞✉♥❣ ♥➔② ❧✉æ♥ ❝â ♠➦t tr♦♥❣ ✤➲ t❤✐ ✣↕✐ ❤å❝✱ ❚r✉♥❣ ❤å❝ ♣❤ê t❤æ♥❣ q✉è❝ ❣✐❛ ❤❛② tr♦♥❣ ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ ❝→❝ ❝➜♣✳ ❈á♥ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ ❈❛♦ ✤➥♥❣✱ ✣↕✐ ❤å❝ t❤➻ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝ơ♥❣ ✤÷đ❝ ✤➲ ❝➟♣ tr♦♥❣ ❜ë ♠æ♥ ●✐↔✐ t➼❝❤✳ ❚r♦♥❣ ❝✉ë❝ sè♥❣ ❤✐➺♥ ♥❛②✱ ♥❤✐➲✉ ❜➔✐ t♦→♥ ✤➣ ✤÷đ❝ ✤➦t r❛ ✈➔ ✤➸ ❣✐↔✐ q✉②➳t ❝❤ó♥❣ t❛ ♣❤↔✐ ✤÷❛ ✈➲ ❣✐↔✐ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ❤❛② ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ♥➔♦ ✤â ♥❤➡♠ ✤→♣ ù♥❣ ♥❤✉ ❝➛✉ ❝õ❛ ❜↔♥ t❤➙♥ ✈➔ ①➣ ❤ë✐✳ ❇➯♥ ❝↕♥❤ ✤â✱ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ ♥❤ú♥❣ ♥ë✐ ❞✉♥❣ ❝ì ❜↔♥ ✈➔ q✉❛♥ trå♥❣ ❝õ❛ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ❜➟❝ ♣❤ê t❤ỉ♥❣ tr✉♥❣ ❤å❝✳ ❈â ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✱ tr♦♥❣ ✤â ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❝ì ❜↔♥ ❝â t❤➸ ❦➸ ✤➳♥ ♥❤÷ ❧➔ ❧ơ② t❤ø❛ ❤❛✐ ✈➳✱ ✤➦t ➞♥ ♣❤ư✱ ①➨t t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❤➔♠ sè✱✳✳✳✣➸ ❣✐↔✐ q✉②➳t ✤÷đ❝ ❝→❝ ❜➔✐ t♦→♥ ♥➔② ✤á✐ ❤ä✐ sü t❤ỉ♥❣ ♠✐♥❤✱ t÷ ❞✉② ♥❤↕② ❜➨♥✱ ✈➟♥ ❞ư♥❣ ❝→❝ ❦✐➳♥ t❤ù❝ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ✤➣ ❤å❝✳ ❈❤➼♥❤ ✈➻ sü ①✉➜t ❤✐➺♥ t❤÷í♥❣ ①✉②➯♥ tr♦♥❣ ❝→❝ ❦ý t❤✐✱ ❧➔ ♠ët ❣✐→♦ ✈✐➯♥ ❚♦→♥ t÷ì♥❣ ❧❛✐✱ tỉ✐ ♠✉è♥ t➻♠ ❤✐➸✉ ❦ÿ ❤ì♥ ♥❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝ì ❜↔♥ ✤➸ ❣✐↔✐ q✉②➳t ♥❤ú♥❣ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ♥❤➡♠ ♣❤ư❝ ✈ư ❝❤♦ ♥❣➔♥❤ ♥❣❤➲ s❛✉ ♥➔②✱ ♥➯♥ tỉ✐ ❝❤å♥ ✤➲ t➔✐ ✧▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❝ì ❜↔♥ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✧ ❝❤♦ ❦❤â❛ ❧✉➟♥ ❝õ❛ ♠➻♥❤✳ ✷✳ ▼ö❝ ✤➼❝❤ ✈➔ ♥❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉ ✲ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉✿ ✣➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❞➜✉ ❤✐➺✉ ♥❤➟♥ ❜✐➳t ❝→❝ ❧♦↕✐ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝â t❤➸ ❣✐↔✐ ✤÷đ❝ ❜➡♥❣ ♥❤ú♥❣ ♣❤÷ì♥❣ ỡ ữợ ữỡ ũ ❤ñ♣ ♥❤➡♠ ❣✐↔✐ q✉②➳t ❜➔✐ t♦→♥ ♥❤❛♥❤ ❣å♥✳ ✲ ◆❤✐➺♠ ✈ư ♥❣❤✐➯♥ ❝ù✉✿ ❍➺ t❤è♥❣ ❤â❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❝ì ❜↔♥ ❣✐↔✐ ✺ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✳ ✸✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ✲ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉✿ P❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✳ ✲ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉✿ P❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ð ❝❤÷ì♥❣ tr➻♥❤ ❚❍P❚✱ ❝→❝ ❦ý t❤✐ ❚❍P❚◗●✱ ❦ý t❤✐ ❍❙● ❝➜♣ ❚❤➔♥❤ ♣❤è✳ ✹✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ✲ ❚❤✉ t❤➟♣✱ t➻♠ ❤✐➸✉ ❝→❝ t➔✐ ❧✐➺✉ ❧✐➯♥ q✉❛♥ ✤➳♥ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✳ ✲ P❤➙♥ t➼❝❤✱ ❤➺ t❤è♥❣ ❝→❝ t➔✐ ❧✐➺✉ ✤➸ tø ✤â tê♥❣ ❤ñ♣✱ ❝❤å♥ ❧å❝ ♥❤ú♥❣ ♥ë✐ ❞✉♥❣ ❝➛♥ t❤✐➳t ✤÷❛ ✈➔♦ ❦❤â❛ ❧✉➟♥✳ ✲ ❚➻♠ ❤✐➸✉ ♥ë✐ ❞✉♥❣ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ð ❝➜♣ tr✉♥❣ ❤å❝ ♣❤ê t❤ỉ♥❣✳ ✲ P❤÷ì♥❣ ♣❤→♣ tê♥❣ ❦➳t ❦✐♥❤ ♥❣❤✐➺♠ ✿ tr❛♦ t t ỵ ữợ s ự t tứ trt tr ợ ữợ tứ ✤â tê♥❣ ❤ñ♣✱ ❝❤å♥ ❧å❝ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝â ❧✐➯♥ q✉❛♥ ✤➸ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✳ ✺✳ Þ ♥❣❤➽❛ t❤ü❝ t✐➵♥ ❝õ❛ ✤➲ t➔✐ ✲ ✣➲ t➔✐ ♥➔② ❝â t❤➸ sû ❞ư♥❣ ♥❤÷ ♠ët t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❣✐➔♥❤ ❝❤♦ ❤å❝ s✐♥❤ ♣❤ê t❤æ♥❣ tr✉♥❣ ❤å❝✱ ❤å❝ s✐♥❤ ❣✐ä✐ t♦→♥✱ ❣✐→♦ ✈✐➯♥ ♣❤ê t❤æ♥❣ tr✉♥❣ ❤å❝ ✈➔ s✐♥❤ ✈✐➯♥ t➻♠ ❤✐➸✉ ✈➲ ♥ë✐ ❞✉♥❣ ♥➔②✳ ✻✳ ❈➜✉ tró❝ ❧✉➟♥ ✈➠♥ ◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉ ✈➔ ❦➳t ❧✉➟♥✱ ♥ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ t❤➔♥❤ ❜❛ ❝❤÷ì♥❣✳ ❼ ❈❤÷ì♥❣ ✶✿ ❈ì s ỵ tt ữỡ tr ữỡ tr ❼ ❈❤÷ì♥❣ ✷✿ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❝ì ❜↔♥ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✳ ❼ ❈❤÷ì♥❣ ✸✿ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❝ì ữỡ tr é ị ì r ữỡ t tr ởt sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✿ ✣↕✐ ❝÷ì♥❣ ✈➲ ♣❤÷ì♥❣ tr➻♥❤✱ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ♥❤➜t ✈➔ ❜➟❝ ❤❛✐ ♠ët ➞♥✱ ♠ët sè ♣❤÷ì♥❣ tr➻♥❤ q✉② ✈➲ ❜➟❝ ♥❤➜t ❤♦➦❝ ❜➟❝ ❤❛✐ ♠ët ➞♥✳✳✳ ❈→❝ ❦➳t q✉↔ tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤õ ②➳✉ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tø ❝→❝ ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱❬✷❪✱❬✸❪✳ ✶✳✶ ✣↕✐ ❝÷ì♥❣ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✶✳✶✳✶ P❤÷ì♥❣ tr➻♥❤ ♠ët ➞♥ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❈❤♦ ❤❛✐ ❤➔♠ sè y = f (x) ✈➔ y = g(x) ❝â t➟♣ ①→❝ ✤à♥❤ ❧➛♥ ❧÷đt ❧➔ Df ✈➔ Dg ✳ ✣➦t D = Df ∩Dg ✳ ▼➺♥❤ ✤➲ ❝❤ù❛ ❜✐➳♥ ✧f (x) = g(x)✧ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ♠ët ➞♥❀ x ❣å✐ ❧➔ ➞♥ sè ✭❤❛② ➞♥✮ ✈➔ D ❣å✐ ❧➔ t➟♣ ①→❝ ✤à♥❤ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✳ ❙è x0 ∈ D ❣å✐ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ f (x) = g(x) ♥➳✉ ✧f (x0 ) = g(x0 ) ú ú ỵ ✣➸ t❤✉➟♥ t✐➺♥ tr♦♥❣ t❤ü❝ ❤➔♥❤✱ t❛ ❦❤æ♥❣ ❝➛♥ ✈✐➳t rã t➟♣ ①→❝ ✤à♥❤ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♠➔ ❝❤➾ ❝➛♥ ♥➯✉ ✤✐➲✉ ❦✐➺♥ ✤➸ x ∈ D✳ ✣✐➲✉ ❦✐➺♥ ✤â ❣å✐ ❧➔ ✤✐➲✉ ❦✐➺♥ ①→❝ ✤à♥❤ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✱ ❣å✐ t➢t ❧➔ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✳ ✭✐✐✮ ❑❤✐ ❣✐↔✐ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ✭tù❝ ❧➔ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✮✱ ♥❤✐➲✉ ❦❤✐ t❛ ❝❤➾ ❝➛♥✱ ❤♦➦❝ ❝❤➾ ❝â t❤➸ t➼♥❤ ❣✐→ trà ❣➛♥ ✤ó♥❣ ❝õ❛ ♥❣❤✐➺♠ ✭✈ỵ✐ ✤ë ❝❤➼♥❤ ①→❝ ♥➔♦ ✤â✮✳ ●✐→ trà ✤â ❣å✐ ❧➔ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✳ ✼ ✭✐✐✐✮ ❈→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ f (x) = g(x) ❧➔ ❤♦➔♥❤ ✤ë ❣✐❛♦ ✤✐➸♠ ❝õ❛ ỗ t số y = f (x) y = g(x)✳ ✶✳✶✳✷ P❤÷ì♥❣ tr➻♥❤ t÷ì♥❣ ✤÷ì♥❣ ❍❛✐ ♣❤÷ì♥❣ tr➻♥❤ ✭❝ị♥❣ ➞♥✮ ✤÷đ❝ ❣å✐ ❧➔ t÷ì♥❣ ✤÷ì♥❣ ♥➳✉ ❝❤ó♥❣ ❝â ❝ị♥❣ ♠ët t➟♣ ♥❣❤✐➺♠✳ ◆➳✉ ♣❤÷ì♥❣ tr➻♥❤ f1(x) = g1(x) tữỡ ữỡ ợ ữỡ tr f2(x) = g2(x) t t❛ ✈✐➳t f1(x) = g1(x) ⇔ f2(x) = g2 (x)✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ ✶✳✶✳✸ P❤➨♣ ❜✐➳♥ ✤ê✐ t÷ì♥❣ ✤÷ì♥❣ ✣➸ ❣✐↔✐ ♠ët ♣❤÷ì♥❣ tr➻♥❤✱ t❤ỉ♥❣ t❤÷í♥❣ t❛ ❜✐➳♥ ✤ê✐ ♣❤÷ì♥❣ tr➻♥❤ ✤â t❤➔♥❤ ♠ët ♣❤÷ì♥❣ tr➻♥❤ t÷ì♥❣ ✤÷ì♥❣ ✤ì♥ ❣✐↔♥ ❤ì♥✳ ❈→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ♥❤÷ ✈➟② ✤÷đ❝ ❣å✐ ❧➔ tữỡ ữỡ ỵ ♣❤÷ì♥❣ tr➻♥❤ f (x) = g(x) ❝â t➟♣ ①→❝ ✤à♥❤ D❀ y = h(x) ❧➔ ♠ët ❤➔♠ sè ①→❝ ✤à♥❤ tr➯♥ D ✭h(x) ❝â t❤➸ ❧➔ ❤➡♥❣ sè✮✳ ❑❤✐ ✤â tr D ữỡ tr tữỡ ữỡ ợ ộ ♣❤÷ì♥❣ tr➻♥❤ s❛✉✿ ✭✶✮ f (x) + h(x) = g(x) + h(x)❀ ✭✷✮ f (x)h(x) = g(x)h(x) ♥➳✉ h(x) = ợ x D Pữỡ tr q✉↔ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳ f1 (x) = g1 (x) ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❤➺ q✉↔ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ f (x) = g(x) ♥➳✉ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ♥â ❝❤ù❛ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ f (x) = g(x)✳ ❑❤✐ ✤â✱ t❛ ✈✐➳t f (x) = g(x) ⇒ f1(x) = g1(x)✳ ✣à♥❤ ỵ ữỡ ởt ữỡ tr➻♥❤✱ t❛ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➺ q✉↔ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦✳ f (x) = g(x) ⇒ [f (x)]2 = [g(x)]2 ú ỵ ❝õ❛ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ❧✉ỉ♥ ❝ị♥❣ ❞➜✉ t❤➻ ❦❤✐ ❜➻♥❤ ♣❤÷ì♥❣ ❤❛✐ ✈➳ ❝õ❛ ♥â✱ t❛ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤ t÷ì♥❣ ✤÷ì♥❣✳ ✭✐✐✮ ◆➳✉ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ❞➝♥ ✤➳♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➺ q✉↔ t❤➻ s❛✉ ❦❤✐ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❤➺ q✉↔✱ t❛ ♣❤↔✐ t❤û ❧↕✐ ❝→❝ ♥❣❤✐➺♠ t➻♠ ✤÷đ❝ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ✤➸ ♣❤→t ❤✐➺♥ ✈➔ ❧♦↕✐ ❜ä ♥❣❤✐➺♠ ♥❣♦↕✐ ❧❛✐✳ ✶✳✶✳✺ P❤÷ì♥❣ tr➻♥❤ ♥❤✐➲✉ ➞♥ P❤÷ì♥❣ tr➻♥❤ ♥❤✐➲✉ ➞♥ ❧➔ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣ F = G✱ tr♦♥❣ ✤â F ✈➔ G ❧➔ ♥❤ú♥❣ ❜✐➸✉ t❤ù❝ ❝õ❛ ♥❤✐➲✉ ❜✐➳♥✳ ❈❤➥♥❣ ❤↕♥✱ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✻✳ 2x2 + 4xy − y = −x + 2y + ❧➔ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ❤❛✐ ➞♥ ✭x ✈➔ y✮❀ x + y + z = 3xyz ❧➔ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ❜❛ ➞♥ ✭x✱ y ✈➔ z ✮✳ ◆➳✉ ♣❤÷ì♥❣ tr➻♥❤ ❤❛✐ ➞♥ x ✈➔ y trð t❤➔♥❤ ♠➺♥❤ ✤➲ ✤ó♥❣ ❦❤✐ x = x0 ✈➔ y = y0 ✭✈ỵ✐ x0 ✈➔ y0 ❧➔ sè✮ t❤➻ t❛ ❣å✐ ❝➦♣ sè (x0; y0) ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♥â✳ ❑❤→✐ ♥✐➺♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜❛ ➞♥✱ ❜è♥ ➞♥✱✳✳✳ ❝ơ♥❣ ✤÷đ❝ ❤✐➸✉ t÷ì♥❣ tü✳ ✶✳✷ P❤÷ì♥❣ tr➻♥❤ ❜➟❝ ♥❤➜t ✈➔ ❜➟❝ ❤❛✐ ♠ët ➞♥ ✶✳✷✳✶ P❤÷ì♥❣ tr➻♥❤ ❜➟❝ ♥❤➜t ❑➳t q✉↔ ❣✐↔✐ ✈➔ ❜✐➺♥ ❧✉➟♥ ♣❤÷ì♥❣ tr➻♥❤ ❞↕♥❣ ax + b = ✤÷đ❝ tâ♠ t➢t ♥❤÷ s❛✉✳ ✶✮ a = ✿ P❤÷ì♥❣ tr➻♥❤ ❝â ♠ët ♥❣❤✐➺♠ ❞✉② ♥❤➜t x = − ab ✳ ✷✮ a = ✈➔ b = ✿ P❤÷ì♥❣ tr➻♥❤ ✈ỉ ♥❣❤✐➺♠✳ ✸✮ a = ✈➔ b = ✿ P❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ ✤ó♥❣ ✈ỵ✐ ♠å✐ x ∈ R✳ ✹✵ ✲ ❱ỵ✐ u = 3, v = 4✱ t❛ ❝â  12   x=  x+y =3 ⇔  x = 4y  y= ✲ ❱ỵ✐ u = 4, v = 3✱ t❛ ❝â x+y =4 ⇔ x = 3y x=3 y = ❙♦ s→♥❤ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉✱ t❛ t❤➜② ❝↔ ❤❛✐ ♥❣❤✐➺♠ tr➯♥ ✤➲✉ t❤ä❛ ♠➣♥✳ ❱➟② ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❝â ❤❛✐ ♥❣❤✐➺♠ ❧➔ (x, y) = ( 12 , ), (3, 1)✳ 5 ❱➼ ❞ư ✸✳✶✳✺✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤  6(x + z) 5(x + y)   + =4    x + y + 6xy x + z + 5xz     6(y + z) 4(x + y) + =5 y + z + 4yz x + y + 6xy      4(x + z) 5(y + z)   +   x + z + 5xz y + z + 4yz = ✭✣➲ ❝❤å♥ ✤ë✐ t✉②➸♥ tr÷í♥❣ P❚◆❑✱ ❚P❍❈▼✮ P❤➙♥ t➼❝❤✳ ❱➲ ❤➻♥❤ t❤ù❝ ❜➔✐ t♦→♥ ♥➔② ❦❤→ ♣❤ù❝ t↕♣✳ ❚✉② ♥❤✐➯♥✱ ♥➳✉ q✉❛♥ s→t ❦➽✱ ❝❤ó♥❣ t❛ s➩ ❞➵ ❞➔♥❣ t❤➜② ✤÷đ❝ ❝→❝ ➞♥ ♣❤ư ❧➛♥ ❧÷đt ❧➔ y+z x+z x+y , , x + y + 6xy y + z + 4yz x + z + 5xz ❚ø ✤â✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ trð t❤➔♥❤ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ♥❤➜t ❜❛ ➞♥ t❤ỉ♥❣ t❤÷í♥❣✳ ▲í✐ ❣✐↔✐✳ ✣➦t a = x +xy++y6xy , b = y +yz++z4yz , c = x +xz++z5xz ✳ ❍➺  ✤➣ ❝❤♦ trð t❤➔♥❤   5a + 6c = 4a + 6b = ⇔   5b + 4c = a = ,b = ,c = 16 ✹✶  x+y   =    x + y + 6xy       7(x + y) = 6xy y+z ❉♦ ✤â✱  y + z + 4yz = ⇔  y + z = 12yz    7(x + z) = 45xz  x + z     x + z + 5xz = 16    1 −33 −14      + = =    x =    x y    x 14 33        1   14 14 = + = 12 ⇔ y= ⇔ ⇔ y 45 y z 45             14 123 1 45       z = =    + =  x z  z 123 14 14 14 ❱➟② ❤➺ ✤➣ ❝❤♦ ❝â ♥❣❤✐➺♠ ❧➔ (x, y, z) = ( −14 , , )✳ 33 45 123 ❱➼ ❞ư ✸✳✶✳✻✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤   x2 + + y + xy = 4y y  x+y−2= + x2 ✭✣➲ t❤✐ ❍❙● t➾♥❤ P❤ó ❚❤å✮ P❤➙♥ t➼❝❤✳ ✲ Ð ❱P ❝õ❛ P❚✭✷✮ ❝â +y x2 ✳ ✲ Ð ❱❚ ❝õ❛ P❚✭✶✮ ❝â x2 + ✈➔ ❱P ❝õ❛ P❚✭✶✮ ❝â y✳ ❉♦ ✤â✱ ♠ët s✉② ♥❣❤➾ t❤ỉ♥❣ t❤÷í♥❣ ❧➔ t❛ ❝❤✐❛ ❤❛✐ ✈➳ ❝õ❛ P❚✭✶✮ ❝❤♦ y ✤➸ ①✉➜t ❤✐➺♥ ♣❤➛♥ t❤û ♥❣❤à❝❤ ✤↔♦ ❝õ❛ +y x2 ✳ P❚✭✶✮ ⇔ x y+ + y + x = ✭❙❛✐ ❧➛♠ t❤÷í♥❣ ❣➦♣  ð ✤➙② ❧➔ ❝❤ó♥❣ t❛ q✉➯♥ ①➨t tr÷í♥❣ ❤đ♣ ②❂✵✮ ❍➺ ✤➣ ❝❤♦ ✈✐➳t ❧↕✐ ❧➔   x +1 +x+y =4 y y   x+y−2= + x2 ❚ø ✤➙②✱ t❛ t❤➜② ❤❛✐ ➞♥ ♣❤ư ❝õ❛ ❜➔✐ t♦→♥ ❝❤➼♥❤ ❧➔ ▲í✐ ❣✐↔✐✳ x2 + y ✈➔ x + y✳ ❉➵ t❤➜② y = ❦❤æ♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ♥➯♥ t❛ ❝❤➾ ①➨t y = 0✳ ❑❤✐ ✤â✱ P❚✭✶✮ ❝õ❛ ❤➺ t÷ì♥❣ ✤÷ì♥❣ ✹✷ x2 + x2 + + x + y = ✣➦t u = ,v = x + y y  y   a+b=4  b=4−a b=4−a ❚❛ ❝â ❤➺  ⇔ ⇔ ⇔  2−a= = (a − 1)2 b−2= a a    x +1 =1 x2 + x − = y ❱ỵ✐ a = 1, b = ⇒  ⇔ y =3−x  x+y =3 x = 1, y = x = −2, y = ⇔ b=3 a = ❱➟② ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❝â ✷ ♥❣❤✐➺♠ ❧➔ (x, y) = (1, 2), (−2, 5)✳ ✸✳✶✳✷ ❇➔✐ t➟♣ ✈➟♥ ❞ư♥❣ ●✐↔✐ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ✶✮ √ √ 2x + 2y = √ √ 2x + + 2y + = ✷✮ x + y + x2 + y = xy(x + 1)(y + 1) = 12 ✸✮ 9x = 2y = x x2 + y +   x2 + y − + 2y = x    ✹✮ x2 + y + x + y = x(x + y + 1) + y(y + 1) = ✺✮ x2 + xy − 3x + y = x4 + 3x2 y − 5x2 + y = ✻✮ √ √ x 3x + y + 5x + 4y = √ 12 5x + 4y + x − 2y = 35 ✹✸ ✼✮  √   x− y+2= √   y + 2(x − 2) x + = − ✽✮ 2x2 − x(y − 1) + y = 3y x2 + xy − 3y = x − 2y     4xy + 4(x2 + y ) + ✾✮    2x + ✶✵✮ =7 (x + y)2 = x+y  8(2z + x) 7(2x + y)   + =6    x + y + 8xy x + z + 7xz     8(2y + z) 6(2x + y) + =7 y + z + 6yz x + y + 8xy      6(2z + x) 7(2y + z)   +   x + z + 7xz y + z + 6yz = ✸✳✷ ❙û ❞ư♥❣ t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❤➔♠ sè ✸✳✷✳✶ ❱➼ ❞ư ♠✐♥❤ ❤å❛ ❱➼ ❞ư ✸✳✷✳✶✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ √ 2(2x + 1)3 + 2x + = (2y − 3) y − √ √ 4x + + 2y + = ✭✣➲ t❤✐ ❝❤å♥ ✤ë✐ t trữớ P ữỡ ỗ P t➼❝❤✳ ✲ ▼➜✉ ❝❤èt ❝õ❛ ❜➔✐ t♦→♥ ♥➔② ❧➔ t❤➜② ✤÷đ❝ ❞↕♥❣ f (u) = f (v) ð P❚✭✶✮✳ ❚ø ✤â✱ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❤➔♠ sè ✈➔ ❝❤➾ r❛ u = v✳ ✲ ◆➳✉ q✉❛♥ s→t ❦➽✱ t❛ t❤➜② ❱❚ ❝õ❛ P❚✭✶✮ ❝â ❞↕♥❣ 2a3 + a ✈➔√ ❤➔♠ sè f (t) = 2t3 + t ❧➔ ❤➔♠ ✤ì♥ ✤✐➺✉✳ ❉♦ ✤â✱ ❝❤ó♥❣ t❛ ❤② ✈å♥❣ (2y − 3) y − = √ √ 2( y − 2)3 + y − ✤➸ P❚✭✶✮ ❝â ❞↕♥❣ f (u) = f (v) ♠➔ t❛ ❝➛♥✳ ❚❤➟t ✈➟②✱ t❛ ❝â ✹✹ √ (2y − 3) y − √ = (2y − + 1) y − √ √ √ √ = 2(y − 2) y − + y − = 2( y − 2)3 + y − 2✳ ▲í✐ ❣✐↔✐✳ −1 ✣✐➲✉ ❦✐➺♥✿ x ≥ , y ≥ 2✳ ❳➨t ❤➔♠ sè✿ f (t) = 2t3 + t, t ∈ (0, +∞) f (t) = 6t2 + > ♥➯♥ ✤➙② ❧➔ ỗ ứ P t õ f (2x + 1) = f ( y − 2) √ ⇔ 2x + = y − ❚❤❛② ✈➔♦ P❚ ✭✷✮✱ t❛ ✤÷đ❝✿ √ √ 4y − + 2y + = (∗) √ √ ❳➨t ❤➔♠ sè g(y) = 4y − + 2y + − 6, y ∈ (2, +∞) 1 +√ g (y) = > 0✱∀y ∈ (2, +∞) ♥➯♥ ✤➙② 2y + (4y − 8)3 ❧➔ ❤➔♠ ỗ ỡ ỳ g(6) = 4.6 + 2.61+ − = ♥➯♥ ✭✯✮ ❝â ✤ó♥❣ ♠ët ♥❣❤✐➺♠ ❧➔ y = ❱ỵ✐ y = 6✱ t❛ ❝â x = 12 ✳ ❱➟② ❤➺ ✤➣ ❝❤♦ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t (x, y) = ( 12 , 6)✳ ❱➼ ❞ư ✸✳✷✳✷✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ P❤➙♥ t➼❝❤✳ √ x2 + x = 2y √ y + y = 2x ✭✣➲ t❤✐ ❍❙● t➾♥❤ ❇➳♥ ❚r❡✮ ❉➵ t❤➜② ❤➺ ✤➣ ❝❤♦ ❝â ❞↕♥❣ f (x) = y f (y) = x ⇒ ❱✐➺❝ ❝➛♥ ❧➔♠ ❝❤➾ ❧➔ ❝❤ù♥❣ ♠✐♥❤ f (t) ❧➔ ❤➔♠ ✤ì♥ ✤✐➺✉✱ s✉② r❛ x = y ✈➔ ❣✐↔✐ ♠ët tr♦♥❣ ❤❛✐ ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ ❤➺ ✤➸ t➻♠ ♥❣❤✐➺♠✳ ▲í✐ ❣✐↔✐✳ ✣✐➲✉ ❦✐➺♥✿ x, y ≥ 0✳ ❉➵ t❤➜② ♥➳✉ x = t❤➻ y = ✈➔ ♥❣÷đ❝ ❧↕✐ ♥➯♥ t❛ ❝â ♥❣❤✐➺♠ (x, y) = (0, 0)✳ ✹✺ √ t2 + t f (t) = ,t > ❉♦ ✈➟②✱ t❛ ❝❤➾ ①➨t x, y > 0✳ ❳➨t ❤➔♠ sè f (t) = t + √ > 0, t > ỗ t ❍➺ ✤➣ ❝❤♦ ✤÷đ❝ ✈✐➳t ❧↕✐ x = f (y) y = f (x) ❚❤❛② ✈➔♦√❤➺ ✤➣ ❝❤♦✱ t❛ ❝â ⇒ x = y✳ x2 + x = 2x √ √ ⇔x x+1=2 x √ √ ⇔ ( x − 1)(x + x − 1) =   x = 1, y = √ √ ⇔ − − x= ,y = 2 √ √ 3− 3− (x, y) = (0, 0), (1, 1), ( , )✳ 2 ❱➟② ❤➺ ✤➣ ❝❤♦ ❝â ✸ ♥❣❤✐➺♠ ❧➔ ❱➼ ❞ö ✸✳✷✳✸✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ √ 2x + y = − 2x − y √ √ x + + − y = ✭✣➲ t❤✐ ❍❙● t➾♥❤ ❱➽♥❤ P❤ó❝✮ P❤➙♥ t➼❝❤✳ ✲ ◆➳✉ q✉❛♥ s→t ❦➽✱ t❛√t❤➜② P❚✭✶✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❤❛✐ t❤❡♦ P❚✭✶✮⇔ 2x + √ y + 2x + y − = 0✳ ❚ø ✤â✱ s✉② r❛ 2x + y = ⇔√ y = −√2x ❚❤❛② ✈➔♦ P❚✭✷✮✱ t❛ ✤÷đ❝✿ x + + 2x = 4✳ ▲í✐ ❣✐↔✐✳ ✣✐➲✉ ❦✐➺♥✿ 2x + y ≥ 0, y ≤ 1✳ Pữỡ tr tự t tữỡ ữỡ ợ (2x  √+ y) + 2x + y − = 2x + y = ⇔√ 2x + y = −3 ⇔ 2x + y = ⇔ y = − 2x ❚❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ t❤ù ❤❛✐ ❝õ❛ ❤➺✱ t❛ ✤÷đ❝ √ √ x+6+ 2x = (∗) √ 2x + y ✳ ✹✻ ❉➵ t❤➜② x = ❦❤æ♥❣√❧➔ ♥❣❤✐➺♠√ ❝õ❛ (∗) ❳➨t ❤➔♠ sè f (x) = x + + 2x − = 1 + √ > x õ ỗ f (x) = (x + 6)2 √ √ 2x ▼➦t ❦❤→❝✱ ✈ỵ✐ f (2) = + + 2.2 = ♥➯♥ x = ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ (∗)✳ ❱ỵ✐ x = 2✱ t❛ ❝â y = −3✳ ❱➟② ❤➺ ✤➣ ❝❤♦ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❧➔ (x, y) = (2, −3)✳ 3 ❱➼ ❞ư ✸✳✷✳✹✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ x 121 x2 + 2x = − 27   x + y + xy − 3x − 4y − =    ✭✣➲ t❤✐ ❍❙● t➾♥❤ P t ị ỗ rã r➔♥❣✳ Ð P❚✭✶✮ ❝õ❛ ❤➺✱ ❞➵ ❞➔♥❣ t❤➜② ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❦❤ỉ♥❣ q✉→ ✷ ♥❣❤✐➺♠✳ ▼ët ❜➯♥ ❧➔ ❤➔♠ ✤❛ t❤ù❝ ❜➟❝ ❤❛✐✱ ♠ët ❜➯♥ ❧➔ ❤➔♠ ♠ô✱ ❞♦ ✈➟② ❝❤ó♥❣ t❛ ♥➯♥ sû ❞ư♥❣ t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❤➔♠ sè ✤➸ ❣✐↔✐ q✉②➳t ✭❜ð✐ ✈➻ ❝❤÷❛ ❝â tổ qt ố ợ ữỡ tr Ð P❚✭✷✮ ❝õ❛ ❤➺✱ t❛ ❝♦✐ ✤â ❧➔ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❤❛✐ t❤❡♦ y✳ ✣➸ ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ t❤➻ ∆ ≥ ✭t❤❡♦ x✮✳ ❚ø ✤➙② t❛ ❝â ữủ x ứ ỵ tr t ❞➵ ❞➔♥❣ ❝❤➾ r❛ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥✳ ▲í✐ ❣✐↔✐✳ Pữỡ tr tự tữỡ ữỡ ợ y + (x − 4)y + x2 − 3x + = (∗) ✣➸ P❚✭✯✮ ❝â ♥❣❤✐➺♠ ∆x = (x − 4)2 − 4(x2 − 3x + 4) = −3x2 + 4x ≥ ⇔0≤x≤ ✳ x ▼➦t ❦❤→❝✱ ①➨t ❤➔♠ sè f (x) = x2 + 2x + 27 (0 ≤ x ≤ ) x f (x) = 2x + + 27 ln 27 > 0✱ ∀ ≤ x ≤ ✳ ❉♦ ✤â✱ ❤➔♠ ♥➔② ỗ f (x) f ( ) x 4 16 24 121 + +9= ✳ ⇒ x2 + 2x + 27 ≤ ( )2 + + 27 = 3 9 ❚ø ✤➥♥❣ t❤ù❝ tr➯♥ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ t❤ù ♥❤➜t ❝õ❛ ❤➺✱ t❛ s✉② r❛ x = 43 ✳ ❚❤❛② ✈➔♦ (∗) 4 y + ( − 4)y + ( )2 − 3.( ) + = 3 16 =0 ⇔ y2 − y + 4 ⇔ (y − )2 = ⇔ y = ✳ 3 ❱➟② ❤➺ ✤➣ ❝❤♦ ♥❣❤✐➺♠ (x, y) = ( 43 , 34 ) ✸✳✷✳✷ ❇➔✐ t➟♣ ✈➟♥ ❞ö♥❣ ●✐↔✐ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✉    ✶✮  ✷✮ ✸✮ ✹✮ x x2 + 3x = 26 − 64  x2 + y + xy + x + 5y + 25 = √ x + 2y = − 2x − 4y √ √ + x + 4y + 3x + 9y = √   √1 + √y = x + x x y √  √ y( x + − 1) = 3x2 +    ( )2x +x = y       ( )2y +y = z     2z +z   = x   (4) ✹✽ ✺✮    x + 3x − + ln(x − x + 1) = y y + 3y − + ln(y − y + 1) = z   z + 3z − + ln(z − z + 1) = x ✸✳✸ ▼ët sè ❜➔✐ t♦→♥ ❝❤å♥ ❧å❝ ❚÷ì♥❣ tü✱ ð ♠ư❝ ♥➔② tr➻♥❤ ❜➔② t❤➯♠ ♠ët sè ❜➔✐ t➟♣ ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✳ ❚❤ỉ♥❣ q✉❛ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ t÷ì♥❣ ✤÷ì♥❣ ❤đ♣ ❧➼ ✤➸ ❝â ❧í✐ ❣✐↔✐ ✤➭♣ ❝❤♦ ❜➔✐ t♦→♥✳ 2x2 y + 3xy = 4x2 + 9y ❇➔✐ ✶✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ 7y + = 2x2 + 9x ✭✣➲ t❤✐ ❝❤å♥ ✤ë✐ t✉②➸♥ ◆❤❛ ❚r❛♥❣✱ ❑❤→♥❤ ❍á❛✮ ▲í✐ ❣✐↔✐✳ ❚ø P❚✭✶✮✱ t❛ ❝â 4x2 y= 2x + 3x − + 9x − 4x2 2x2 + 9x − ⇒ = 2x + 3x − ❚ø P❚✭✷✮✱ t❛ ❝â y = 2x ⇔ 28x2 = (2x2 + 9x − 6)(2x2 + 3x − 9) ⇔ (x  + 2)(2x − 1)(2x + 9x − 27) = x = −2    ⇔x=2  √  −9 ± 33 x= 2x2 + 9x − −16 ✲ ◆➳✉ x = −2✱ t❛ ❝â y = = 7 2x2 + 9x − −1 ✲ ◆➳✉ x = ✱ t❛ ❝â y = = 7 √ −9 ± 33 2x2 + 9x − ✲ ◆➳✉ x = ✈ỵ✐ 2x + 9x = 27✱ t❛ ❝â y = = 3✳ ✹✾ −1 ❱➟② ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❝â ✹ ♥❣❤✐➺♠ ❧➔ (x, y) = (−2, −16 ), ( , ), 7 √ −9 ± 33 ( , 3)✳ ❇➔✐ ✷✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ▲í✐ ❣✐↔✐✳ 2y(x2 − y ) = 3x x(x2 + y ) = 10y ✭✣➲ t❤✐ ❝❤å♥ ✤ë✐ t✉②➸♥ ❚❍P❚ ❈❤✉②➯♥ ▲❛♠ ❙ì♥✱ ❚❤❛♥❤ ❍â❛✮ ❚❛ t❤➜② ♥➳✉ x = t❤➻ y = ✈➔ ♥❣÷đ❝ ❧↕✐ ♥➯♥ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❝â ♥❣❤✐➺♠ (x, y) = (0, 0)✳ ❳➨t tr÷í♥❣ ❤đ♣ xy = 0✳ ▲➜② P❚✭✶✮ ❝❤✐❛ ❝❤♦ P❚✭✷✮✱ t❛ ✤÷đ❝ 2y(x2 − y ) 3x = x(x2 + y ) 10y 2 ⇔ 20y (x − y ) = 3x2 (x2 + y ) ⇔ 3x4 − 17x2 y + 20y = ⇔ 3x4 − 5x2 y − 12x2 y + 20y = ⇔ x2 (3x2 − 5y ) − 4y (3x2 − 5y ) = 2 2 ⇔ (x  − 4y )(3x − 5y ) = x2 = 4y ⇔ x = y ✲ ◆➳✉ x2 = 4y2✱ ❤➺ ✤➣ ❝❤♦ trð t❤➔♥❤ 2y.3y = 3x ⇔ x.5y = 10y 2y = x ⇔ xy = ✲ ◆➳✉ x2 = 35 y2✱ ❤➺ ✤➣ ❝❤♦ trí t❤➔♥❤ 2y = x ⇔ 2y = x = ±2 y = ±1  15   √ x = ±   135 4y = 9x 4y = 9x ⇔ ⇔ √  4xy = 15 16y = 135  135  y=± √ 15 ❱➟② ❤➺ ✤➣ ❝❤♦ ❝â ✺ ♥❣❤✐➺♠ ❧➔ (x, y) = (0, 0), (2, 1), (−2, −1), ( √4 , 135 ), 2 135    2y y = 3x ⇔   x y = 10y ✺✵ √ 15 135 (− √ ,− )✳ 2 135 ❇➔✐ ✸✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ x3 − y = 35 2x2 + 3y = 4x − 9y ✭✣➲ t❤✐ ❍❙● t➾♥❤ ❨➯♥ ❇→✐✮ ▲í✐ ❣✐↔✐✳ P❤÷ì♥❣ tr➻♥❤ t❤ù ❤❛✐ tữỡ ữỡ ợ (6x2 12x + 8) + (9y + 12y + 27) = 35 ❚❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ t❤ù ♥❤➜t ❝õ❛ ❤➺✱ t❛ ✤÷đ❝✿ x3 − y = (6x2 − 12x + 8) + (9y + 12y + 27) ⇔ (x − 2)3 = (y + 3)3 ⇔x=y+5 ❚❤❛② ❧↕✐ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ t❤ù ❤❛✐ ❝õ❛ ❤➺✱ t❛ ✤÷đ❝✿ 2(y + 5)2 + 3y = 4(y + 5) − 9y ⇔ 5y + 25y + 30 = y = −2 ⇔ y = −3 ✲ ❱ỵ✐ y = −2 t❤✐ x = 3✳ ✲ ❱ỵ✐ y = −3 t❤➻ x = 2✳ ❚❤û ❧↕✐ t❛ t❤➜② ❝↔ ❤❛✐ ♥❣❤✐➺♠ ✤➲✉ t❤ä❛ ♠➣♥✳ ❱➟② ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❝â ✷ ♥❣❤✐➺♠ ❧➔ (x, y) = (3, −2), (2, −3)✳ ❇➔✐ ✹✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ▲í✐ ❣✐↔✐✳ ✣✐➲✉ ❦✐➺♥✿ x, y = 0✳ ▲➜② P❚✭✶✮ ✰ P❚✭✷✮✱ t❛ ✤÷đ❝  1  2   x + 2y = 2(x + y ) 1    − = y − x2 x 2y ✭✣➲ t❤✐ ❝❤å♥ ✤ë✐ t✉②➸♥ t➾♥❤ ◗✉↔♥❣ ◆✐♥❤✮ = x2 + 3y ⇔ = x3 + 3xy x ✺✶ ▲➜② P❚✭✶✮ ✲ P❚✭✷✮✱ t❛ ✤÷đ❝✿ = 3x2 + y ⇔ = 3x2 y + y y tữỡ ữỡ ợ = x3 + 3xy = y + 3x2 y ⇔ = x3 + 3xy + 3x2 y + y = x3 + 3xy − y − 3x2 y = (x + y)3 = (x − y)3 √ x+y = 33 ⇔ x−y =1 √  3+1   x= ⇔ √    y = − ⇔ ❚❤û ❧↕✐✱ t❛ t❤➜② ♥❣❤✐➺♠ ♥➔② t❤ä❛ ♠➣♥✳ √ √ 3+1 3−1 ❱➟② ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❧➔ (x, y) = ( , ) ❇➔✐ ✺✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ▲í✐ ❣✐↔✐✳ ✣✐➲✉ ❦✐➺♥✿ x, y = 0✳  1 2 2    x + 2y = (3x + y )(x + 3y )  1   − = 2(y − x4 ) x 2y ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ tữỡ ữỡ ợ 4 2  x = x + 5y + 10x y    = 5x4 + y + 10x2 y y t P ỗ ▼✐♥❤✮ ✺✷ ⇔ = x5 + 10x3 y + 5xy = 5x4 y + 10x2 y + y ⇔ = x5 + 5x4 y + 10x3 y + 10x2 y + 5xy + y = x5 − 5x4 y + 10x3 y − 10x2 y + 5xy − y = (x + y)5 ⇔ = (x − y)5 √ x+y = 53 ⇔ x−y =1 √  3+1   x= ⇔ √    y = − ❚❤û ❧↕✐✱ t❛ t❤➜② ♥❣❤✐➺♠ ♥➔② t❤ä❛ ♠➣♥✳ √ √ 3+1 3−1 ❱➟② ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❧➔ (x, y) = ( , ) 5 ✺✸ ❑➌❚ ▲❯❾◆ ✣➸ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✱ ♥❣♦➔✐ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ tæ✐ ❝â t❤❛♠ ❦❤↔♦ t❤➯♠ ❝→❝ ❜➔✐ t➟♣ ✈➔ ỗ tr trt ổ tỹ ❤✐➺♥ ✤÷đ❝ ❝→❝ ❝ỉ♥❣ ✈✐➺❝ s❛✉ ✤➙②✳ ✶✳ ❍➺ t❤è♥❣ ỵ tt ởt số ữỡ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✳ ✷✳ ●✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ♠✐♥❤ ❤å❛ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ tr♦♥❣ ❝→❝ ❦ý t❤✐ ❍❙● ❝➜♣ t➾♥❤✱ t❤➔♥❤ ♣❤è ✈➔ ❚❍P❚◗●✳ ✸✳ ❚r➻♥❤ ❜➔② ♠ët sè ❜➔✐ t♦→♥ ❤❛② ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✳ ▼➦❝ ❞ị ✤➣ ❝è ❣➢♥❣ ❤➳t sù❝ ♥❤÷♥❣ ❞♦ t❤í✐ ❣✐❛♥ ✈➔ ❦❤↔ ♥➠♥❣ ❝â ❤↕♥ ♥➯♥ ❧✉➟♥ ✈➠♥ ❝ô♥❣ ❝â ♥❤✐➲✉ t❤✐➳✉ sõt t ữủ sỹ õ õ qỵ t ❝ỉ✱ ❝→❝ ❜↕♥ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ ❝❤➾♥❤ ❤ì♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ✺✹ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ✣♦➔♥ ◗✉ý♥❤ ✭❚ê♥❣ ❈❤õ ❜✐➯♥✮ ✭✷✵✶✸✮✱ ❙→❝❤ ❣✐→♦ ❦❤♦❛ ✣↕✐ sè ✶✵ ♥➙♥❣ ❝❛♦✱ ◆❳❇ ●✐→♦ ❞ö❝ ❱✐➺t ◆❛♠✳ ❬✷❪ ✣♦➔♥ ◗✉ý♥❤ ✭❚ê♥❣ ❈❤õ ❜✐➯♥✮ ✭✷✵✶✸✮✱ ❙→❝❤ ❣✐→♦ ❦❤♦❛ ✣↕✐ sè ✈➔ ●✐↔✐ t➼❝❤ ✶✶ ♥➙♥❣ ❝❛♦✱ ◆❳❇ ●✐→♦ ❞ö❝ ❱✐➺t ◆❛♠✳ ❬✸❪ ✣♦➔♥ ◗✉ý♥❤ ✭❚ê♥❣ ❈❤õ ❜✐➯♥✮ ✭✷✵✶✸✮✱ ❙→❝❤ ❣✐→♦ ❦❤♦❛ ●✐↔✐ t➼❝❤ ✶✷ ♥➙♥❣ ❝❛♦✱ ◆❳❇ ●✐→♦ ❞ö❝ ❱✐➺t ◆❛♠✳ ❬✹❪ ◆❣✉②➵♥ ❱➠♥ ▼➟✉ ✭❈❤õ ❜✐➯♥✮ ✭✷✵✶✵✮✱ ❑✛ ②➳✉ tr↕✐ ❤➧ ❍ị♥❣ ❱÷ì♥❣✱ ❱♥▼❛t❤✳❝♦♠ ❬✺❪ ▲➯ P❤ó❝ ▲ú ✭✷✵✶✶✮✱ ❚✉②➸♥ ❝❤å♥ ❝→❝ ❜➔✐ t♦→♥ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ tr♦♥❣ ✤➲ t❤✐ ❍❙● ❝→❝ t➾♥❤✱ t❤➔♥❤ ♣❤è ♥➠♠ ✷✵✶✵✲✷✵✶✶✳ ❬✻❪ ◆❣✉②➵♥ ❚r✉♥❣ ◆❣❤➽❛✱ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ✈æ t✛✳ ... x1 < x2 ⇒ f (x1 ) > f (x2 ) số ỗ ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ D ✤÷đ❝ ❣å✐ ❝❤✉♥❣ ❧➔ ❤➔♠ sè ỡ tr D ỵ số y = f (x) ❝â ✤↕♦ ❤➔♠ tr➯♥ D✳ ✭❛✮ ◆➳✉ f (x) > ✈ỵ✐ ♠å✐ x t❤✉ë❝ D t số f (x) ỗ tr D ◆➳✉ f (x)