Bài toán cực trị của hàm số

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ❚➔♦ ❚❤à ⑩♥❤ ❚➙♠ ❇⑨■ ❚❖⑩◆ ❈Ü❈ ❚❘➚ ❈Õ❆ ❍⑨▼ ❙➮ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ ❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✼ ❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ❚➔♦ ❚❤à ⑩♥❤ ❚➙♠ ❇⑨■ ❚❖⑩◆ ❈Ü❈ ❚❘➚ ❈Õ❆ ❍⑨▼ ❙➮ ❈❤✉②➯♥ ♥❣➔♥❤✿ ✣↕✐ sè ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ ◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈ ❚❙✳ ◆●❯❨➍◆ ❚❍➚ ❑■➋❯ ◆●❆ ❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✼ ▲í✐ ❝↔♠ ì♥ ❚r♦♥❣ t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥✱ ❡♠ ✤➣ ♥❤➟♥ ✤÷đ❝ sü ❣✐ó♣ ✤ï ♥❤✐➺t t➻♥❤ ❝õ❛ ❝→❝ t❤➛② ❝æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥ tr♦♥❣ ❦❤♦❛✳ ◗✉❛ ✤➙②✱ ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ❝→❝ t❤➛②✱ ❝ỉ tr♦♥❣ tê ✣↕✐ sè ✈➔ ✤➦❝ ❜✐➺t ❧➔ ❝æ ❣✐→♦ ❚❙✳ ữớ ữợ ✤➲ t➔✐ ✈➔ t➟♥ t➻♥❤ ❝❤➾ ❜↔♦✱ ❣✐ó♣ ✤ï ❡♠ ❤♦➔♥ t❤✐➺♥ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ♥➔②✳ ❉♦ t❤í✐ ❣✐❛♥ ✈➔ ❦✐➳♥ t❤ù❝ ❝â ❤↕♥✱ ❦❤â❛ ❧✉➟♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ❝â ♥❤ú♥❣ ❤↕♥ ❝❤➳ ✈➔ t❤✐➳✉ sât ♥❤➜t ✤à♥❤✳ ❊♠ ữủ sỹ õ õ ỵ qỵ t ổ s õ ❧✉➟♥ ❝õ❛ ❡♠ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✼ ❙✐♥❤ ✈✐➯♥ ❚➔♦ ❚❤à ⑩♥❤ ❚➙♠ ▲í✐ ❝❛♠ ✤♦❛♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❝❤②➯♥ ♥❣➔♥❤ ✣↕✐ sè ✈ỵ✐ ✤➲ t➔✐ ✧❇➔✐ t♦→♥ ❝ü❝ trà ❝õ❛ ❤➔♠ sè✧ ❧➔ ❦➳t q ự r tổ ữợ sỹ ữợ t➟♥ t➻♥❤ ❝õ❛ ❝æ ❚❙✳ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ✈➔ ❦❤ỉ♥❣ ❝â sü trị♥❣ ❧➦♣ ✈ỵ✐ ❜➜t ❦➻ ❝ỉ♥❣ tr➻♥❤ ❦❤♦❛ ❤å❝ ♥➔♦ ❦❤→❝✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✼ ❙✐♥❤ ✈✐➯♥ ❚➔♦ ❚❤à ⑩♥❤ ❚➙♠ ▼ư❝ ❧ư❝ ▲í✐ ♠ð ✤➛✉ ✶ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✸ ✶✳✶ ❚➟♣ ỗ ỗ t t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✷ ❚➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❍➔♠ sè ✤ì♥ ✤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✷ ❚➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ❈ü❝ trà ❝õ❛ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✶ ✣à♥❤ ♥❣❤➽❛ ❝ü❝ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✷ ❚➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ●✐→ trà ❧ỵ♥ ♥❤➜t ✭●❚▲◆✮✱ ❣✐→ trà ♥❤ä ♥❤➜t ✭●❚◆◆✮ ❝õ❛ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹✳✶ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹✳✷ ❚➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ❝ü❝ trà ❝õ❛ ❤➔♠ sè ✸ ✸ ✹ ✹ ✹ ✹ ✹ ✹ ✻ ✼ ✼ ✼ ✶✷ ✷✳✶ P❤÷ì♥❣ ♣❤→♣ ✤↕♦ ❤➔♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✶✳✶ P❤÷ì♥❣ ♣❤→♣ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✐ ❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✷✳✷ ✷✳✸ ✷✳✹ ✷✳✺ ✷✳✶✳✷ P❤÷ì♥❣ ♣❤→♣ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❙û ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✶ ❙û ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✷ ❙û ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❇✉♥❤✐❛❝♦♣①❦✐ ✳ ✳ ✳ ✳ ✷✳✷✳✸ ❙û ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ trà t✉②➺t ✤è✐ ❝ì ❜↔♥ ✳ P❤÷ì♥❣ ♣❤→♣ ♠✐➲♥ ❣✐→ trà ❝õ❛ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Pữỡ sỷ ỗ ó ✳ ✳ ✳ ✳ P❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë✱ ✈❡❝tì ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ▼ët sè s❛✐ ❧➛♠ ❦❤✐ ❣✐↔✐ ❜➔✐ t♦→♥ ❝ü❝ trà ❝õ❛ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶ ❙û ❞ö♥❣ s❛✐ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✳✷ ▼ët sè ❞↕♥❣ s❛✐ ❧➛♠ t❤÷í♥❣ ❣➦♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷ ❙❛✐ ❧➛♠ ❞♦ ❦❤ỉ♥❣ ♣❤➙♥ ❜✐➺t ✤÷đ❝ ✤✐➲✉ ❦✐➺♥ ❝➛♥✱ ✤✐➲✉ ❦✐➺♥ ✤õ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷✳✷ ❈→❝ ✈➼ ❞ö ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✶ ✷✶ ✷✼ ✸✵ ✸✸ ✸✻ ✹✵ ✹✹ ✹✹ ✹✹ ✹✻ ✻✷ ✻✷ ✻✸ ❑➳t ❧✉➟♥ ✻✽ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✻✾ ✐✐ ❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲í✐ ♠ð ✤➛✉ ❉↕♥❣ t♦→♥ ✧❚➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t✱ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❤➔♠ sè tr➯♥ ♠ët t➟♣ ①→❝ ✤à♥❤ ♥➔♦ ✤â✧ ❧➔ ❞↕♥❣ t♦→♥ ❤❛② ✈➔ ❦❤â tr♦♥❣ t♦→♥ ❝➜♣✳ ❉↕♥❣ t♦→♥ ♥➔② ❝á♥ ✤÷đ❝ ❣å✐ ❧➔ ❞↕♥❣ t♦→♥ t➻♠ ❝ü❝ trà ❝õ❛ ❤➔♠ sè✳ ◆â t❤÷í♥❣ ①✉➜t ❤✐➺♥ tr♦♥❣ ❝→❝ ❦➻ t❤✐ t✉②➸♥ ❝❤å♥ ❤å❝ s✐♥❤ ❣✐ä✐ ✈➔ ❝→❝ ❦➻ t❤✐ tèt ♥❣❤✐➺♣ tr✉♥❣ ❤å❝ ♣❤ê t❤æ♥❣✳ ❱✐➺❝ ❣✐↔✐ ❜➔✐ t♦→♥ ♥➔② ✤á✐ ❤ä✐ ♥❣÷í✐ ❧➔♠ ♣❤↔✐ ✈➟♥ ❞ư♥❣ ❦✐➳♥ t❤ù❝ ❤đ♣ ❧➼✱ ♥❤✐➲✉ ❦❤✐ ❦❤→ ✤ë❝ ✤→♦ ✈➔ ❜➜t ♥❣í✳ ✣✐➲✉ ✤â ❝â t→❝ ❞ư♥❣ r➧♥ ❧✉②➺♥ t÷ ❞✉② t♦→♥ ❤å❝ ♠➲♠ ❞➫♦✱ ❧✐♥❤ ❤♦↕t ✈➔ s→♥❣ t↕♦✳ ✣è✐ ✈ỵ✐ ❜➔✐ t♦→♥ t➻♠ ❝ü❝ trà ❝õ❛ ❤➔♠ sè ❦❤æ♥❣ ❝â ❝→❝❤ ❣✐↔✐ ♠➝✉ ♠ü❝ ♠➔ ♠é✐ ❜➔✐ t♦→♥ ❧↕✐ ❝â ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❦❤→❝ ♥❤❛✉ ✈➔ ♠ët sè ❜➔✐ t♦→♥ ❝â t❤➸ ❝â ♥❤✐➲✉ ❝→❝❤ ❣✐↔✐✳ ❱➻ ❝→❝ ❧➼ ❞♦ tr➯♥ ✈➔ ✈ỵ✐ ♥✐➲♠ ②➯✉ t❤➼❝❤ t♦→♥ ❤å❝ tỉ✐ ✤➣ ❝❤å♥ ✤➲ t➔✐ ✧❇➔✐ t♦→♥ ❝ü❝ trà ❝õ❛ ❤➔♠ sè✧ ✤➸ ❧➔♠ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❝õ❛ ♠➻♥❤✳ ◆ë✐ ❞✉♥❣ ❦❤â❛ ❧✉➟♥ ❝❤✐❛ ❧➔♠ ❜❛ ❝❤÷ì♥❣ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ❤➔♠ sè✳ ❈❤÷ì♥❣ ✷✳ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ❝ü❝ trà ❝õ❛ ❤➔♠ sè✳ ◆➯✉ r❛ ♠ët ✈➔✐ ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ❝ü❝ trà ❝õ❛ ❤➔♠ sè✳ ❈❤÷ì♥❣ ✸✳ ▼ët sè s❛✐ ❧➛♠ ❦❤✐ ❣✐↔✐ ❜➔✐ t♦→♥ ❝ü❝ trà ❝õ❛ ❤➔♠ sè✳ ❈❤➾ r❛ ♠ët sè s❛✐ ❧➛♠ ❦❤✐ ❣✐↔✐ ❜➔✐ t♦→♥ ❝ü❝ trà ❝õ❛ ❤➔♠ sè ✈➔ ❝→❝❤ ❦❤➢❝ ♣❤ö❝✳ ❉♦ t❤í✐ ❣✐❛♥ ❝â ❤↕♥ ✈➔ ♥➠♥❣ ❧ü❝ ❜↔♥ t❤➙♥ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤â❛ ❧✉➟♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ s❛✐ ①ât✳ ✶ ❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚ỉ✐ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü ✤â♥❣ ❣â♣ þ ❦✐➳♥ ❝õ❛ ❝→❝ t❤➛② ❝æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✼ ❙✐♥❤ ✈✐➯♥ ❚➔♦ ❚❤à ⑩♥❤ ❚➙♠ ữỡ tự ỗ ỗ t t D ữủ t ỗ ợ x, y D✱ ♠å✐ λ ∈ [0, 1] t❛ ❝â λx + (1 − λ)y ∈ D ❍➔♠ ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ỗ tr x, y D λ ∈ [0, 1] t❛ ❝â f (x) D ♥➳✉ ✈ỵ✐ ♠å✐ f [λx + (1 − λ)y] ≤ λf (x) + (1 − λ)f (y) ❍➔♠ f (x) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ❧ã♠ tr➯♥ t➟♣ ①→❝ ✤à♥❤ D ♥➳✉ f (x) ỗ tự D t ỗ x, y D [0, 1] t❛ ❝â f [λx + (1 − λ)y] ≥ λf (x) + (1 − λ)f (y) ✸ ❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✶✳✶✳✷ ❚➼♥❤ ❝❤➜t ❈❤♦ y = f (x) ❧✐➯♥ tö❝ ❝â ✤↕♦ ❤➔♠ tỵ✐ ❝➜♣ ❤❛✐ tr➯♥ [a, b] ✈➔ f ♠å✐ x ∈ (a, b)✳ ❑❤✐ ✤â✱ y = f (x) ỗ tr [a, b] y = f (x) ❧✐➯♥ tư❝ ❝â ✤↕♦ ❤➔♠ tỵ✐ ❝➜♣ ❤❛✐ tr➯♥ [a, b] ✈➔ f ♠å✐ x ∈ (a, b)✳ ❑❤✐ ✤â✱ y = f (x) ❧➔ ❤➔♠ ❧ã♠ tr➯♥ [a, b]✳ (x) > 0✱ (x) < 0✱ ✶✳✷ ❍➔♠ sè ✤ì♥ ✤✐➺✉ ✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛ ●✐↔ sû ❤➔♠ sè y = f (x) ①→❝ ✤à♥❤ tr➯♥ K ✳ ❑❤✐ ✤â✿ ◆➳✉ ✈ỵ✐ ♠å✐ x1, x2 ∈ K, x1 > x2✱ t❛ ❝â f (x1) > f (x2) t❤➻ ❤➔♠ sè y = f (x) ❣å✐ ❧➔ ✤ì♥ ✤✐➺✉ t➠♥❣ ỗ tr K ợ 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✱ ✈ỵ✐ ♠å✐ x ∈ K t❤➻ f (x) ỗ tr K f (x) < 0✱ ✈ỵ✐ ♠å✐ x ∈ K t❤➻ f (x) ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ K ✳ ✶✳✸ ❈ü❝ trà ❝õ❛ ❤➔♠ sè ✶✳✸✳✶ ✣à♥❤ ♥❣❤➽❛ ❝ü❝ trà ❛✳ ✣à♥❤ ♥❣❤➽❛ ✶ ✹ ❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✤÷đ❝ f (x, y, z) = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = y = z✳ ❇➔✐ ✷✳ ❈❤♦ x, y, z ❧➔ ❝→❝ sè t❤ü❝ ❧ỵ♥ ❤ì♥ −1✳ ❚➻♠ ●❚◆◆ ❝õ❛ P = + x2 + y2 + z2 + + + y + z + z + x2 + x + y ▲í✐ ❣✐↔✐ s❛✐ ❧➛♠✳ ◆➳✉ x < 0✱ t❛ t❤❛② x ❜ð✐ (−x) t❤➻ ❤❛✐ ❤↕♥❣ tû ✤➛✉ ❝õ❛ P ❦❤æ♥❣ ✤ê✐ ❝á♥ ❤↕♥❣ tû t❤ù ❜❛ ❣✐↔♠✳ ❚ø ✤â ❦❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t ❣✐↔ sû x ≥ y ≥ z ≥ 0✳ ❚ø (x − 1)2 ≥ s✉② r❛ x2 + ≥ 2x✱ s✉② r❛ 3(x2 + 1) ≥ 2(x2 + x + 1)✳ ✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = 1✳ ❉♦ ✤â + x2 + x2 ≥ ≥ + y + z2 + x + x2 + y2 + z2 ❚÷ì♥❣ tü t❛ ❝ơ♥❣ ❝â + z + x2 ≥ ; + x + y2 ≥ 32 ✳ ❚ø ✤â s✉② r❛ P ≥ 2✳ ❉➜✉ ✧❂✧ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = y = z = 1✳ P❤➙♥ t➼❝❤ s❛✐ ❧➛♠✳ ❈→❝ ❜✐➳♥ x, y, z tr♦♥❣ P ❝â ❞↕♥❣ ❤♦→♥ ✈à ✈á♥❣ q✉❛♥❤ ♠➔ ❦❤ỉ♥❣ ❝â ✈❛✐ trá ♥❤÷ ♥❤❛✉ ữủ t ợ ♥❤➜t ✭❤♦➦❝ ♥❤ä ♥❤➜t✮ ♠➔ t❤æ✐✳ ❉♦ ✤â ✤♦↕♥ ❧➟♣ ❧✉➟♥✿ ❑❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t ❣✐↔ sû x ≥ y ≥ z ≥ 0✳ ❚ø (x − 1)2 ≥ s✉② r❛ 3(x2 + 1) ≥ 2(x2 + x + 1)✳ ✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = 1✳ ❉♦ ✤â + x2 + x2 ≥ ≥ + y + z2 + x + x2 ✺✺ ✭✸✳✶✵✮ ❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚÷ì♥❣ tü t❛ ❝ô♥❣ ❝â ✈➔ + y2 ≥ + z + x2 ✭✸✳✶✶✮ + z2 ≥ + x + y2 ✭✸✳✶✷✮ ❧➔ ❦❤ỉ♥❣ ✤ó♥❣✳ ❑❤æ♥❣ t❤➸ tø ✭✸✳✶✵✮ s✉② r❛ ✭✸✳✶✶✮ ✈➔ ✭✸✳✶✷✮ ❜➡♥❣ ♣❤➨♣ t÷ì♥❣ tü ✈➻ ✈❛✐ trá ❝õ❛ ❝→❝ ❜✐➳♥ x, y, z tr♦♥❣ P ❦❤ỉ♥❣ ♥❤÷ ♥❤❛✉✳ ▲í✐ ❣✐↔✐ ✤ó♥❣✳ ❈â + y2 ≥ 2y ✈ỵ✐ ♠å✐ y ♥➯♥ 2(1 + x2 ) 2(1 + x2 ) + x2 = ≥ + y + z2 + 2y + 2z 2(1 + z ) + (1 + y ) ❚÷ì♥❣ tü + y2 2(1 + y ) + z2 2(1 + z ) ≥ ; ≥ + z + x2 2(1 + x2 ) + (1 + z ) + x + y 2(1 + y ) + (1 + x2 ) ❙✉② r❛ + y2 + z2 + x2 + + + y + z + z + x2 + x + y 2(1 + x2 ) 2(1 + y ) 2(1 + z ) ≥ + + = M 2(1 + z ) + (1 + y ) 2(1 + x2 ) + (1 + z ) 2(1 + y ) + (1 + x2 ) P = ✣➦t + x2 = a; + y2 = b; + z2 = c, (a, b, c > 0)✳ ▲ó❝ ✤â M = 2c2a+ b + 2a2b+ c + 2b2c+ a ✳ ✣➦t N = 2c c+ b + 2a a+ c + 2b b+ a ✈➔ H = 2c b+ b + 2a c+ c + 2b a+ a ✳ ❑❤✐ ✤â 2N + H = 3✳ + c 2b + a 2c + b ⑩♣ ❞ư♥❣ ❇✣❚ ❈ỉ✲s✐✱ t❛ ❝â M + N = 2a + + ≥3 2c + b 2a + c 2b + a ✺✻ ❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ s✉② r❛ ✭✸✳✶✸✮ 2M + 2N ≥ ▲↕✐ ❝â 2H + M2 = 2b + a 2c + b 2a + c + + ≥ 3✱ 2c + b 2a + c 2b + a H+ s✉② r❛ ✭✸✳✶✹✮ M ≥ ❈ë♥❣ ✈➳ ✈ỵ✐ ✈➳ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✭✸✳✶✸✮ ✈➔ ✭✸✳✶✹✮ t❛ ❝â 9M 15 + (2N + H) ≥ ▼➔ 2N + H = ♥➯♥ M ≥ 2✳ ❚ø ✤â s✉② r❛ P ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = y = z = 1✳ ≥ 2✳ ❉➜✉ ✧❂✧ ①↔② r❛ ❉✳ ▼ët sè ❞↕♥❣ s❛✐ ❧➛♠ ❦❤→❝ t❤÷í♥❣ ♠➢❝ ♣❤↔✐ ❇➔✐ ✶✳ ❈❤♦ ❤❛✐ sè x, y t❤ä❛ ♠➣♥ x > y ✈➔ xy = 1✳ ❚➻♠ ●❚◆◆ ❝õ❛ ❤➔♠ sè x2 + y f (x, y) = x−y ▲í✐ ❣✐↔✐ s❛✐✳ x2 + y x2 − 2xy + y + 2xy (x − y)2 + 2xy ❚❛ ❝â f (x, y) = x − y = = ✳ x−y x−y (x − y)2 ❉♦ x > y ✈➔ xy = ♥➯♥ f (x, y) = x − y + x2xy =x−y+ ✳ −y x−y ❇✐➳t r➡♥❣ ♥➳✉ a > t❤➻ a + a1 ≥ ✭❇✣❚ ❈æ✲s✐✮ ❉♦ ✤â f (x, y) = x −2 y + x −2 y + x −2 y ≥ + x −2 y ✳ ❱➟② f (x, y) ❝â ❣✐→ trà ♥❤ä ♥❤➜t ❦❤✐ x −2 y + x −2 y = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ [(x − y) − 2]2 = ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤  ♥➔② t❛ ✤÷đ❝ ♥❣❤✐➺♠ x − y = 2✳  x − y = ❉♦ ✤â t❛ ❝â ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✉  xy = ♥❣❤✐➺♠ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ❧➔ (x, y) = (1 + ✺✼ √ 2; −1 + √ 2) ✈➔ ❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ √ 2; −1 − √ √ 2)✳ ọ r ữ ợ x = +2 , y = −2 t❤➻ ❝â x > y✱ xy = −4 = √ ✈➔ f (x, y) = 2 < 3✳ (x, y) = (1 − √ P❤➙♥ t➼❝❤ s❛✐ ❧➛♠✳ ❈❤ù♥❣ ♠✐♥❤ f ≥ m ✭❤❛② f ≤ m✮✱ ❦❤➥♥❣ ✤à♥❤ ●❚◆◆ ✭❤❛② ●❚▲◆✮ ❝õ❛ f ❜➡♥❣ m ♠➔ ❦❤æ♥❣ ❝❤➾ r❛ m ❧➔ ❤➡♥❣ sè✳ ❘ã r➔♥❣ ❧í✐ ❣✐↔✐ s❛✐✿ ❱➻ x−y x−y f (x, y) ≥ + ♠➔ ❝❤÷❛ ❧➔ ❤➡♥❣ sè✳ ❙❛✐ ❧➛♠ s 2 ữợ ✤→♥❤ ❣✐→ f ≥ m ♥❤÷♥❣ m ❝❤÷❛ ❧➔ ❤➡♥❣ sè✳ ▲í✐ ❣✐↔✐ ✤ó♥❣✳ (x − y)2 + 2xy x2 + y = = (x − y) + f (x, y) = x−y x−y x−y √ ≥ (x − y) = 2 x−y ✭⑩♣ ❞ö♥❣ ❇✣❚ ❈ỉ✲s✐ ❝❤♦ ❤❛✐ sè ❞÷ì♥❣ x − y ✈➔ x −2 y ✮✳    x − y = ❉➜✉ ✧❂✧ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐  x−y ✳  xy = √ √ √ √ 6+ 6− x= ;y = 2 ●✐↔✐ ❤➺ ♥➔② t❛ t➻♠ ✤÷đ❝ ❜➔✐✳ ❇➔✐ ✷✳ ❚➻♠ ●❚◆◆ ❝õ❛ ❤➔♠ sè g(x) = ▲í✐ ❣✐↔✐ s❛✐ ❧➛♠✳ ✺✽ t❤ä❛ ♠➣♥ ✤➲ x2 − x + + x2 − x − ❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚❛ ❝â g(x) = x2 − x + + x2 − x − 1 11 = x − x + ( ) + 2 11 = (x − ) + + (x − 1 + x − x + ( ) − 2 ) − −9 11 11 + = + = 5✳ 4 4 ❉➜✉ ✧❂✧ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ (x − ) = ❱➟② g(x) = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = 12 ✳ ❙✉② r❛ g(x) ≥ P❤➙♥ t➼❝❤ s❛✐ ❧➛♠✳ ❍✐➸✉ s❛✐ ♥❤✐➲✉ ❧♦↕✐ t tự ữ tr õ ữợ ❣✐↔✐ s❛✐ ❧➛♠ ❤❛② x = 12 ✳ A2 + m ≥ |m|✳ 11 11 −9 (x − ) + + (x − ) − ≥ + = 4 4 9 (x − ) − ≥ − 4 −9 (x − ) − ≥ ✳ 4 ❈❤➥♥❣ ❤↕♥ ♥➳✉ x = t t ợ x ữ ổ t❤➸ s✉② r❛ 9 −9 −9 (x − ) − = (0 − ) − = − = |−2| < 4 4 ◆❤➟♥ ①➨t ✸✳✸✳ ❚ø a ≥ b ❝❤➾ s✉② r❛ ✤÷đ❝ |a| ≥ |b| ❦❤✐ a ≥ b ≥ 0✳ ▲í✐ ❣✐↔✐ ✤ó♥❣✳ ✺✾ ❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ 11 11 g(x) = (x − ) + + (x − ) − = (x − ) + + − (x − ) 4 4 2 11 11 ≥ (x − ) + + − (x − ) = + = 4 4 ❉♦ ✤â g(x) = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ [(x − 12 ) + 11 ].[ − (x − ) ] ≥ 4 ✵ ❤❛② −1 ≤ x ≤ ✭✈➻ (x − 21 ) + 114 ≥ 0✱ ✈ỵ✐ ♠å✐ x✮✳ ❇➔✐ ✸✳ ❚➼♥❤ ●❚◆◆ ❝õ❛ P = (x2 − 1)(x2 + 1)✳ ▲í✐ ❣✐↔✐ s❛✐✳ ❚❛ ❝â x2 ≥ 0✱ ✈ỵ✐ ♠å✐ x s✉② r❛ x2 − ≥ −1 ✈➔ x2 + ≥ 1✳ ❙✉② r❛ P = (x2 − 1)(x2 + 1) ≥  (−1).1 = −1 ⇒ P ≥ −1✳  x2 − = −1 ❉➜✉ ✧❂✧ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐  x2 + = ✳ ❱➟② P = −1 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = 0✳ P❤➙♥ t➼❝❤ s❛✐ ❧➛♠✳ ❱➟♥ ❞ö♥❣ s❛✐ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❇✣❚ ♥❤÷ ♥❤➙♥ ❤❛✐ ❇✣❚ ❝ị♥❣ ❝❤✐➲✉ ♠➔ ❦❤ỉ♥❣ ❝â ✤✐➲✉ ❦✐➺♥ ❤❛✐ ✈➳ ❝ị♥❣ ❦❤ỉ♥❣ ➙♠✳ ❈❤é s❛✐ ❝õ❛ ❧í✐ ❣✐↔✐ tr➯♥ ❧➔ ✤➣ ♥❤➙♥ ❤❛✐ ✈➳ ❝õ❛ ❇✣❚ ❝ò♥❣ ❝❤✐➲✉ tr♦♥❣ ❦❤✐ ❝â ♥❤ú♥❣ ✈➳ ♥❤➟♥ ❣✐→ trà ➙♠✱ ❝❤➥♥❣ ❤↕♥ > ✈➔ −2 > −3 ♥❤÷♥❣ 5.(−2) < 3.(−3)✳ ▲í✐ ❣✐↔✐ ✤ó♥❣✳ ▲í✐ ❣✐↔✐ ✤ó♥❣ ❦❤→ ✤ì♥ ❣✐↔♥✿ P = (x2 − 1)(x2 + 1) = x4 − ≥ −1 s✉② r❛ P ≥ −1✳ ❉➜✉ ✧❂✧ ①↔② r❛ x4 = ❤❛② x = 0✳ ❱➟② P = −1 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = 0✳ ❇➔✐ ✹✳ ❚➻♠ ●❚◆◆ ❝õ❛ ❤➔♠ sè f (x) = √x2 − x + 1+ x2 − √3x + ✻✵ ❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ợ x R s ữ số tr➯♥ ✈➲ ❞↕♥❣ f (x) = √ 2 (x − ) + ( ) + 2 √ (x − ) +( ) 2 √ √ ❚r♦♥❣ ❤➺ trö❝ tå❛ ✤ë ❖①② ①➨t ❝→❝ ✤✐➸♠ A( , ); B( 23 , 12 ); C(x, 0)✳ ❑❤✐ ✤â f (x) = CA + CB ✳ ❱➻ CA + CB ≥ AB ✱ tr♦♥❣ ✤â √ √ √ 3 2( − 1) − ) +( − ) = ( 2 2 √ AB = ❙✉② r❛ √ √ 2( − 1) f (x) = ✳ P❤➙♥ t➼❝❤ s❛✐ ❧➛♠✳ ❙û ❞ư♥❣ ♠➦t ♣❤➥♥❣ tå❛ ✤ë ♥❤÷♥❣ ữ ũ ủ rữợ t t ợ ❧↕✐ ♠ët ❦➳t q✉↔ ✤ó♥❣ s❛✉✿ ❚r♦♥❣ ♠➦t ♣❤➥♥❣ ❝❤♦ ❤❛✐ ✤✐➸♠ A, B ✈➔ ✤÷í♥❣ t❤➥♥❣ (d) ✤✐ q✉❛ ✤✐➸♠ C ✳ ❑❤✐ ✤â✿ ❛✳ ◆➳✉ A, B ❝ò♥❣ ♣❤➼❛ s♦ ✈ỵ✐ (d) t❤➻ CA + CB ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t ✭●❚◆◆✮ ❦❤✐ C ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ AB ✈ỵ✐ (d) ✭tr♦♥❣ ✤â B ❧➔ ✤✐➸♠ ✤è✐ ①ù♥❣ ❝õ❛ B q✉❛ (d)✮✱ ❧ó❝ ✤â CA + CB = AB ✳ ❜✳ ◆➳✉ A, B ❦❤→❝ ♣❤➼❛ s♦ ✈ỵ✐ (d) t❤➻ CA + CB ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t ✭●❚◆◆✮ ❦❤✐ C ❧➔ ❣✐❛♦ ✤✐➸♠ ❝õ❛ AB √✈ỵ✐ (d)✱√❧ó❝ ✤â CA + CB = AB ✳ ❚r♦♥❣ ❧í✐ ❣✐↔✐ tr➯♥ ✤➣ ❝❤å♥ A( 21 , 23 ); B( 23 , 12 ) ❧➔ ❤❛✐ ✤✐➸♠ ❝ị♥❣ ♣❤➼❛ s♦ ✈ỵ✐ trư❝ ❤♦➔♥❤✳ ✣♦↕♥ AB ❦❤ỉ♥❣ ❝➢t trư❝ ❖①✱ ❞♦ ✤â ❞➜✉ ✧❂✧ ð ❜➜t ✤➥♥❣ t❤ù❝ CA + CB ≥ AB ổ r ổ tỗ t C tr ❖① s❛♦ ❝❤♦ C A + C B = AB ✱ ♥❣❤➽❛ ❧➔ CA + CB > AB ♥➯♥ ✻✶ ❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✈✐➺❝ ❦➳t ❧✉➟♥ √ √ 2( − 1) f (x) = ❑❤➢❝ ♣❤ö❝ s❛✐ ❧➛♠✳ ❧➔ s❛✐ ❧➛♠✳ √ √ ❳➨t ❤➺ trö❝ tå❛ ✤ë ❖①②✱ tr➯♥ ✤â ❝❤å♥A( , ); B( 23 , − 12 ); C(x, 0)✳ ❚❛ ❝â f (x) = CA + CB ≥ AB ✳ √ √ √ ✭tr♦♥❣ ✤â AB = ( − ) + ( − 23 ) = 2✳✮ √ ◆➯♥ f (x) ≥ 2✱ ✈ỵ✐ ♠å✐ x ∈ R✳ √ ✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ x = − 1✳ √ √ ❉♦ ✤â f (x) = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = − 1✳ ✸✳✷ ❙❛✐ ❧➛♠ ❞♦ ❦❤ỉ♥❣ ♣❤➙♥ ❜✐➺t ✤÷đ❝ ỵ tt ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✤➸ ❤➔♠ sè ✤↕t ❝ü❝ trà ◆➳✉ ❤➔♠ sè y = f (x) ❝â ✤↕♦ ❤➔♠ ✈➔ ✤↕t ❝ü❝ trà t↕✐ x0 t❤➻ f (x0) = 0✳ ❜✳ ✣✐➲✉ ❦✐➺♥ ✤õ ✤➸ ❤➔♠ sè ✤↕t ❝ü❝ trà ❈❤♦ ❤➔♠ sè y = f (x) ❝â ✤↕♦ ❤➔♠ ❝➜♣ ✶ tr➯♥ ♠ët ❦❤♦↔♥❣ ❝❤ù❛ x0✱ f (x0 ) = ✈➔ f ❝â ✤↕♦ ❤➔♠ ❝➜♣ ❤❛✐ t↕✐ x0 ✐✳ ◆➳✉ f (x0) > t❤➻ x = x0 ❧➔ ✤✐➸♠ ❝ü❝ t✐➸✉✳ ✐✐✳ ◆➳✉ f (x0) < t❤➻ x = x0 ❧➔ ✤✐➸♠ ❝ü❝ ✤↕✐ ✭❈á♥ ♥➳✉ f (x0) = t❤➻ t❛ ❝❤÷❛ ❦➳t ❧✉➟♥ ✤÷đ❝ ❣➻✮✳ ✻✷ ❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✸✳✷✳✷ ❈→❝ ✈➼ ❞ö ♠✐♥❤ ❤å❛ ❇➔✐ ✶✳ ❈❤♦ ❤➔♠ sè y = x x = 0✳ + mx✳ ❚➻♠ ♠ ✤➸ ❤➔♠ sè ✤↕t ❝ü❝ t✐➸✉ t↕✐ ▲í✐ ❣✐↔✐ s❛✐ ❧➛♠ ❦❤✐ t➻♠ ❝ü❝ trà✳ ❚❳✣✿ D = R y = 4x3 + m; y = 12x2   y (0) = ❍➔♠ sè ✤↕t ❝ü❝ t✐➸✉ t↕✐ x = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐  ✭✯✮ y (0) > ❑❤✐ ✤â t❛ t➼♥❤ ✤÷đ❝✿ ✶✳ y (0) = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ 4.0 + m = ❤❛② m = 0✳ ✷✳ y (0) = 0✳ ❚❛ ❝â y (0) = tr♦♥❣ ❦❤✐ ②➯✉ ❝➛✉ ❝õ❛ ❤➺ ❧➔ y (0) > 0✳ ❱➟② ❤➔♠ sè tr➯♥ ❦❤æ♥❣ ❝â ❝ü❝ t✐➸✉ t↕✐ x = 0✳ ◆❤➟♥ ①➨t ✸✳✹✳ ◆❤÷♥❣ t❤ü❝ ❝❤➜t ❦❤✐ m = t❤➻ ❤➔♠ sè tr➯♥ ✈➝♥ ❝â ❝ü❝ t✐➸✉ t↕✐ x = P❤➙♥ t➼❝❤ s❛✐ ❧➛♠✳ ❚r♦♥❣ ❧í✐ ❣✐↔✐ tr➯♥ ❜↕♥❤å❝ s✐♥❤ ✤â ✤➣ sû ❞ö♥❣ t➼♥❤ ❝❤➜t ♥➔②✿ x =  y (0) = ❧➔ ✤✐➸♠ ❝ü❝ t✐➸✉ s✉② r❛  ✈➔ ❝♦✐ ♥â ❧➔ ✤ó♥❣✳ ✭✣➙② ❝❤➼♥❤ y (0) > ❧➔ ❤➺ ✭✯✮ ♠➔ ❜↕♥ ✤â ❧➟♣ ð tr➯♥✮✳ ❚→❝ ❣✐↔ ✤➣ t❤ø❛  ♥❤➟♥ ♠➺♥❤ ✤➲✿ ✧❍➔♠ sè ❝â ✤✐➸♠ ❝ü❝ t✐➸✉ t↕✐ x = x0  y (x0 ) = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐  ✧ ❧➔ ✤ó♥❣✳ y (x0 ) > ✣✐➲✉ ❣➻ ❦❤✐➳♥ t→❝ ❣✐↔ ❞➵ ♠➢❝ s❛✐ ❧➛♠ tr➯♥❄ ◆❣✉②➯♥ ♥❤➙♥ ❧➔ ♥➡♠ ð ❤❛✐ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➣ ♥➯✉ tr♦♥❣ ỵ tt ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❦❤ỉ♥❣ sû ❞ư♥❣ ✧❦❤✐ ✈➔ ❝❤➾ ❦❤✐✧ ♠➔ ❝❤➾ sû ❞ö♥❣ ✧♥➳✉✳✳✳t❤➻✧ tr♦♥❣ ♠➺♥❤ ✤➲✳ ❚ù❝ ❧➔ ♠➺♥❤ ✤➲ tr➯♥ ❝❤➾ ✤ó♥❣ ✈ỵ✐ ❝❤✐➲✉ t❤✉➟♥✱ ❝á♥ ♥❣÷đ❝ ❧↕✐ tù❝ ❧➔ ✧♠➺♥❤ ✤➲ ✤↔♦✧ ❝õ❛ ♥â t❤➻ ❦❤ỉ♥❣ ❦❤➥♥❣ ✤à♥❤ ✤÷đ❝ ❧➔ ♥â ✤ó♥❣✳ ❚❤❡♦ ♥❤÷ ✈➟② t❤➻ ❜↕♥ ❤å❝ s✐♥❤ tr➯♥ ✤➣ ❤✐➸♥ ♥❤✐➯♥ ❝♦✐ ♠➺♥❤ ✤➲ ✤↔♦ ❝õ❛ ♠➺♥❤ ✤➲ tr➯♥ ❧➔ ✤ó♥❣ ✈➔ →♣ ❞ư♥❣ ❜➻♥❤ t❤÷í♥❣✳ ❈❤➼♥❤ ✈➻ ✈➟② ♠➔ ✤➣ ❞➝♥ ✤➳♥ s❛✐ ❧➛♠ ✤→♥❣ t✐➳❝ ❦❤✐ t➻♠ ✤✐➲✉ ❦✐➺♥ ❝õ❛ t❤❛♠ sè m tr♦♥❣ ❜➔✐ t♦→♥ ❝ü❝ trà✳ ❱➟② ❝❤ó♥❣ t❛ ❝â t❤➸ ✤÷❛ r❛ ♠ët ❦❤➥♥❣ ✤à♥❤ ♥❤÷ s❛✉✿ ❑❤ỉ♥❣ ✤÷đ❝ sû ❞ư♥❣ ♠➺♥❤ ✤➲ s❛✉ ✭♠➺♥❤ ✤➲ ✤↔♦✮ ❝õ❛ ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ ❤➔♠ sè ✤↕t ❝ü❝ trà ✈➔♦ t➻♠ ❝ü❝ trà✿   y (x0 ) = ◆➳✉ x0 ❧➔ ✤✐➸♠ ❝ü❝ ✤↕✐ t❤➻  y (x0 ) <   y (x0 ) = ◆➳✉ x0 ❧➔ ✤✐➸♠ ❝ü❝ t✐➸✉ t❤➻  ▲í✐ ❣✐↔✐ ✤ó♥❣✳ ✳ ✳ y (x0 ) > ❚❛ ❝â ❤❛✐ ❝→❝❤ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ❞↕♥❣ ♥❤÷ ♥➔②✿ ❈→❝❤ ✶✳ ❙û ❞ư♥❣ ❜↔♥❣ ❜✐➳♥ t❤✐➯♥✳ ❈→❝❤ ✷✳ ❚➻♠ t❤❛♠ sè m t❤ä❛ ♠➣♥ ❤❛✐ ✤✐➲✉ ❦✐➺♥ ♥â✐ tr➯♥✳ ❙❛✉ ✤➙②✱ ❝❤ó♥❣ t❛ s➩ tr➻♥❤ ❜➔② ❝→❝❤ sỷ t ữợ y = 4x3 + m ữợ số t ỹ t t↕✐ x = s✉② r❛ y (0) = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ 4.0 + m = ❤❛② m = ữợ ợ m = t ❝â y = 4x3; y = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = 0✳ ❱ỵ✐ x = s✉② r❛ y = ❚❛ ❝â ❜↔♥❣ ❜✐➳♥ t❤✐➯♥ ✻✹ ❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ x −∞ ✵ − y +∞ + +∞ +∞ y ❉ü❛ ✈➔♦ ❇❇❚ t❛ t❤➜② ❤➔♠ sè ✤↕t ❝ü❝ t✐➸✉ t↕✐ x = 0; yCT = ❱➟② ✈ỵ✐ m = t❤➻ ❤➔♠ sè ✤↕t ❝ü❝ t✐➸✉ t↕✐ x = 0✳ +m ❇➔✐ ✷✳ ❚➻♠ m ✤➸ ❤➔♠ sè y = x +x 2x (C) ✤↕t ❝ü❝ t✐➸✉ t↕✐ x = +2 ▲í✐ ❣✐↔✐ s❛✐ ❧➛♠✳ ❚❳✣✿ D = R\{✲2} y =1− ✣➸ ❤➔♠ m 2m ; y = (x + 2)2 (x + 2)3 sè ✤↕t ❝ü❝ t✐➸✉ t↕✐ x =   y (2) =  y (2) > t❤➻   1 − m = 16 ⇒ 2m   >0 64 ❑❤✐ m = 16✱ t❛ ❝â✿ y = x + x 16 ✱ s✉② r❛ y +2 ✈➔ ❝❤➾ ❦❤✐ x = ❤♦➦❝ x = −6 ❇❇❚ x −∞ −6 + y ⇒ m = 16 = 1− +∞ ✵ − ✵ ❈✣ 16 y =0 (x + 2)2 + +∞ y ❈❚ −∞ ❱➟② m = 16 ❧➔ ❣✐→ trà ❝➛♥ t➻♠✳ ✻✺ ❦❤✐ ❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ P t s tr ỗ ữợ ♥❤÷ s❛✉✿ ✶✳ ●✐↔ sû ❤➔♠ sè ✤↕t ❝ü❝ t✐➸✉ t↕✐ x = s✉② r❛ y (2) = 0; y (2) > 0✱ s✉② r❛ m = 16✳ ✷✳ ợ m = 16 tr ữủ số t ❝ü❝ t✐➸✉ t↕✐ x = ✭♥❤í ❜↔♥❣ ❜✐➳♥ t❤✐➯♥✮✳ ữợ õ ỳ s t r ữợ t ❜✐➳♥ t❤✐➯♥ s❛✐✳ ❚✉② ♥❤✐➯♥ s❛✐ ❧➛♠ tr➛♠ trå♥❣ ♥❤➜t tr ữợ t ró s t tữỡ tỹ ợ t s m ✤➸ ❤➔♠ sè y = mx4 + (C) ✤↕t ❝ü❝ t✐➸✉ t↕✐ x = 0✳ ❚÷ì♥❣ tü ❧í✐ ❣✐↔✐ ❝õ❛ ❜➔✐ t♦→♥ ✤➛✉✱ t❛ ❧➔♠ ♥❤÷ s❛✉✿ ❚❛ ❝â y = 4mx3; y = 12mx2✳ ❍➔♠ sè ✤↕t ❝ü❝ t✐➸✉ t↕✐ x = ♥➯♥ y (0) = 0; y (0) > 0✱ ✤✐➲✉ ♥➔② ❞➝♥ ✤➳♥ ❦❤æ♥❣ ❝â ❣✐→ trà ♥➔♦ ❝õ❛ mt❤ä❛ ♠➣♥✳ ✭❉♦ ✤â ❦❤æ♥❣ ❝➛♥ ữợ t r số tr➯♥ s➩ ✤↕t ❝ü❝ t✐➸✉ t↕✐ x = ✈ỵ✐ ♠é✐ sè ❞÷ì♥❣ m✳ ◆❤÷ ✈➟② ❧í✐ ❣✐↔✐ ❝õ❛ ❜➔✐ t♦→♥ ❜❛♥ ✤➛✉ s❛✐ ð ❝❤é ♥➔♦❄ ✣➸ tr↔ ❧í✐ ọ trữợ t t ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ❤➔♠ sè ✤↕t ❝ü❝ trà ỵ tt ứ õ õ t t s ❧➛♠ ❝õ❛ ❧í✐ ❣✐↔✐ tr➯♥ ♥➡♠ ð ❝❤é✿ t→❝ ❣✐↔ ❦❤ỉ♥❣ ♣❤➙♥ ❜✐➺t ✤÷đ❝ ✤➙✉ ❧➔ ✤✐➲✉ ❦✐➺♥ ❝➛♥✱ ✤➙✉ ❧➔ ✤✐➲✉ ❦✐➺♥ ✤õ✳ ✣➙② ❧➔ s❛✐ ❧➛♠ ♠➔ ♥❤✐➲✉ ❤å❝ s✐♥❤ t❤÷í♥❣ ♠➢❝ ♣❤↔✐ ❦❤✐ ❣✐↔✐ ❜➔✐ t♦→♥ ✧ t➻♠ m ✤➸ ❤➔♠ sè ✤↕t ❝ü❝ ✤↕✐ ✭❝ü❝ t✐➸✉✮ t↕✐ ♠ët ✤✐➸♠✧✳ ▲í✐ ❣✐↔✐ ✤ó♥❣✳ ●✐↔ sû ❤➔♠ sè ✤↕t ❝ü❝ t✐➸✉ t↕✐ x = 2✳ ❑❤✐ ✤â✱ t❤❡♦ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✻✻ ❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❝õ❛ ❝ü❝ trà✱ t❛ ❝â y (2) = 0✱ s✉② r❛ m = 16✳ ❱ỵ✐ m = 16✱ t❛ ❦✐➸♠ tr❛ ✤÷đ❝ ❤➔♠ sè ✤↕t ❝ü❝ t✐➸✉ t↕✐ x = ✭❝â t❤➸ ❞ò♥❣ ❜↔♥❣ ❜✐➳♥ t❤✐➯♥ ❤♦➦❝ ✤✐➲✉ ❦✐➺♥ ✤õ ❝õ❛ ❝ü❝ trà✱ t✉② ♥❤✐➯♥ ♥➯♥ ❞ò♥❣ ✧✤✐➲✉ ❦✐➺♥ ✤õ✧ ❝❤♦ ♥❤❛♥❤✮✳ ❱➟② m = 16 ❧➔ ❣✐→ trà ❞✉② ♥❤➜t t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉ ❜➔✐ t♦→♥✳ ✻✼ ❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❑➳t ❧✉➟♥ ❑❤â❛ ❧✉➟♥ ✧❇➔✐ t♦→♥ ❝ü❝ tr số ỗ s ✲ ✣÷❛ r❛ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❝ì ❜↔♥ t➻♠ ❝ü❝ trà ❝õ❛ ❤➔♠ sè✱ ❝ư t❤➸ ❧➔✿ ♣❤÷ì♥❣ ♣❤→♣ ✤↕♦ ❤➔♠✱ sû ❞ư♥❣ ❜➜t ✤➥♥❣ t❤ù❝✱ ♣❤÷ì♥❣ ♣❤→♣ ♠✐➲♥ ❣✐→ trà ❝õ❛ ❤➔♠ sè✱ ♣❤÷ì♥❣ ♣❤→♣ sû ❞ư♥❣ ❤➔♠ ỗ ó ữỡ tồ tỡ ♠ët sè s❛✐ ❧➛♠ ❞➵ ❣➦♣ ♣❤↔✐ ❦❤✐ ❣✐↔✐ ❜➔✐ t♦→♥ t➻♠ ❝ü❝ trà ❝õ❛ ❤➔♠ sè✳ P❤➙♥ t➼❝❤ ✈➔ ✤÷❛ r❛ ❝→❝❤ ❦❤➢❝ ♣❤ư❝ ✤➸ tr→♥❤ ♥❤ú♥❣ s❛✐ ❧➛♠ õ ỗ tớ ữ r ú tớ ❣✐❛♥ ❝â ❤↕♥ ♥➯♥ ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ❝ü❝ trà ❝õ❛ ❤➔♠ sè ❝❤÷❛ ✤÷đ❝ ✤➲ ❝➟♣ ✤➳♥✳ ❑➼♥❤ ♠♦♥❣ ✤÷đ❝ sü ✤â♥❣ ❣â♣ ❝õ❛ t❤➛② ❝ỉ✱ ❜↕♥ ❜➧ ✤➸ ❦❤â❛ ❧✉➟♥ ✤÷đ❝ ❤♦➔♥ ❝❤➾♥❤ ❤ì♥✳ ✻✽ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ P❤❛♥ ❍✉② ❑❤↔✐✱ ❈→❝ ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t✱ ❣✐→ trà ♥❤ä ♥❤➜t✱ ◆❳❇ ❍➔ ◆ë✐✱ ✷✵✵✷✳ ❬✷❪ ❚r➛♥ P❤÷ì♥❣✱ ❚✉②➸♥ t➟♣ ❝→❝ ❝❤✉②➯♥ ✤➲ ❧✉②➺♥ t❤✐ ✣↕✐ ❤å❝ ♠æ♥ ❚♦→♥ ❤➔♠ sè✱ ◆❳❇ ❍➔ ◆ë✐✱ tờ tr ỗ trt ❬✺❪ P❤❛♥ ✣ù❝ ❈❤➼♥❤ ✲ P❤↕♠ ❱➠♥ ✣✐➲✉ ✲ ✣é ❱➠♥ ❍➔ ✈➔ ❈s✱ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❝❤å♥ ❧å❝ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ❝➜♣ ✲ ❚➟♣ ✷✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✱ ✷✵✵✸✳ ✻✾ ... ❦❤✐ x1 = x2 = = xn✳ ❜✳ ❈ì sð ỵ ỷ t tự s t ❣✐→ trà ❧ỵ♥ ♥❤➜t✱ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❤➔♠ số t ữợ s t số y = f (x)✳ ❚➼♥❤ y ❇✷✳ ❚➼♥❤ y = f (x)✳ ✸✼ = f (x)✳ ❚⑨❖ ❚❍➚ ⑩◆❍ ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣... (x) ✤↕t ❝ü❝ ✤↕✐ ✤à❛ ♣❤÷ì♥❣ t↕✐ x0 ∈ D t❤➻ t❛ ❝â✿ f (x0) ≤ M ✱ ✈ỵ✐ M = max f (x) ❱➟② ●❚▲◆ ✭●❚◆◆✮ số f (x) ữ xD trũ ợ ❝ü❝ ✤↕✐ ✤à❛ ♣❤÷ì♥❣ ✭❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣✮ tr➯♥ ♠✐➲♥ ①→❝ ✤à♥❤ D ♥➔♦ ✤â✳ ◆❤➟♥... ❚❹▼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ max f (x) ≤ max f (x) ✳ x∈A x∈B f (x) ≥ f (x) ✳ x∈A ỵ xB sỷ số f (x) ①→❝ ✤à♥❤ tr➯♥ ♠✐➲♥ D✱ t❛ ❧✉æ♥ ❝â max f (x) = −min(−f (x)) x∈D x∈D ❚➼♥❤ ❝❤➜t ♥➔② ❝❤♦ ♣❤➨♣