✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ◆ỉ♥❣ ❍÷ì♥❣ ◆❛ ✣❆ ❚❍Ù❈ ✣➮■ ❳Ù◆● ❱⑨ ❈⑩❈ ❍➏ P❍×❒◆● ❚❘➐◆❍ ❱⑨ ❇❻❚ ✣➃◆● ❚❍Ù❈ ▲■➊◆ ◗❯❆◆ ▲❯❾◆ ò Pì PP P số ữớ ữợ ●❙✳ ❚❙❑❍✳ ◆●❯❨➍◆ ❱❿◆ ▼❾❯ ❚❍⑩■ ◆●❯❨➊◆ ✲ ◆❿▼ ✷✵✶✸ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✶ ▼ư❝ ❧ö❝ ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✶ ✣❛ t❤ù❝ ✤↕✐ sè ✈➔ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ì ❜↔♥ ✶ ✷ ✺ ✶✳✶ ❚➼♥❤ ❝❤➜t ❝õ❛ ✤❛ t❤ù❝ ✤↕✐ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸ ✶✳✷✳✶ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ♥❤✐➲✉ ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✷✳✷ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ❜❛ ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✷✳✸ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ❤❛✐ ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ▼ët sè ❞↕♥❣ ❜✐➸✉ ❞✐➵♥ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ ✈➔ ❤➺ ❞↕♥❣ ✤è✐ ①ù♥❣ ✷✳✶ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ n ➞♥ ✭n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ > 3, n ∈ N✮ ✷✵ ✷✵ ✷✳✶✳✶ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✶✳✷ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ❜❛ ➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✶✳✸ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ❤❛✐ ➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✷ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ ✈á♥❣ q✉❛♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✸ ▼ët sè ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸ ❇➜t ✤➥♥❣ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ✸✳✶ ✸✼ ❇➜t ✤➥♥❣ t❤ù❝ ❝õ❛ ❝→❝ ❞↕♥❣ ✤❛ t❤ù❝ ❜➟❝ ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✶✳✶ ❚➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✶✳✷ ❇➔✐ t➟♣ →♣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✷ ❇➜t ✤➥♥❣ t❤ù❝ ❝õ❛ ❝→❝ ❞↕♥❣ ✤❛ t❤ù❝ ❜➟❝ ❝❛♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✸ ❇➜t ✤➥♥❣ t❤ù❝ ❝õ❛ ❝→❝ ❞↕♥❣ ♣❤➙♥ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✷ ỵ t ỡ sð ❧➼ ❧✉➟♥✿ ❚♦→♥ ❤å❝ ❧➔ ♠æ♥ t❤➸ t❤❛♦ ❝õ❛ tr➼ t✉➺✱ ❧➔ ♠ỉ♥ ❦❤♦❛ ❤å❝ ❣✐ó♣ ❤å❝ s✐♥❤ ♣❤→t tr✐➸♥ ♥➠♥❣ ❧ü❝ t÷ ❞✉②✱ ❦❤↔ ♥➠♥❣ ❞ü ✤♦→♥ ♣❤➙♥ t➼❝❤ tê♥❣ ❤đ♣✱ ♣❤→t ❤✐➺♥✱ t✐➳♣ t❤✉✱ ❣❤✐ ♥❤ỵ ❦❤✐ tr➻♥❤ ❜➔② ♠ët ✈➜♥ ✤➲ ♠ët ❝→❝❤ ❦❤♦❛ ❤å❝✱ ❧æ ❣✐❝✱ ❝❤➦t ❝❤➩✳ ✶✳✷ ❈ì sð t❤ü❝ t➳✿ ❚r♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ❤å❝ ð tr✉♥❣ ❤å❝ ♣❤ê t❤æ♥❣ t❤➻ ✤❛ t❤ù❝ ❝â ✈❛✐ trá ✈➔ ✈à tr➼ r➜t q✉❛♥ trå♥❣ ✈➻ ♥â ❦❤ỉ♥❣ ♥❤ú♥❣ ❧➔ ♠ët ✤è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ trå♥❣ t➙♠ ❝õ❛ ✣↕✐ sè ♠➔ ❝á♥ ❧➔ ♠ët ❝æ♥❣ ỹ t tr ỵ tt ỵ tt s ỵ tt ✳ ❚r♦♥❣ ❝→❝ ❦ý t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ t♦→♥ q✉è❝ ❣✐❛✱ ♦❧②♠♣✐❝ t♦→♥ ❦❤✉ ✈ü❝ ✈➔ q✉è❝ t➳ t❤➻ ❝→❝ ❜➔✐ t♦→♥ ✈➲ ✤❛ t❤ù❝ ❝ơ♥❣ ✤÷đ❝ ①❡♠ ♥❤÷ ♥❤ú♥❣ ❞↕♥❣ ❜➔✐ t♦→♥ ❦❤â ð ❜➟❝ tr✉♥❣ ❤å❝ ♣❤ê t❤æ♥❣✳ ❚r♦♥❣ ❧➽♥❤ ✈ü❝ ♣❤ù❝ t↕♣ ❝õ❛ ✤↕✐ sè ✤è✐ ✈ỵ✐ ❤å❝ s✐♥❤ ♣❤ê t❤ỉ♥❣ t❤÷í♥❣ ❧➔ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❝❛♦✱ ♣❤➙♥ t➼❝❤ ❝→❝ ✤❛ t❤ù❝ ♥❤✐➲✉ ❜✐➳♥ ❜➟❝ ❝❛♦ t❤➔♥❤ ♥❤➙♥ tû✱ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ✤➥♥❣ t❤ù❝ ❜➜t ✤➥♥❣ t❤ù❝ ❝❤ù❛ ♥❤✐➲✉ ❜✐➳♥ sè✳ ✳ ✳ ▼ët tr÷í♥❣ ❤đ♣ q✉❛♥ trå♥❣ ✈➔ t❤÷í♥❣ ❣➦♣ tr♦♥❣ ❝→❝ ❜➔✐ t♦→♥ ❝õ❛ ❝→❝ ❧➽♥❤ ✈ü❝ ♥â✐ tr➯♥ ❧➔ ❦❤✐ ❝→❝ ❜✐➳♥ sè ❝õ❛ ✤❛ t❤ù❝ ❝â ✈❛✐ trá ✈➔ ✈à tr➼ ♥❤÷ ♥❤❛✉✳ ❈❤ó♥❣ t❛ ❣å✐ ✤❛ t❤ù❝ tr♦♥❣ tr÷í♥❣ ❤đ♣ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ✈➔ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ❧✐➯♥ q✉❛♥ ✧ tr➻♥❤ ❜➔② ♠ët sè ✈➜♥ ✤➲ ❧✐➯♥ ♥➔② ❧➔ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣✳ ▲✉➟♥ ✈➠♥ ✧ q✉❛♥ ✤➳♥ ♥❤✐➲✉ ❜➔✐ t♦→♥ ❦❤â ❝â ❝❤ù❛ ②➳✉ tè ✤è✐ ①ù♥❣ ♥➳✉ ❜✐➳t ỵ tt tự ố ự s ❧➔♠ ❝❤♦ ❜➔✐ t♦→♥ trð ♥➯♥ ✤ì♥ ❣✐↔♥ ❤ì♥✳ ▲✉➟♥ ợ t ỡ s ỵ tt ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ✈➔ ù♥❣ ❞ö♥❣ ❝õ❛ ♥â tr♦♥❣ số sỡ ỵ tt ữủ tr ởt ỡ t ữợ q ♥↕♣✱ tø tr÷í♥❣ ❤đ♣ ❤❛✐ ❜✐➳♥✱ ❜❛ ❜✐➳♥✱ ✤➳♥ ♥❤✐➲✉ ❜✐➳♥✳ ❈→❝ ✈➼ ❞ư →♣ ❞ư♥❣ ❝ơ♥❣ ✤÷đ❝ tr➻♥❤ ❜➔② tø ✤ì♥ ❣✐↔♥ ✤➳♥ ♣❤ù❝ t↕♣✳ ❈→❝ ❜➔✐ t♦→♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ❝❤õ ②➳✉ ❧➔ ❝→❝ ❜➔✐ t♦→♥ ❦❤â✱ ♥❤✐➲✉ ❜➔✐ t♦→♥ ✤÷đ❝ tr➼❝❤ r❛ tø ❝→❝ ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ q✉è❝ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✸ ❣✐❛✱ ❖❧②♠♣✐❝ t♦→♥ q✉è❝ t➳✱ ■▼❖✳ ✳ ✳ ✣➲ t➔✐ q✉❛♥ t➙♠ ✤➳♥ ♥❤✐➲✉ ✤è✐ tữủ tr õ t ũ ủ ợ tỹ t ♠➔ ❜↔♥ t❤➙♥ ✤❛♥❣ ❝ỉ♥❣ t→❝✳ ✷✳ ▼ư❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ✈➔ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ❧✐➯♥ q✉❛♥ ✧ ♥❤➡♠ t❤➸ ❤✐➺♥ rã ✈❛✐ trá q✉❛♥ trå♥❣ ❝õ❛ ✤↕✐ sè tr♦♥❣ ▲✉➟♥ ✈➠♥ ✧ t♦→♥ ❤å❝✳ ▲✉➟♥ ✈➠♥ ♥➔② ❧➔ ❝❤✉②➯♥ ✤➲ tê♥❣ q✉❛♥ ✈➲ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ t❤æ♥❣ q ỵ ❜➔✐ t➟♣ →♣ ❞ư♥❣✳ ✸✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ❚❤❛♠ ❦❤↔♦ ✈➔ ♥❣❤✐➯♥ ❝ù✉ tø ❝→❝ t➔✐ ❧✐➺✉✱ ❣✐→♦ tr➻♥❤ ❝õ❛ ●❙✲❚❙❑❍ ◆❣✉②➵♥ ❱➠♥ ▼➟✉ ✈➔ ❝→❝ s→❝❤ ❝❤✉②➯♥ ✤➲ ✈➲ ✤❛ t❤ù❝✱ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❝→❝ ❜➔✐ ❜→♦ t♦→♥ ❤å❝ ✈✐➳t ✈➲ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣✱ ♥❤➡♠ ❤➺ t❤è♥❣ ❝→❝ ❞↕♥❣ t♦→♥ ✈➲ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣✳ ◆❣❤✐➯♥ ❝ù✉ trü❝ t✐➳♣ tø ❝→❝ t t ữợ ỗ ụ ữ tr ợ ị tỹ t ❝õ❛ ✤➲ t➔✐ ❚↕♦ ✤÷đ❝ ♠ët ✤➲ t➔✐ ♣❤ị ❤đ♣ ỗ ữù s tr ♣❤ê t❤æ♥❣✱ ✤➲ t➔✐ ✤â♥❣ ❣â♣ t❤✐➳t t❤ü❝ ❝❤♦ ✈✐➺❝ ❞↕② ✈➔ ❤å❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣✱ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ tr÷í♥❣ ♣❤ê t❤ỉ♥❣✱ ✤❡♠ ❧↕✐ ♥✐➲♠ ✤❛♠ ♠➯ s→♥❣ t↕♦ tø ♥❤ú♥❣ ❜➔✐ t♦→♥ ❝ì ❜↔♥ ♥❤➜t✳ ✺✳ ❈➜✉ tró❝ ❝õ❛ ❧✉➟♥ ✈➠♥ ỗ t t t❤❛♠ ❦❤↔♦ ✈➔ ✸ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✿ ✣❛ t❤ù❝ ✤↕✐ sè ✈➔ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ì ❜↔♥✳ ❈❤÷ì♥❣ ✷✿ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ ✈➔ ❤➺ ❞↕♥❣ ✤è✐ ①ù♥❣✳ ❈❤÷ì♥❣ ✸✿ ❇➜t ✤➥♥❣ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣✳ ❉ị ✤➣ r➜t ❝è ❣➢♥❣✱ ♥❤÷♥❣ ❝❤➢❝ ❝❤➢♥ ♥ë✐ ❞✉♥❣ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ tr→♥❤ ❦❤ä✐ t❤✐➳✉ sât✱ ❡♠ r➜t ♠♦♥❣ ✤÷đ❝ sü ❣â♣ þ ❝õ❛ ❝→❝ t❤➛② ❝æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ ✤➸ ❡♠ t✐➳♣ tö❝ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❧✉➟♥ ✈➠♥ ♥➔② ữủ t ữợ sỹ ữợ ●❙✳ ❚❙❑❍✳ ◆●❯❨➍◆ ❱❿◆ ▼❾❯✳ ❊♠ ①✐♥ ✤÷đ❝ tä ❧á♥❣ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ♥❤➜t tỵ✐ ❚❤➛② ✈➲ sü ❣✐ó♣ ✤ï ♥❤✐➺t t➻♥❤ tø ❦❤✐ ①➙② ❞ü♥❣ ✤➲ ❝÷ì♥❣✱ ✈✐➳t ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚✐➳♣ t❤❡♦ ❡♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ♣❤↔♥ ❜✐➺♥ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✹ ✤➣ ✤å❝ õ ỵ t ♠➻♥❤✱ ❡♠ ①✐♥ ✤÷đ❝ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ♥❤➜t ✤➳♥ ❦❤♦❛ ❚♦→♥ ✲ ❚✐♥ ❝õ❛ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ♥ì✐ ❡♠ ✤➣ ♥❤➟♥ ✤÷đ❝ ♠ët ❤å❝ ✈➜♥ s❛✉ ✤↕✐ ❤å❝ ❝➠♥ ❜↔♥✳❳✐♥ ❝↔♠ ì♥ ỗ tổ s ❤ë ✈➔ ❣✐ó♣ ✤ï tr♦♥❣ t❤í✐ ❣✐❛♥ ❡♠ ❤å❝ ❝❛♦ ❤å❝ ✈➔ ✈✐➳t ❧✉➟♥ ✈➠♥✳ ▲í✐ ❝✉è✐ ❡♠ ①✐♥ ❝❤ó❝ ọ t ổ ỗ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✺ ❈❤÷ì♥❣ ✶ ✣❛ t❤ù❝ ✤↕✐ sè ✈➔ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ì ❜↔♥ ✶✳✶ ❚➼♥❤ ❝❤➜t ❝õ❛ ✤❛ t❤ù❝ ✤↕✐ sè ✣à♥❤ ♥❣❤➽❛ ✶✳✶ ✳ ✭①❡♠ ❬✶❪✲❬✹❪✮ ▼ët ✤❛ t❤ù❝ ❜➟❝ n ❝õ❛ ➞♥ x ❧➔ ❜✐➸✉ t❤ù❝ ❝â ❞↕♥❣✿ Pn (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 , tr♦♥❣ ✤â✱ ❝→❝ ❤➺ sè an , an−1 , , a0 ❧➔ ♥❤ú♥❣ sè t❤ü❝ ✭❤♦➦❝ sè ♣❤ù❝✮ ✈➔ an = 0, n ∈ N ❚❛ ❦➼ ❤✐➺✉✿ ✐✮ ❇➟❝ ❝õ❛ ✤❛ t❤ù❝ Pn (x) ❧➔ ❞❡❣Pn ❉♦ ✈➟② an ❧➔ ❤➺ sè ❝❛♦ ♥❤➜t ✭❝❤➼♥❤✮ ❝õ❛ ✐✐✐✮ a0 ❧➔ ❤➺ sè tü ❞♦ ❝õ❛ ✤❛ t❤ù❝✱ n ✐✈✮ an x ❧➔ ❤↕♥❣ tû ❝❛♦ ♥❤➜t✳ ✐✐✮ ✣à♥❤ ♥❣❤➽❛ ✶✳✷ ✭①❡♠ ❬✶❪✲❬✸❪✮ ✳ deg Pn (x) = n ✤❛ t❤ù❝✱ ❈❤♦ ✤❛ t❤ù❝ Pn (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 , ✈ỵ✐ an = α ∈ C ✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✤❛ t❤ù❝ Pn (x) ♥➳✉ Pn (α) = ◆➳✉ k k+1 tỗ t k N, k > s❛♦ ❝❤♦ Pn (x)✳✳(x − α) ✈➔ Pn (x) ✳✳(x − α) t❤➻ α ✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❜ë✐ k ❝õ❛ ✤❛ t❤ù❝ Pn (x)✳ ✣➦❝ ❜✐➺t k = t❤➻ α ✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ✤ì♥✱ k = t❤➻ α ✤÷đ❝ ❣å✐ ❧➔ ❑❤✐ ✤â✱ ♥❣❤✐➺♠ ỵ trữớ ỵ ●❛✉ss✮ ▼å✐ ✤❛ t❤ù❝ ❜➟❝ n ≥ tr➯♥ C ✤➲✉ ❝â ✤ó♥❣ n ♥❣❤✐➺♠ ♥➳✉ ♠é✐ ♥❣❤✐➺♠ ✤÷đ❝ t➼♥❤ ♠ët sè ❧➛♥ ❜➡♥❣ ❜ë✐ ❝õ❛ ♥â✳ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✻ ❇ê ✤➲ ✶✳✶✳ Pn (z) = ❈→❝ ♥❣❤✐➺♠ ♣❤ù❝ t❤ü❝ sü ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤❛ t❤ù❝ t❤ü❝ ①✉➜t ❤✐➺♥ t❤❡♦ tø♥❣ ❝➦♣ ♥❣❤✐➺♠ ❧✐➯♥ ❤đ♣✳ ❚❤➟t ✈➟②✱ ♥➳✉ a∈C ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ Pn (z) = t❤➻ Pn (a) = 0✳ õ t õ ỵ = Pn (a) = Pn (a) ✳ ✭①❡♠ ❬✶❪✲❬✸❪✮ ▼å✐ ✤❛ t❤ù❝ ợ số tỹ õ t ữợ ❞↕♥❣✿ Pn (x) = a0 (x − α1 )n1 (x − αr )nr (x2 + p1 x + q1 )m1 (x2s + ps x + qs )ms , tr♦♥❣ ✤â✱ r s mi = n, p2i − 4qi < 0, i = 1, s ni +2 i=1 i=1 ✈➔ α0 , α1 , , αr ; p1 , q1 , ps , qs R ứ ỵ ✶✳✷ t❛ ❝â ❦➳t q✉↔ q✉❛♥ trå♥❣ s❛✉ ✤➙②✳ ❍➺ q✉↔ ✶✳✶✳ n ✈➔ k ●✐↔ sû Pn (x) ❝ò♥❣ t ỵ tự ✳ n ✭①❡♠ ❬✶❪✲❬✹❪✮ ▼é✐ ✤❛ t❤ù❝ ❜➟❝ ❝â n k ♥❣❤✐➺♠ t❤ü❝✱ ✤➲✉ ❝â ❦❤æ♥❣ q✉→ k≤n n t❤➻ ♥❣❤✐➺♠ t❤ü❝✳ ✶✳✷ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ì ❜↔♥ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ❧➔ ❝ỉ♥❣ ❝ư ❤ú✉ ❤✐➺✉ ✤➸ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè ❜➟❝ ❝❛♦✱ ✤➦❝ ❜✐➺t ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❤➺ sè ✤è✐ ①ù♥❣ ữỡ tr ỗ q ✳ ✣❛ t❤ù❝ f (z) = a0 z n + a1 z n−1 + · · · + an−1 z + an (a0 = 0) ✤÷đ❝ ❣å✐ ❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣✱ ♥➳✉ ❝→❝ ❤➺ sè ❝→❝❤ ✤➲✉ ❤❛✐ ✤➛✉ ❜➡♥❣ ♥❤❛✉✱ ♥❣❤➽❛ ❧➔✿ a0 = an , a1 = an−1 , a2 = an−2 , ❱➼ ❞ö ✶✳✶✳ ❈→❝ ✤❛ t❤ù❝ s❛✉ ✤➙② ❧➔ ✤❛ t❤ù❝ ❤➺ sè ✤è✐ ①ù♥❣✿ z − 3z + 2z + 2z − 3z + 1, 2z + z − 6z + 4z + 3z + 4z − 6z + z + Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ỵ f (z) tự z znf ✣à♥❤ ♥❣❤➽❛ ✶✳✹ ❜➟❝ ✭①❡♠ ❬✸❪✮ ✳ n ❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ = f (z), ợ z = Pữỡ tr P (x) = an xn + an−1 xn−1 + an−2 xn−2 + · · · + a1 x2 + a1 x + a0 = 0, an = (1) ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣✱ ♥➳✉ ❤➺ sè ❝õ❛ ♥❤ú♥❣ sè ❤↕♥❣ ❝→❝❤ ✤➲✉ ✤➛✉ ✈➔ ❝✉è✐ ❜➡♥❣ ♥❤❛✉✱ tù❝ ❧➔ ✲ ◆➳✉ ✲ ◆➳✉ an = a0 ; an−1 = a1 ; an−2 = a2 ; n = 2k + 1, t❛ ❣å✐ ✭✶✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ ❜➟❝ ❧➫✳ n = 2k t❛ ❣å✐ ✭✶✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ ❜➟❝ ❝❤➤♥✳ ❚❛ ❝â ❝→❝ ❦➳t q✉↔ s❛✉ ✤➙②✳ ▼➺♥❤ ✤➲ ✶✳✶ x = −1 ✭①❡♠ ❬✸❪✮ ✳ ▼å✐ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ ❜➟❝ ❧➫ ởt ú ỵ tr ố ự ứ ỵ t s✉② r❛ ♥➳✉ (deg P (x) = 2k + 1) P (x) = ❧➔ ♣❤÷ì♥❣ t❤➻ P (x) = ⇔ (x − 1)Q (x) = 0, ð ✤➙② deg Q (x) = 2k, ▼➺♥❤ ✤➲ ✶✳✷ ❜➡♥❣ ❝→❝❤ ✤➦t ❱➼ ❞ư ✶✳✷✳ ▲í✐ ❣✐↔✐✳ Q(x) ✈➔ ✳ ✭①❡♠ ❬✸❪✮ y =x+ x ❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ợ ữỡ tr ố ự ✱ ♣❤÷ì♥❣ tr➻♥❤ q✉② ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ 2k ✱ k✳ x4 + 2x3 − 6x2 + 2x + = ❳➨t ♣❤÷ì♥❣ tr➻♥❤ x4 + 2x3 − 6x2 + 2x + = ✣➙② ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ ❜➟❝ ❝❤➤♥✳ ❘ã r➔♥❣ (1) x=0 ❦❤æ♥❣ ♣❤↔✐ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✶✮ ♥➯♥ + 2(x + ) − = x2 x 1 ⇔ x+ −2+2 x+ − = x x ⇔ x+ +2 x+ − = (2) x x (1) ⇔ x2 + Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✽ ✣➦t y =x+ ✱ x ❦❤✐ ✤â ✭✷✮ ⇔ y + 2y − = x + =2 y=2 x ⇔ y = −4 ⇔ x + = −4 x ⇔ x2 − 2x + = ⇔ x2 + 4x + = ❱➟② ♣❤÷ì♥❣ tr➻♥❤ ✭✶✮ ❝â ❜❛ ♥❣❤✐➺♠ ❱➼ ❞ö ✶✳✸✳ x = 1; x = −2 + √ x=1 √ x = −2 ± 3; x = −2 − √ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ x5 − 4x4 + 3x3 + 3x2 − 4x + = (1) ▲í✐ ❣✐↔✐✳ ✣➙② ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ ❜➟❝ ❧➫✱ ♥➯♥ ✭✶✮ ❝❤➢❝ ❝❤➢♥ ❝â ♠ët ♥❣❤✐➺♠ ữủ ỗ t t x = −1 x4 − 5x3 + 8x2 − 5x + = (x + 1) (x4 − 5x3 + 8x2 − 5x + 1) = ⇔ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ x4 − 5x3 + 8x2 − 5x + = ❉♦ x=0 ❦❤æ♥❣ ♣❤↔✐ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✷✮ ♥➯♥ 1 (2) ⇔ x + −5 x + +8 = ⇔ x + x x x 2 ✣➦t y =x+ ✱ x ❜✳ ◆➳✉ −5 x + +6 = (3) x ✈➔ tø ✭✸✮ t❛ ❝â✿ y = y = y − 5y + = ⇔ ❛✳ ◆➳✉ (2) = ⇔ x2 − 2x + = ⇔ x = x √ ± x + = ⇔ x2 − 3x + = ⇔ x = x x+ ❱➟② ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❝â ✹ ♥❣❤✐➺♠ ❧➔ √ √ 3+ 3− x = 1; x = −1; x = ;x = 2 Soá hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✾ ❱➼ ❞ư ✶✳✹✳ ❈❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ ❜➟❝ ❝❤➤♥ s❛✉ ✤➙②✿ x4 + 2x3 + 4x2 + 2x + = (1) ❈❤ù♥❣ ♠✐♥❤ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ✈ỉ ♥❣❤✐➺♠✳ ▲í✐ ❣✐↔✐✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ x4 + 2x3 + 4x2 + 2x + = ❉♦ x=0 ❦❤æ♥❣ ♣❤↔✐ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✶✮✱ t❛ ❝â (1) ⇔ x2 + ✣➦t x2 +2 x+ x + = (2) ✳ x y =x+ ❚❛ ❝â✿ |y| = x + 1 = |x| + ≥ 2, x ❑❤✐ ✤â (2) ⇔ y + 2y + = (3) ❉♦ ∆ = − = −1 < ❱➟② ✭✸✮ ✈æ ♥❣❤✐➺♠✳ ❑➨♦ t❤❡♦ ✭✶✮ ✈æ ♥❣❤✐➺♠✳ ❙✉② r❛ ✤♣❝♠✳ ◆❤➟♥ ①➨t ✶✳✶✳ ❚❛ ❝â ❝→❝❤ ❧➔♠ ❦❤→❝ ♥❤÷ s❛✉✿ (1) ⇔ x2 x2 + 2x + + x2 + 2x + + 2x2 = ⇔ x2 + (x + 1)2 + 2x2 = x + = (4) ⇔ x = (5) ❱➻ ✭✹✮✱✭✺✮ ✈æ ♥❣❤✐➺♠✳ ❙✉② r❛ ✤♣❝♠✳ ❱➼ ❞ö ✶✳✺✳ ●✐↔ sû ❛✱ ❜ ❧➔ ❝→❝ sè s❛♦ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ú♥❣ ❜➟❝ ❝❤➤♥ x + ax + bx2 + ax + = ❝â ♥❣❤✐➺♠✱ t➻♠ ❣✐→ trà ❜➨ ♥❤➜t ❝õ❛ a2 + b2 ✳ ▲í✐ ❣✐↔✐✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ x4 + ax3 + bx2 + ax + = 0.(1) ✭✶✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ ❜➟❝ ❝❤➤♥✳ ❚❤❡♦ ❣✐↔ t❤✐➳t ✭✶✮ ❝â ♥❣❤✐➺♠✱ ♥➯♥ ❣å✐ x0 = 0, ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭✶✮✳ ❘ã r➔♥❣ tø ✭✶✮ t❛ ❝â x20 + x20 + a x0 + ⇔ x0 + x0 x0 + a x0 + Số hóa trung tâm học liệu +b=0 x0 + b − = (2) http://www.lrc-tnu.edu.vn/ ✹✸ a3 b3 c3 a2 c b2 a c2 b ⇔a +b +c + + + + + + ≥ a2 + b2 + c2 b c a b c a a3 b3 c3 a2 c b2 a c2 b ⇔ + + + + + ≥ a2 + b2 + c2 ✭✸✮ b c a b c a 2 ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❆▼ ✲ ●▼✿ a2 c b2 a c2 b + bc ≥ 2ac; + ac ≥ 2ba; + ba ≥ 2bc, b c a t❛ ✤÷đ❝ a2 c b2 a c2 b + + ≥ ab + bc + ca b c a ❙✉② r❛ a3 b3 c3 a2 c b2 a c2 b a3 b3 c3 + + + + + ≥ + + + ab + bc + ca b c a b c a b c a (3) ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❆▼ ✲ ●▼✿ 3 a3 b c + ab ≥ 2a , + bc ≥ 2b , + ca ≥ 2c2 , b c a s✉② r❛ a3 b c + + + ab + bc + ca ≥ a2 + b2 + c2 b c a (4) ❚ø ✭✸✮ ✈➔ ✭✹✮ s✉② r❛ a3 b3 c3 a2 c b2 a c2 b + + + + + ≥ a2 + b2 + c2 , b c a b c a ✭✤♣❝♠✮ ❇➔✐ t♦→♥ ✸✳✼ ✳ ✭❇❛❧❦❛♥ ✷✵✶✵✮ ❈❤♦ a, b, c ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ t❤ä❛ ♠➣♥ a4 + b4 + c4 ≥ a3 + b3 + c3 ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✿ √ a3 b3 c3 √ +√ +√ ≥ b4 + b2 c2 + c4 c4 + c2 a2 + a4 a4 + a2 b2 + b4 ▲í✐ ❣✐↔✐✳ ❚ø ❣✐↔ t❤✐➳t s✉② r❛ a4 + b4 + c4 ≥ 1, a3 + b3 + c3 ♥➯♥ t❛ q✉✐ ❜➔✐ t♦→♥ ự t tự ỗ √ √ a4 + b4 + c4 a3 b3 c3 +√ +√ ≥ 3 a + b3 + c3 b4 + b2 c2 + c4 c4 + c2 a2 + a4 a4 + a2 b2 + b4 Soá hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✹✹ ❙û ❞ư♥❣ ❦➽ t❤✉➟t ❣❤➨♣ ✤è✐ ①ù♥❣✱ t❛ s➩ ❝❤➾ r❛ r➡♥❣✿ √ a3 3a √ ≥ a + b3 + c3 b4 + b2 c2 + c4 ⇔ 3a2 b4 + b2 c2 + c4 ≤ a3 + b3 + c3 (1) ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❆▼ ✲ ●▼ t❛ ❝â 3a2 b4 = 3.ab.ab.b2 ≤ a3 b3 + a3 b3 + b6 3a2 c4 = 3.ac.ac.c2 ≤ a3 c3 + a3 c3 + c6 3a2 b2 c2 = 3.a2 bc.bc ≤ a6 + b3 c3 + b3 c3 ❈ë♥❣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✈➳ t❤❡♦ ✈➳ t❛ t❤✉ ✤÷đ❝ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✮✳ ❉♦ ✤â √ a3 3a √ ≥ a + b3 + c3 b4 + b2 c2 + c4 ❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ a = b = c = ✸✳ ▼ët sè ❦➽ t❤✉➟t ❝❤ù♥❣ ♠✐♥❤ t tự ỗ t a, b, c > ✭■r❛♥✲✷✵✶✵✮ ❈❤♦ ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ 1 1 1 1 + + + ≥ + + + a2 b2 c2 (a + b + c)2 25 a b c a + b + c ▲í✐ ❣✐↔✐✳ ❑❤✐ t❛ t❤❛② ◆❤➟♥ ①➨t✿ ❇➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ t❤✉➛♥ ♥❤➜t✳ (a; b; c) ❜ð✐ (ta; tb; tc) t❤➻ ❜➜t ✤➥♥❣ t❤ù❝ ❦❤æ♥❣ t❤❛② ✤ê✐✳ ❉♦ ✤â ❦❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ ❣✐↔ sû a + b + c = (a, b, c > 0) ❇➜t ✤➥♥❣ t❤ù❝ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ✈✐➳t ❧↕✐✿ 1 1 + + + ≥ + + +1 a2 b2 c2 25 a b c ✣➦t (1) (2) 1 = x; = y; = z (x; y; z > 0) a b c 1 ⇒x+y+z = + + ≥ = a b c a+b+c Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✹✺ ❇➜t ✤➥♥❣ t❤ù❝ ✭✷✮✿ x2 + y + z + ≥ ❉♦ (x + y + z + 1)2 25 x2 + y + z ≥ (x + y + z)2 ♥➯♥ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ♥➳✉ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ (x + y + z)2 + ≥ (x + y + z + 1)2 25 ✣➦t t=x+y+z t❤➻ t ≥ 9✳ ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ t + ≥ (t + 1)2 ⇔ 4t2 − 42t + 54 ≥ ⇔ (t − 9) t − 25 ✣✐➲✉ ♥➔② ❤♦➔♥ t♦➔♥ ✤ó♥❣ ∀t ≥ 9✳ ❉♦ ✤â ❜➔✐ t♦→♥ ✤➣ ❣✐↔✐ ①♦♥❣✳ ✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❇➔✐ t♦→♥ ✸✳✾✳ ❈❤♦ a, b, c ≥ a = b = c ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✿ (a + b)2 (b + c)2 (c + a)2 (a + b + c) ≥ abc 3 (1) ▲í✐ ❣✐↔✐✳ (a; b; c) ❜ð✐ (ta; tb; tc) t❤➻ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✮ ❦❤æ♥❣ t❤❛② ♠➜t tê♥❣ q✉→t ❣✐↔ sû a + b + c = ❈→❝❤ ✶✳ ❑❤✐ t❤❛② ✤ê✐✱ ♥➯♥ ❦❤æ♥❣ (a + b)2 (b + c)2 (c + a)2 (1) ⇔ ≥4 abc ⇔ (a + b)2 (b + c)2 (c + a)2 ≥√ 64abc ⇔ (a + b) (b + c) (c + a) ≥ abc ❱➻ a+b+c=3 (2) ♥➯♥ (a + b) (b + c) (c + a) = (3 − c) (3 − a) (3 − b) = 27 − (a + b + c) + (ab + bc + ca) − abc = 27 − 9.3 + (ab + bc + ca) − abc = (ab + bc + ca) − abc Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✹✻ ⑩♣ ❞ư♥❣ ❜➜t ✤➥♥❣ t❤ù❝ q✉❡♥ t❤✉ë❝✿ (x + y + z)2 ≥ (xy + yz + zx) , ∀x, y, z ∈ R ❚❛ ❝â √ (ab + bc + ca) ≥ 3abc (a + b + c) = 9abc ⇒ ab + bc + ca ≥ abc ❉♦ ✤â √ √ √ √ (a + b) (b + c) (c + a) ≥ abc − abc = abc + abc − abc ❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ❆▼ ✲ ●▼✱ t❛ ❝â √ √ 3 = a + b + c ≥ abc ⇒ ≥ abc ⇒ − abc ≥ ❙✉② r❛✿ √ (a + b) (b + c) (c + a) ≥ abc ❇➜t ✤➥♥❣ t❤ù❝ ✭✷✮ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ✈➔ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ a = b = c = ❱➟② ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✮ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✱ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ a = b = c ❈→❝❤ ✷✳ ❈❤✉➞♥ ❤â❛ a+b+c= ✳ ❇➜t ✤➥♥❣ t❤ù❝ ✭✶✮ trð t❤➔♥❤ ✿ (a + b)2 (b + c)2 (c + a)2 ≥ ⇔ (a + b)2 (b + c)2 (c + a)2 ≥ abc abc (2) ❇✐➳♥ ✤ê✐ ✈➔ →♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❆▼ ✲ ●▼ t❛ ✤÷đ❝ ✿ (a + b)2 (b + c)2 (c + a)2 = [(a + b + c) (ab + bc + ca) − abc]2 = (a + b + c) (ab + bc + ca) + (a + b + c) (ab + bc + ca) − abc 9 √ 3 ≥ (ab + bc + ca) + abc.3 (abc) − abc = (ab + bc + ca) 9 = (ab + bc + ca)2 4 ≥ 3abc (a + b + c) = 3abc = abc.(✤♣❝♠) 9 Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✹✼ ❇➔✐ t♦→♥ ✸✳✶✵ ✳ ✭◆❤➟t ❜↔♥ ✶✾✾✼✮ ❈❤♦ a, b, c ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣✳ ❈❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ (b + c − a)2 (c + a − b)2 (a + b − c)2 + + ≥ (b + c)2 + a2 (c + a)2 + b2 (a + b)2 + c2 ▲í✐ ❣✐↔✐✳ ◆❤➟♥ ①➨t r➡♥❣ ❜➔✐ t♦→♥ ♥➔② ❝â tr♦♥❣ ❝✉è♥ s→❝❤ ✧❚✉②➸♥ t➟♣ ❝→❝ ❜➔✐ t♦→♥ tø ♥❤ú♥❣ ❝✉ë❝ t❤✐ t↕✐ ❚r✉♥❣ ◗✉è❝✧✱ ✤÷đ❝ ❣✐↔✐ ❦❤→ ♣❤ù❝ t↕♣ ❜➡♥❣ ❝→❝❤ sû ❞ư♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❙❝❤✉r✳ Ð ✤➙② t❛ ✤÷❛ r❛ ❧í✐ sỷ t ỗ ❜✐➸✉ t❤ù❝ t❤❛♠ ❣✐❛ tr♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝✳ ❈→❝❤ ✶✿ ✣➦t 2x = b + c − a 2y = c + a − b 2z = a + b − c a = y + z ⇔ b=z+x c=x+y ❜➜t ✤➥♥❣ t❤ù❝ 4x2 4y ⇔ + (2x + y + z)2 + (y + z)2 (2y + z + x)2 + (z + x)2 4z + 2 ≥ (2z + x + y) + (x + y) 2 x y ⇔ + 2x + y + z + 2xy + 2xz + 2yz 2y + x2 + z + 2xy + 2yz + 2zx z2 + ≥ 2z + x2 + y + 2xy + 2yz + 2zx 10 ❉♦ 2xy ≤ x2 + y , 2yz ≤ y + z , 2zx ≤ x2 + z ♥➯♥ x2 y2 z2 VT ≥ + + 4x + 3y + 3z 4y + 3z + 3x2 4z + 3x2 + 3y ✣➦t x1 = x2 , y1 = y , z1 = z , x1 , y1 , z1 > x1 y1 z1 VT ≥ + + 4x1 + 3y1 + 3z1 4y1 + 3z1 + 3x1 4z1 + 3x1 + 3y1 ❈→❝ ♣❤➙♥ t❤ù❝ õ tỷ số số ỗ ❜➟❝✱ ❦❤æ♥❣ ♠➜t tê♥❣ q✉→t✱ ❣✐↔ sû x1 + y1 + z1 = VT ≥ x1 y1 z1 + + x1 + y1 + z1 + Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✹✽ ✣➦t f (t) = ⇒ f (t) t −6 , t > 0, f (t) = , f (t) = < t+3 (t + 3)2 (t + 3)3 (0, +∞) ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ x1 + y1 + z1 f (x1 ) + f (y1 ) + f (z1 ) ≥ 3.f = 3f 3 ⇒ V T ≥ 3 = ✭✤♣❝♠✮ 10 +3 ỗ tr ỗ t❛ ❝â✿ ❈→❝❤ ✷✳ ❇➜t ✤➥♥❣ t❤ù❝ (b + c − a)2 (c + a − b)2 (a + b − c)2 12 ⇔ − + − + − ≥ − (b + c)2 + a2 (c + a)2 + b2 (a + b)2 + c2 (b + c) a (c + a) b (a + b) c ⇔ + + ≤ (b + c)2 + a2 (c + a)2 + b2 (a + b)2 + c2 ❈→❝ ♣❤➙♥ t❤ù❝ ð ✈➳ tr→✐ ❝â tỷ số số ỗ ổ t tờ q✉→t✱ ❣✐↔ sû a + b + c = ❇➜t ✤➥♥❣ t❤ù❝ ✈✐➳t ❧↕✐ t❤➔♥❤ (1 − a) a (1 − b) b (1 − c) c + + ≤ 2 − 2a + 2a − 2b + 2b − 2c + 2c ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❆▼ ✲ ●▼✿ (a + 1)2 2a (1 − a) ≤ , s✉② r❛✿ (a + 1)2 (1 − a) (3 + a) − 2a + 2a ≥ − = 4 ❉♦ ✤â (1 − a) a (1 − a) a 4a ≤ = (1 − a) (3 + a) − 2a + 2a2 3+a ❚÷ì♥❣ tü✱ (1 − b) b 4b (1 − c) c 4c ≤ , ≤ − 2b + 2b2 + b − 2c + 2c2 3+c ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ ✤➲ ❜➔✐ t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ 4a 4b 4c 1 + + ≤ ⇔ + + ≥ 3+a 3+b 3+c 3+a 3+b 3+c 10 Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✹✾ ⑩♣ ❞ư♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❆▼ ✲ ●▼✿ 1 + + 3+a 3+b 3+c (3 + a + + b + + c) ≥ ❙✉② r❛✿ 1 + + 3+a 3+b 3+c 1 + + ≥ , ✤♣❝♠✳ 3+a 3+b 3+c 10 10 ❉♦ ✤â ❇➔✐ t♦→♥ ✸✳✶✶ ✳ ✭▼♦❧❞♦✈❛ ✶✾✾✾✮ ❈❤♦ a, b, c > ≥ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✿ ab bc ca a b c + + ≥ + + c(c + a) a(a + b) b(b + c) a + c b + a c + b ▲í✐ ❣✐↔✐✳ ❈→❝ ✈➳ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❧➔ ❝→❝ ❜✐➸✉ t❤ù❝ ❝ị♥❣ ❜➟❝ ✭❜➟❝ ❦❤ỉ♥❣✮✳ ❑❤ỉ♥❣ ♠➜t tê♥❣ q✉→t✱ t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ abc = ❑❤✐ ✤â ❜➜t ✤➥♥❣ t❤ù❝ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ trð t❤➔♥❤ (b + c)(b + a) (c + a)(c + b) (a + b)(a + c) + + c2 a2 b2 (b + a)(b + c) (c + a)(c + b) (a + b)(a + c) ≥ + + bc ca ab 1 b2 c2 a2 ⇔ (ab + bc + ca) + + + + + c a b c a b 1 b c a ≥ (ab + bc + ca) + + + + + bc ca ab c a b ❉♦ 1 1 1 + + ≥ + + a2 b2 c2 bc ca ab ✈➔ b c a2 b c a + + ≥ + + c2 a2 b2 c a b ♥➯♥ t❛ ❝â ✤♣❝♠✳ ❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❇➔✐ t♦→♥ ✸✳✶✷ a = b = c ✳ ✭❱▼❖ ✲ ✷✵✵✹✱ ❇↔♥❣ ❆✮ ❳➨t ❝→❝ sè t❤ü❝ ❞÷ì♥❣ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (x + y + z)3 = 32xyz ❍➣② t➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ✈➔ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝✿ x4 + y + z P = (x + y + z)4 Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ x, y, z t❤ä❛ ✺✵ ▲í✐ ❣✐↔✐✳ α ❈→❝❤ ✶✳ ◆❤➟♥ ①➨t r➡♥❣ ✈ỵ✐ ởt số tỹ ữỡ tũ ỵ t ổ õ P (x, y, z) = P (αx, αy, αz) ✈➔ ♥➳✉ x, y, z t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ✤➲ ❜➔✐ t❤➻ αx, αy, αz ❝ô♥❣ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤â✳ ❱➻ t❤➳ ❦❤æ♥❣ ♠➜t tê♥❣ q✉→t✱ ❝â t❤➸ ❣✐↔ sû ❦❤✐ ✤â x + y + z = 4, xyz = ❇➔✐ t♦→♥ trð t❤➔♥❤✿ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ✈➔ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ x4 + y + z ) 256 ❦❤✐ x, y, z > t❤❛② ✤ê✐ s❛♦ ❝❤♦ x + y + z = 4, vxyz = 4 ✣➦t Q = x + y + z ✈➔ t = xy + yz + zx P = ❚❛ ❝â Q = (x2 + y + z )2 − 2(x2 y + y z + z x2 ) ⇔ Q = 42 − 2t − t2 − 2xyz(x + y + z) ⇔ Q = 2t2 − 64t + 44 + 32 = 2(t2 − 32t + 144) ❚ø ❣✐↔ t❤✐➳t t❛ ❝â✿ y + z = − x, yz = ❉♦ ✤â x (2) t = x(4 − x) + x ✭✸✮ ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❆▼ ✲ ●▼✿ √ y + z ≥ yz ⇔ x3 − 8x2 + 16x − ≥ x ⇔ (x −√2) x2 − 6x + ≥ ⇔ − ≤ x ≤ 2, ❞♦ x ∈ (0; 4) ⇒ (4 − x)2 ≥ ❳➨t ❤➔♠ sè ❚❛ ❝â t = x(4 − x) + (1) x tr➯♥ ✤♦↕♥ 3− √ 5; ✳ −2(x − 1) x2 − x − t (x) = , x2 ✈➔ 1± t (x) = ⇔ x = ∨ x = Số hóa trung tâm học liệu √ http://www.lrc-tnu.edu.vn/ ✺✶ ❙✉② r❛ √ 5−1 5≤t≤ 2 ❱➻ ❤➔♠ sè f (t) = t − 32t + 144 ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ √ √ 5−1 5−1 5; ⊂ (0; 16) ♥➯♥ tr➯♥ 5; t❛ ❝â✿ 2 f (t) = f √ 5−1 ❦❤♦↔♥❣ (0; 16) ✈➔ √ 383 − 165 = , max f (t) = f (5) = ✈➔ √ Q = 383 − 165 5✱ max Q = 18 √ √ 383 − 165 ❱➟② P = ✱ ✤↕t ✤÷đ❝ ❝❤➥♥❣ ❤↕♥ ❦❤✐ x = − 256 √ 1+ y=z= max P = ✱ ✤↕t ✤÷đ❝ ❝❤➥♥❣ ❤↕♥ ❦❤✐ x = 2, y = z = 128 t ủ ợ t ữủ ❑❤æ♥❣ ♠➜t tê♥❣ q✉→t✱ t❛ ❣✐↔ sû s✉② r❛ xyz = ✳ 32 ✣➦t x+y+z = t = ab + bc + ca = xy + yz + zx✳ ✈➔ ✈➔ tø ❣✐↔ t❤✐➳t ❑❤✐ ✤â x4 + y + z = (1 − 2t)2 − t2 − 2xyz = 2(1 − t)2 − ❚❤➳ ♥➯♥ ✤➸ t➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t✱ ♥❤ä ♥❤➜t ❝õ❛ ♥❤➜t✱ ♥❤ä ♥❤➜t ❝õ❛ P✱ t t = xy + yz + zx = y(1 − y) + ❉♦ t❛ ❝➛♥ t➻♠ ❣✐→ trà ❧ỵ♥ x + z = − y, xz = , (x + z)2 ≥ 4xz 32y (1 − y)2 ≥ 32y ♥➯♥ 8y √ 3− ●✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ♥➔② ❝❤♦ t❛ ♥❣❤✐➺♠ ≥y≥ ✱ t❛ ❝❤➾ √ 3− ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ t(y) ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ ❦❤♦↔♥❣ ; ✈➔ ❞♦ ✤â t❛ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✺✷ ❝â✿ √ 3− t ≥ t(y) t ứ õ t ữủ tr ợ t ♥❤ä ♥❤➜t ❝õ❛ P✳ ✸✳✸ ❇➜t ✤➥♥❣ t❤ù❝ ❝õ❛ ❝→❝ ❞↕♥❣ ♣❤➙♥ t❤ù❝ ❱➼ ❞ö ✸✳✶✳ ❈❤♦ a, b, c ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ abc = 1✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ 1 + + ≥ a(b + 1) b(c + 1) c(a + 1) ▲í✐ ❣✐↔✐✳ ❱➻ x y a = ,b = , y z a, b, c ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ ❝â abc = ♥➯♥ t❛ ❝â t❤➸ ✤➦t z ✈➔ c = ✱ ợ x, y, z số tỹ ữỡ ✤â✱ ❜➜t ✤➥♥❣ y t❤ù❝ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ trð t❤➔♥❤ 1 + + ≥ x y y z z x +1 +1 +1 y z z x x y yz zx xy ⇔ + + ≥ x(y + z) y(z + x) z(x + y) yz zx xy ⇔ + + ≥ xy + zx yz + xy zx + yz ❤❛② ✈ỵ✐ u v w + + ≥ , v+w w+u u+v u = yz, v = zx, w = xy ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣✳ ✣➙② ❝❤➼♥❤ ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ◆❡s❜✐t ❝❤♦ ❜❛ sè ❞÷ì♥❣✳ ❉♦ ✤â✱ t õ ự ú ỵ ố ✈ỵ✐ ♠ët sè ❜➔✐ t♦→♥ ❝❤ó♥❣ t❛ ❧↕✐ ❣➦♣ ❝→❝ ❜✐➸✉ t❤ù❝ ❱➼ ❞ö ✸✳✷✳ ❈❤♦ a b c , , ✱ b c a a b c tr♦♥❣ ✤â a, b, c ❧➔ ❝→❝ sè t❤ü❝ ❦❤→❝ ✵✳ ❑❤✐ ✤â✱ t❛ ✤➦t x = ,y = ,z = b c a a b c ✤➸ ✤÷❛ ❜➔✐ t♦→♥ ✈➲ ❝→❝ ❜✐➳♥ ♠ỵ✐ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ xyz = = 1✳ b c a a, b, c ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ b c a + + ≤ a + 2b b + 2c c + 2a Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✺✸ ▲í✐ ❣✐↔✐✳ ❇➜t ✤➥♥❣ t❤ù❝ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ tữỡ ữỡ ợ a +2 b t a b c x = ,y = ,z = b c a t❤➻ + + c ≤1 b + +2 a c x, y, z ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ ❝â t➼❝❤ ✭✸✳✺✮ xyz = 1✳ ❑❤✐ ✤â✱ ❜➜t tự ữủ t ữợ 1 + + ≤1 x+2 y+2 z+2 ❤❛② (x + 2)(y + 2) + (y + 2)(z + 2) + (z + 2)(x + 2) ≤ (x + 2)(y + 2)(z + 2) ⇔ (xy + yz + zx) + 4(x + y + z) + 12 ≤ xyz + 4(x + y + z) +2(xy + yz + zx) + ⇔ ≤ xyz + xy + yz + zx ⇔ ≤ xy + yz + zx (✈➻ xyz = 1) ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❆▼✲●▼ ❝❤♦ ❜❛ sè ❞÷ì♥❣✱ t❛ ❝â xy + yz + zx ≥ (xyz)2 = 3 ❚ø ✤â✱ s✉② r❛ ❜➜t ✤➥♥❣ t❤ù❝ ✭✸✳✺✮ ✤ó♥❣ ✈➔ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❱➼ ❞ö ✸✳✸✳ ❈❤♦ a, b, c ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ab bc ca a b c + + ≥ + + c(c + a) a(a + b) b(b + c) c + a a + b b + c ▲í✐ ❣✐↔✐✳ t tự ự tữỡ ữỡ ợ b a c b a c a b c + + ≥ + + c c+a a a+b b b+c c+a a+b b+c ❤❛② b c a 1 1 c + a + ≥ c +a + c +1 a +1 b b +1 +1 b +1 + a b a b c c ✣➦t x= a b c ,y = ,z = b c a t❤➻ Số hóa trung tâm học liệu x, y, z ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ ❝â t➼❝❤ http://www.lrc-tnu.edu.vn/ ✭✸✳✻✮ xyz = 1✳ ✺✹ ❑❤✐ ✤â y z x 1 + + ≥ + + z+1 x+1 y+1 z+1 x+1 y+1 ⇔ y(x + 1)(y + 1) + z(y + 1)(z + 1) + x(z + 1)(x + 1) ≥ (x + 1)(y + 1) + (y + 1)(z + 1) + (z + 1)(x + 1) 2 ⇔ (xy + yz + zx2 ) + (x2 + y + z ) + (xy + yz + zx) + (x + y + z) ≥ (xy + yz + zx) + 2(x + y + z) + 2 2 ⇔ (xy + yz + zx ) + (x + y + z ) ≥ (x + y + z) + ⇔ (x − 1)2 + (y − 1)2 + (z − 1)2 + (x + y + z − 3) +(xy + yz + zx2 − 3) ≥ (3.6) ⇔ ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❆▼✲●▼ ❝❤♦ ❜❛ sè ❞÷ì♥❣✱ t❛ ❝â √ x + y + z ≥ 3 xyz = ✈➔ xy + yz + zx2 ≥ 3 (xyz)3 = ❚ø ✤â s✉② r❛ ❜➜t ✤➥♥❣ t❤ù❝ ✭✸✳✻✮ ✤ó♥❣✳ ❉♦ ✤â✱ t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ▼ët sè ❜➔✐ t♦→♥ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ ❝❤ù❛ ❜❛ ❜✐➳♥ ❞↕♥❣ ✤è✐ ①ù♥❣ ❚❛ ❝â t❤➸ sû ❞ö♥❣ ♣❤➨♣ ✤ê✐ ❜✐➳♥✿ x = a + b + c; y = ab + bc + ca; z = abc ❈→❝ ✤➥♥❣ t❤ù❝ t❤÷í♥❣ sû ❞ö♥❣ xy − z = (a + b)(b + c)(c + a) x2 + y = (a + b)(b + c) + (b + c)(c + a) + (c + a)(a + b) x2 − 2y = a2 + b2 + c2 x3 − 3xy + 3z = a3 + b3 + c3 ✭✸✳✼✮ ✭✸✳✽✮ ❈→❝ ❜➜t ✤➥♥❣ t❤ù❝ t❤÷í♥❣ sû ❞ö♥❣ x2 ≥ 3y x3 ≥ 27z y ≥ 3xz xy ≥ 9z x3 − 4xy + 9z ≥ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✭✸✳✾✮ ✭✸✳✶✵✮ ✺✺ ❱➼ ❞ö ✸✳✹✳ 3✳ ❈❤♦ a, b, c ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ab+bc+ca = ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ 1 a+b+c + + ≥ + a+b b+c c+a a+b+c ▲í✐ ❣✐↔✐✳ ❇➜t ✤➥♥❣ t❤ù❝ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ ✤÷ì♥❣ ✈ỵ✐ (a + b)(b + c) + (b + c)(c + a) + (c + a)(a + b) a + b + c ≥ + (a + b)(b + c)(c + a) a+b+c ✭✸✳✶✶✮ ✣➦t x = a + b + c, y = ab + bc + ca, z = abc✳ ❚ø ✭✸✳✼✮ ✈➔ ✭✸✳✾✮✱ t❛ ❝â ✭✸✳✶✶✮ trð t❤➔♥❤ x2 + y x ≥ + ⇔ (x2 + 3)6x − (x2 + 18)(3x − z) ≥ xy − z x ⇔ 3x3 − 36x + x2 z + 18z ≥ ⇔ 3(x3 − 12x + 9z) + x2 z − 9z ≥ ⇔ 3(x3 − 12x + 9z) + z(x2 − 9) ≥ ❉♦ y=3 ♥➯♥ tø ✭✸✳✶✶✮✱ s✉② r❛ x ≥ 9✱ ❦➳t ❤đ♣ ✈ỵ✐ ✭✸✳✾✮✱ t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✤ó♥❣✱ s✉② r❛ ❜➔✐ t♦→♥ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ a = b = c = 1✳ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✺✻ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✧✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ✈➔ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ❧✐➯♥ q✉❛♥✧ ✤➣ tr➻♥❤ ❜➔② ỳ t q s ợ t ỡ s ỵ t❤✉②➳t ❝õ❛ ❝→❝ ✤↕✐ sè ✈➔ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ì ❜↔♥ ✈➔ ♠ët sè ù♥❣ ❞ư♥❣ ❝õ❛ ♥â tr♦♥❣ ✤↕✐ sè ❝➜♣✳ ✲ ❈→❝ ✈➜♥ ✤➲ ❝õ❛ ỵ tt ữủ tr ởt ỡ t ❝→❝ tr÷í♥❣ ❤đ♣ ❤❛✐ ❜✐➳♥✱ ❜❛ ❜✐➳♥ ✤➳♥ ♥❤✐➲✉ ❜✐➳♥✳ ✲ ❚r➻♥❤ ❜➔② ❝→❝ ❜➔✐ t♦→♥ ❧♦↕✐ ❦❤â✱ ♥❤✐➲✉ ❜➔✐ t♦→♥ ✤÷đ❝ tr➼❝❤ r❛ tø ❝→❝ ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ q✉è❝ ❣✐❛✱ ❖❧②♠♣✐❝ t♦→♥ q✉è❝ t➳✱ ■▼❖✳ ✳ ✳ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/ ✺✼ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ◆❣✉②➵♥ ❱➠♥ ▼➟✉✱ ✣❛ t❤ù❝ ✤↕✐ sè ✈➔ ♣❤➙♥ t❤ù❝ ❤ú✉ t✛✱ ◆❳❇ ●✐→♦ ❞ö❝✱ ✷✵✵✹✳ ❬✷❪ ◆❣✉②➵♥ ❱➠♥ ▼➟✉✱ ❇➜t ✤➥♥❣ t❤ù❝ ✲ ✣à♥❤ ❧➼ ✈➔ →♣ ❞ö♥❣✱ ◆❳❇ ●✐→♦ ❞ö❝✱ ✷✵✵✻✳ ❬✸❪ ◆❣✉②➵♥ ❱➠♥ ▼➟✉✱ ◆❣✉②➵♥ ❱➠♥ ◆❣å❝✱ ❈❤✉②➯♥ ✤➲ ❝❤å♥ ❧å❝✿ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ✈➔ →♣ ❞ö♥❣✱ ◆❳❇ ●✐→♦ ❞ö❝✱ ✷✵✵✽✳ ❬✹❪ P❤❛♥ ❍✉② ❑❤↔✐✱ P❤÷ì♥❣ tr➻♥❤ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè✱ ◆❳❇ ❑❤♦❛ ❤å❝ tü ♥❤✐➯♥ ✈➔ ❝æ♥❣ ♥❣❤➺✱ ✷✵✵✾✳ Số hóa trung tâm học liệu http://www.lrc-tnu.edu.vn/