✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆ ▲➊ ❍×❒◆● ❚❍❷❖ P❍×❒◆● P❍⑩P ❍⑨▼ ❱⑨ Ù◆● ❉Ư◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙ß ❑❍❖❆ ❍➴❈ ❍⑨ ◆❐■ ✲ ◆❿▼ ✷✵✶✺ ✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆ ▲➊ ❍×❒◆● ❚❍❷❖ P❍×❒◆● P❍⑩P ❍⑨▼ ❱⑨ Ù◆● ❉Ư◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙ß ❑❍❖❆ ❍➴❈ ❈❤✉②➯♥ ♥❣➔♥❤ ✿ P❤÷ì♥❣ ♣❤→♣ t♦→♥ ❝➜♣ ▼➣ sè ✿ ✻✵✹✻✵✶✶✸ ◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈ P●❙✳❚❙ ◆❣✉②➵♥ ✣➻♥❤ ❙❛♥❣ ❍⑨ ◆❐■ ✲ ◆❿▼ ✷✵✶✺ ▼ư❝ ❧ư❝ ▲í✐ ♠ð ✤➛✉ ✸ ❇↔♥❣ ❦➼ ❤✐➺✉ ✺ ✶ ❑✐➳♥ t❤ù❝ ỵ ỡ ❤➔♠ ❦❤↔ ✈✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ✤✐➸♠ ❝ü❝ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✷ ỵ rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✸ ✣à♥❤ ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✹ ✣à♥❤ ỵ r ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❈æ♥❣ t❤ù❝ ❚❛②❧♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✶ ❈æ♥❣ t❤ù❝ r ợ số ữ r ổ tự r ợ số ữ P tr ❧ỵ♥ ♥❤➜t✱ ❣✐→ trà ♥❤ä ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✶ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✷ P❤÷ì♥❣ ♣❤→♣ t➻♠ ●❚▲◆✱ ●❚◆◆ ✳ ✳ ✳ ✳ ✳ ✳ ✷ Ù♥❣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❤➔♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✻ ✻ ✻ ✼ ✽ ✽ ✽ ✾ ✶✶ ✶✶ ✶✸ ✶✹ ✷✳✶ P❤÷ì♥❣ ♣❤→♣ ❤➔♠ tr♦♥❣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶✳✶ Ù♥❣ ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ❚❛②❧♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶ ✷✳✶✳✷ Ù♥❣ ỵ ỡ ✳ ✷✳✷ P❤÷ì♥❣ ♣❤→♣ ❤➔♠ tr♦♥❣ ❣✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ✳ ✳ ✳ ✷✳✷✳✶ ❈ì sð ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✷ ⑩♣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ P❤÷ì♥❣ ♣❤→♣ ❤➔♠ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ ✷✳✸✳✶ ❈ì sð ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✷ ⑩♣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✺✶ ✺✶ ✺✷ ✺✼ ✺✼ ✺✼ ✸ ●✐↔✐ ✈➔ ❜✐➺♥ ❧✉➟♥ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ t❤❛♠ sè ✻✸ ✸✳✶ ❈ì sð ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✸✳✷ ⑩♣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ❑➳t ❧✉➟♥ ✻✾ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✼✵ ✷ ▲í✐ ♠ð ✤➛✉ P❤÷ì♥❣ ♣❤→♣ ❤➔♠ ✤â♥❣ ♠ët ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ❣✐↔✐ t➼❝❤ t♦→♥ ❤å❝ ✈➔ t❤÷í♥❣ ✤÷đ❝ ❦❤❛✐ t❤→❝ tr♦♥❣ ❝→❝ ❦➻ t❤✐ ❖❧②♠♣✐❝ q✉è❝ ❣✐❛✱ q✉è❝ t➳✱ ❦ý t❤✐ ❖❧②♠♣✐❝ s✐♥❤ ✈✐➯♥✳ ✣➙② ❧➔ ♠ët ❝ỉ♥❣ ❝ư r➜t ❤✐➺✉ ❧ü❝ tr♦♥❣ ✈✐➺❝ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ sü tỗ t t t ❞↕♥❣ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❦❤→❝ ♥❤❛✉✳ ❱ỵ✐ s✉② ♥❣❤➽ ✤â✱❝❤ó♥❣ tỉ✐ ✤➣ ❝❤å♥ ✤➲ t➔✐✿ ✧P❤÷ì♥❣ ♣❤→♣ ❤➔♠ ✈➔ ù♥❣ ❞ư♥❣✧ ✤➸ ❧➔♠ ❧✉➟♥ ✈➠♥ ❝õ❛ ♠➻♥❤✳ ▲✉➟♥ ✈➠♥ ♥➔② tr➻♥❤ ❜➔② t÷ì♥❣ ✤è✐ ✤➛② ✤õ ❝→❝ t➼♥❤ ❝❤➜t ❤➔♠ ❦❤↔ ✈✐ ✈➔ ù♥❣ ❞ư♥❣ ❝õ❛ ❝❤ó♥❣ ✈➔♦ ✈✐➺❝ ❦❤↔♦ s→t t➼♥❤ ❝❤➜t ♥❣❤✐➺♠ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✱❜➜t ♣❤÷ì♥❣ tr➻♥❤✳ ❇↔♥ ❧✉➟♥ ỗ ữỡ t t ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✈➔ ♠ư❝ ❧ư❝✿ ❈❤÷ì♥❣ ✶ ✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✿ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❦✐➳♥ t❤ù❝ ❝➛♥ t❤✐➳t ❝❤♦ ❝❤÷ì♥❣ s❛✉ ♥❤÷ ✿ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✈➲ ❤➔♠ ❦❤↔ ✈✐ ❝õ❛ ❤➔♠ ♠ët ❜✐➳♥ ♠➔ trå♥❣ t ỵ ỡ ✈✐ ✈➔ ❝ỉ♥❣ t❤ù❝ ❚❛②❧♦r✳ ❈❤÷ì♥❣ ✷ ✿ ◆❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ t♦→♥ ❝â ù♥❣ ❞ư♥❣ ❦➳t q✉↔ tr♦♥❣ ❝❤÷ì♥❣ ■ t❛ ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❤➔♠✳ ▼ư❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ ✿ Ù♥❣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❤➔♠ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ ✸ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ✤➥♥❣ t❤ù❝✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② s➩ →♣ ❞ư♥❣ ❦❤❛✐ tr✐➸♥ ❚❛②❧♦r ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛✱ ❜➟❝ ❜è♥✱ sû ❞ư♥❣ t ỡ ỵ r ỵ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ✤➥♥❣ t❤ù❝✳ ❈❤÷ì♥❣ ✸ ✿ ●✐↔✐ ✈➔ ❜✐➺♥ ❧✉➟♥ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ t❤❛♠ sè✿ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❝→❝ ù♥❣ ❞ư♥❣✱ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ♥❤÷ ð ❝❤÷ì♥❣ ■■ ❝ë♥❣ t❤➯♠ ♠ët ữỡ ợ ữỡ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ t❤❛♠ sè✳ ✣➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔② ❡♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ tỵ✐ ♥❣÷í✐ t❤➛② ❦➼♥❤ ♠➳♥ P●❙✳❚❙ ◆❣✉②➵♥ ✣➻♥❤ ❙❛♥❣ ✤➣ ❞➔♥❤ tớ ữợ tr sốt tớ ❣✐❛♥ ①➙② ❞ü♥❣ ✤➲ t➔✐ ❝❤♦ ✤➳♥ ❦❤✐ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ tỵ✐ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✲ ❈ì ✲ ❚✐♥ ❤å❝✱ ❇❛♥ ●✐→♠ ❍✐➺✉✱ P❤á♥❣ ❙❛✉ ✤↕✐ ❤å❝ tr÷í♥❣ ✣❍❑❍❚◆ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳ ▼➦❝ ❞ị ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ ♥❤÷♥❣ ❞♦ t❤í✐ ❣✐❛♥ ✈➔ ♥➠♥❣ ❧ü❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❜↔♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✱ r➜t t ổ õ ỵ ỹ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ ♥❣➔② ✷✺ t❤→♥❣ ✾ ♥➠♠ ✷✵✶✺ ❍å❝ ✈✐➯♥ ▲➯ ❍÷ì♥❣ ❚❤↔♦ ✹ ❇↔♥❣ ❝→❝ ❦➼ ❤✐➺✉ ✈✐➳t t➢t N N∗ Z Z+ Z− R R∗ R+ R− i C ❚➟♣ ❝→❝ sè tü ♥❤✐➯♥ ❚➟♣ ❝→❝ sè tü ♥❤✐➯♥ ❦❤→❝ ✵ ❚➟♣ ❝→❝ sè ♥❣✉②➯♥ ❚➟♣ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣ ❚➟♣ ❝→❝ sè ♥❣✉②➯♥ ➙♠ ❚➟♣ ❝→❝ sè t❤ü❝ ❚➟♣ ❝→❝ sè t❤ü❝ ❦❤→❝ ✵ ❚➟♣ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ ❚➟♣ ❝→❝ sè t❤ü❝ ➙♠ ✣ì♥ ✈à ↔♦ ❚➟♣ ❝→❝ sè ♣❤ù❝ ✺ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ❈→❝ ✤à♥❤ ỵ ỡ ✤✐➸♠ ❝ü❝ trà ❈❤♦ ❦❤♦↔♥❣ (a, b) ⊂ R ✱ ❤➔♠ sè f : (a, b) → R✳ ❚❛ ♥â✐ r➡♥❣ ❤➔♠ f ✤↕t ❝÷❝ ✤↕✐ ✤à❛ ♣❤÷ì♥❣ ✭t÷ì♥❣ ù♥❣ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣✮ t↕✐ x0 ∈ (a, b)✱ ♥➳✉ tỗ t ởt số > s (x0 − δ, x0 + δ) ⊂✭❛✱❜✮ ✈➔ f (x) ≤ f (x0)tữỡ ự f (x) f (x0) ợ ♠å✐ x ∈ (x0 − δ, x0 + δ)✳ ❈ü❝ ✤↕✐ ✤à❛ ♣❤÷ì♥❣ ❤♦➦❝ ❝ü❝ t✐➸✉ ✤à❛ ♣❤÷ì♥❣ ❣å✐ ❝❤✉♥❣ ❧➔ ❝ü❝ trà ❝õ❛ ❤➔♠ f ✳ ✣✐➸♠ ✭x0, y(x0)✮ ỹ tr ỵ rt (a, b) ⊂ R ✱ ❤➔♠ sè f : (a, b) → R✳ ◆➳✉ ❤➔♠ sè ✤↕t ❝ü❝ trà t↕✐ x = c tỗ t f (c) t f (c) = ỵ sỷ f : [a, b] → R ❝â ❝→❝ t➼♥❤ ❝❤➜t✿ ✭✶✮f ❧✐➯♥ tö❝ tr➯♥ [a, b]✳ ✭✷✮ f ❦❤↔ ✈✐ tr♦♥❣ ❦❤♦↔♥❣ (a, b)✳ ✭✸✮ f (a) = f (b)✳ õ tỗ t t t ởt c ∈ (a, b) s❛♦ ❝❤♦ f (c) = 0✳ ✶✳✶✳✹ ỵ r sỷ [a, b] R ❝â ❝→❝ t➼♥❤ ❝❤➜t✿ ✭✶✮ f ❧✐➯♥ tö❝ tr➯♥ [a, b]✳ ✭✷✮ f ❦❤↔ ✈✐ tr♦♥❣ ❦❤♦↔♥❣ (a, b)✳ ❑❤✐ õ tỗ t t t ởt (a, b) s❛♦ ❝❤♦ ✿ f (b) − f (a) = f (c)(b a) t ỵ trữớ ủ t ỵ r q✉↔ ●✐↔ sû f : [a, b] → R ❧✐➯♥ tö❝ tr➯♥ ❬❛✱❜❪ ✈➔ ❦❤↔ ✈✐ tr♦♥❣ ❦❤♦↔♥❣ ❬❛✱❜❪✳ ❑❤✐ ✤â✿ ✭❛✮ ◆➳✉ f (x) = ✈ỵ✐ ∀x ∈ (a, b) t❤➻ f ❧➔ ❤➔♠ ❤➡♥❣ tr➯♥ ❬❛✱❜❪✳ ✭❜✮ ◆➳✉ f (x) ≥ 0(f (x) ≤ 0) ✈➔ f (x) = t↕✐ ❤ú✉ ❤↕♥ ✤✐➸♠ tr➯♥ ✭❛✱❜✮ t❤➻ ❢ t➠♥❣ ✭❣✐↔♠✮ t❤ü❝ sü tr➯♥ ❬❛✱❜❪✳ ❈❤ù♥❣ ♠✐♥❤✿ ❛✮ ●✐↔ sû a ≤ x1 ≤ x2 ≤ b✳ ❚❤❡♦ ỵ r tỗ t (a, b) s ❝❤♦✿ f (x2 ) − f (x1 ) = f (c)(x2 − x1 ) ✭✷✮✳ ❱➻ f (c) = 0✱ tø ✤â s✉② r❛ f (x2) = f (x1)✳ ❱➟② ❢ ❧➔ ❤➡♥❣ sè✳ ❜✮ ◆➳✉ f (x) ≥✵ ✈ỵ✐ ♠å✐ ① ∈ (a, b)✱ t❤➻ tø ✭✷✮ ❞♦ f (c) ≥✵ ◆➯♥ f (x2) − f (x1) ≥ 0✳ t ỵ ●✐↔ sû ❝→❝ ❤➔♠ ❢✱❣ ✿ [a, b] → R ❝â ❝→❝ t➼♥❤ ❝❤➜t ✿ ✭✶✮ ❢ ✈➔ ❣ ❧✐➯♥ tö❝ tr➯♥ ❬❛✱❜❪✳ ✭✷✮ ❢✱❣ ❦❤↔ ✈✐ tr➯♥ ✭❛✱❜✮✳ ❑❤✐ õ tỗ t (a, b) s [f (b) − f (a)]g (c) = [g(b) − g(a)]f (c)✳ ✭✸✮ ❍ì♥ ♥ú❛✱ ♥➳✉ g (x) ❦❤→❝ ✵ ✈ỵ✐ ♠å✐ ① ∈ (a, b) t❤➻ ❝æ♥❣ t❤ù❝ ✭✸✮ ❝â ❞↕♥❣✿ f (c) f (b) − f (a) = g (c) g(b)g(a) (4) t ỵ r trữớ ủ r ỵ ợ g(x) = x ✶✳✷ ❈ỉ♥❣ t❤ù❝ ❚❛②❧♦r ✶✳✷✳✶ ❈ỉ♥❣ t❤ù❝ ❚❛②❧♦r ✈ỵ✐ sè ❞÷ ❞↕♥❣ ▲❛❣r❛♥❣❡ ●✐↔ sû f : [a, b] → R ❝â ✤↕♦ ❤➔♠ ✤➳♥ ❝➜♣ ✭♥✰✶✮ tr♦♥❣ ❦❤♦↔♥❣ ✭❛✱❜✮✱ x0 ∈ (a, b)✳ ❑❤✐ ✤â✱ ✈ỵ✐ ∀x ∈ (a, b)✱ t❛ ❝â✿ n f (x) = k=0 f ( k)(x0 ) f (n+1) (c) k (x − x0 ) + (x − x0 )n+1 (1.4) k! (n + 1)! tr♦♥❣ ✤â ❝ ♥➡♠ ❣✐ú❛ ① ✈➔ x0✳ ◆❤➟♥ ①➨t✿ ❱➻ ❝ ♥➡♠ ❣✐ú❛ ① ✈➔ x0 ♥➯♥ ✭✶✳✹✮ ❝â t t ữợ s n f (x) = k=0 f ( k)(x0 ) f (n+1) (x0 + θ(x − x0 )) k (x − x0 ) + (x − x0 )n+1 (1.5) k! (n + 1)! ✽ ✷✳✸ P❤÷ì♥❣ ♣❤→♣ ❤➔♠ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ ✷✳✸✳✶ ❈ì sð ♣❤÷ì♥❣ ♣❤→♣ ❚r♦♥❣ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝✱ ♠ët sè ❜➜t ✤➥♥❣ t❤ù❝ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ❝â t❤➸ sỷ ữỡ t ữợ s • • ✣➦t ❤➔♠ ♣❤ư✳ ✣÷❛ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ ✈➲ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ ❤➔♠ ✈ỵ✐ ❜✐➳♥ ❝❤↕② tr➯♥ ❦❤♦↔♥❣ ♥➔♦ ✤â✳ ✷✳✸✳✷ ⑩♣ ❞ö♥❣ ❱➼ ❞ö ✶ ❈❤♦ a, b, c t❤✉ë❝ (0, 1)✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✿ a b c + + + (1 − a)(1 − b)(1 − c) < b+c+1 c+a+1 a+b+1 ▲í✐ ❣✐↔✐✿ ❳➨t ❤➔♠ f (x) = b + xc + + c + xb + + x + cb + +(1−x)(1−b)(1−c) ❚❛ ❝â ✿ f (x) = b + c1 + − (x + cb + 1)2 − (x + bc + 1)2 − (1 − b)(1 − c) 2b 2c + > ∀x ∈ (0, 1)✳ (x + c + 1)4 (x + b + 1)4 f (x) ỗ tr ✴(0, 1)✳ f ”(x) = ❉♦ ✤â ◆➳✉ f (x) ≤ t❤➻ ✿ + b + c + b2 c2 max f (x) = f (0) = ❳➨t ❤➔♠ sè f (x) = sin x + tan x − 2x✈ỵ✐ < x < π2 ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ f (x) > ✳ ❚❤➟t ✈➙②✿ f (x) = cos x + −2 cos2 x ❱➻0 < x < π2 ♥➯♥ < cos x < f (x) > cos2 x + s✉② r❛ cos x > cos2 x 1 π − ≥ cos − = ∀x ∈ (0, ) cos2 x cos2 x ❉♦ f (x) > ✈ỵ✐ x ∈ (0, π2 ) ♥➯♥ f (x) ❧➔ ❤➔♠ ỗ tr r f (x) > f (0) ❤❛② f (x) > ❤❛② sin x + tan x > 2x ▲➛♥ ❧÷đt t❤❛② x ❜ð✐ ❆✱❇✱ ❈ t❛ ✤÷đ❝ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ❱➼ ❞ö ✹ ✭✣❍ ❦❤è✐ ❆✲✷✵✶✷✮ ❈❤♦ x, y, z ∈ R t❤ä❛ ♠➣♥ x + y + z = 0✳❈❤ù♥❣ ♠✐♥❤✿ |x−y| +3 |y−z| +3 |z−x| − √ 6x2 + 62 + 6z rữợ ❤➳t t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ 3t > t + ∀t > ✭✯✮ ❳➨t ❤➔♠ f (t) = 3t − t − tr➯♥ [0, +∞] ❚❛ ❝â✿ f (t) = 3tln3 − > ∀t > r t ỗ tr [0, +∞] ❑❤✐ ✤â f (t) > f (0) = ⇐ (∗) ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ⑩♣ ❞ư♥❣ ✭✯✮ t❛ ❝â✿ 3|x−y| + 3|y−z| + 3|z−x| ≥ + |x − y| + |y − z| + |z − x|✳ ❙û ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ |a| + |b| ≥ |a + b| t❛ ❝â ✿ (|x − y| + |y − z| + |z − x|)2 = |x − y|2 + |y − z|2 + |z − x|2 + |x − y|(|y − z| + |z − x|) + |y − z|(|z − x| + |x − y|) + |z − x|(|x − y| + |y − z|) ≥ 2(|x − y|2 + |y − z|2 + |z − x|2 )✳ ❉♦ ✤â✿ |x−y|+|y−z|+|z−x| ≥ (|x − y|2 + |y − z|2 + |z − x|2 ) = 6x2 + 6y + 6z − ▼➔ x + y + z = s✉② r❛ |x − y| + |y − z| + |z − x| ≥ 6x2 + 6y2 + 6z 2✳ √ ❙✉② r❛ 3|x−y| + 3|y−z| + 3|z−x| − 6x2 + 62 + 6z ≥ 3✳ ✣✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❱➼ ❞ö ✺ ❈❤♦ a, b > s❛♦ ❝❤♦ a < b✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✿ b−a b b−a < ln < b a a ▲í✐ ❣✐↔✐✿ ❳➨t ❤➔♠ sè f (x) = ln x ✈ỵ✐ x > 0✳ ❚❛ ❝â ✿ f (x) = x1 ỵ r tỗ t c ∈ (a, b) s❛♦ ❝❤♦✿ f (c) = f (b) − f (a) ln b − ln a b−a b ⇔ = ⇔ = ln b−a c b−a c a ❉♦ < a < c < b ♥➯♥ t❛ ❝â✿ 1 b−a b−a b−a b−a b b−a < < ⇔ < < ⇔ < ln < ✳ b c a b c a b a a ❱➟② ❜➜t ✤➥♥❣ t❤ù❝ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✻✵ ❱➼ ❞ö ✻ ❈❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝✿ x3 < sin x x− 3! x>0 ▲í✐ ❣✐↔✐✿ x3 ❇➜t ✤➥♥❣ t❤ù❝ t÷ì♥❣ ✤÷ì♥❣ ✿ x − sin x < 3! t3 ❳➨t ❤➔♠ f (t) = t − sin t ✈➔ g(t) = 3! ❦❤↔ ✈✐ tr➯♥ (0, +∞✮✳ x (0, +) t ỵ tỗ t t0 ∈ (0, x) s❛♦ ❝❤♦✿ ❱ỵ✐ ♠å✐ f (t0 ) f (x) − f (0) − cos t0 x − sin x = ⇔ = g (t0 ) g(x) − g(0) t20 x3 3! 2 sin (t0 /2) t0 sin(t0 /2) = ❱➻ − tcos =( ) c ♥❣❤✐➺♠ ✤ó♥❣ ✈ỵ✐ ♠å✐ x t❤✉ë❝ [a, b] ❦❤✐ ✈➔ ❝❤➾ ❦❤✐✿ f (x) > c [a,b] ✺✮ ❇➜t ữỡ tr f (x) c ú ợ c t❤✉ë❝ [a, b] ❦❤✐ ✈➔ ❝❤➾ ❦❤✐✿ max f (x) ≤ c [a,b] ✻✸ ✸✳✷ ⑩♣ ❞ö♥❣ ❱➼ ❞ö ✶ ❚➻♠ ♠ ✤➸ ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠✿ √ x2 + x + − x2 − x + = m ▲í✐ ❣✐↔✐✿ √ √ ✣➦t f (x) = x2 + x + − x2 − x + √ √ (2x + 1) x2 − x + − (2x − 1) x2 + x + √ √ f (x) = x2 + x + x2 − x + √ √ f (x) = ⇔ (2x + 1) x2 − x + = (2x − 1) x2 + x + 1✭✈æ ♥❣❤✐➺♠✮✳ ❚❛ t❤➜② f (0) = ♥➯♥ f (x) > limx→−∞ f (x) = −1 ∀x limx→+∞ f (x) = −1 ❉♦ ✤â ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ −1 < m < 1✳ ❱➼ ❞ö ✷ ❚➻♠ ♠ ✤➸ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ❝â ✹ ♥❣❤✐➺♠✿ 2|x2 − 5x + 4| = x2 − 5x + m (1) ▲í✐ ❣✐↔✐✿ ✣➦t t = x2 − 5x + ❑❤✐ ✤â ✭✶✮⇔ 2|t| − t = m − ✭✷✮ ❚❤➜② ✭✷✮ ❝â ♥❤✐➲✉ ♥❤➜t ✷ ♥❣❤✐➺♠ ♥➯♥ ♣❤÷ì♥❣ tr➻♥❤ t = x2 − 5x + ♣❤↔✐ ❝â ✷ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t✳❚ù❝ t > −9 õ tữỡ ữỡ ợ t t = m − ⇔m>4 ❍♦➦❝ ✿ t < t = − m > −9 4 ⇔m< 43 ✻✹ ❱➟② ♣❤÷ì♥❣ tr➻♥❤ ❝â ✹ ♥❣❤✐➺♠ ♥➳✉ m > ❤♦➦❝ m < 43 ❱➼ ❞ö ✸ ●✐↔✐ ✈➔ ❜✐➺♥ ❧✉➟♥ t❤❡♦ ♠ sè ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✿ x+m √ =m x2 + ▲í✐ ❣✐↔✐✿ ❚❛ ❝â✿ √ √ x+m + − 1)m ⇔ x( x2 + + 1) = x2 m = m ⇔ x = ( x x2 + x=0 √ ⇔ x2 + + f (x) = =m x ❳➨t f (x) = 2√−12 < ∀x = x x +1 limx→−∞ f (x) = −1 limx→+∞ f (x) = limx→0+ f (x) = +∞ limx→0− f (x) = −∞ ợ t ữỡ tr õ x = ợ tở ữỡ tr õ t x > ợ tở ữỡ tr ❝â t❤➯♠ ♥❣❤✐➺♠ x < 0✳ ❱➼ ❞ö ✹ ●✐↔✐ ✈➔ ❜✐➺♥ ❧✉➙♥ ♣❤÷ì♥❣ tr➻♥❤✿ 2x +2mx+4 − 22x +4mx+1 = x2 + 2mx − ▲í✐ ❣✐↔✐✿ P❤÷ì♥❣ tr➻♥❤ t÷ì♥❣ ✤÷ì♥❣✿ 2x +2mx+4 − 22x +4mx+1 = 2x2 + 4mx + − (x2 + 2mx + 4) ✻✺ ✣➦t u = 2x2 + 4mx + v = x2 + 2mx + ✐♥❞❡♥t ❑❤✐ ✤â 2v − 2u = u − v ⇔ 2u + u = 2v + v ⇔ u = v ✭❉♦ f (t) = 2t + t ❧➔ ❤➔♠ ỗ s r 2x2 + 4mx + = x2 + 2mx + ⇔ x2 + 2mx = ợ ữỡ tr➻♥❤ ❧✉æ♥ ❝â ♥❣❤✐➺♠ x = −m ± m2 + ❱➼ ❞ư ✺ ❚➻♠ ♠ ✤➸ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ❝â ♥❣❤✐➺♠ t❤✉ë❝ [0, + √3]✿ m( x2 − 2x + + 1) + x(2 − x) ≤ (1) ▲í✐ ❣✐↔✐✿ √ ✣➦t t = x2 − 2x + ≥ s✉② r❛ −x(x − 2) = t2 − ❚❛ ❝â t = √ 2x − ✱ t = ⇔ x = 1✳ x − 2x + ▲➟♣ ❜↔♥❣ ❜✐➳♥ t❤✐➯♥✿ ❚ø ❜↔♥❣ ❜✐➳♥ t❤✐➯♥ s✉② r❛ ≤ t ≤ 2 ❑❤✐ ✤â ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ✭✶✮ trð t❤➔♥❤ ✿ m(t+1) ≤ t2 −2 ⇔ m ≤ tt +−12 = f (t) ✭✷✮ t2 + 2t + ❚❛ ❝â f (t) = (t + 1)2 > ∀t [1, 2] õ f (t) ỗ ❜✐➳♥✳ √ ❱➟② ✤➸ ✭✶✮ ❝â ♥❣❤✐➺♠ t❤✉ë❝ [0, + t❤➻ ✭✷✮ ❝â ♥❣❤✐➺♠ t t❤✉ë❝ ❬✶✱✷❪✳ ❚ù❝ ✿ m ≤ max f (t) = f (2) = [1,2] ✳ ✻✻ ❱➼ ❞ö ✻ ❚➻♠ ❝→❝ ❣✐→ trà ❝õ❛ ♠ ✤➸ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ❝â ♥❣❤✐➺♠✿ mx − √ x − ≤ m + (1) ▲í✐ ❣✐↔✐✿ √ ✣➦t t = x − ≥ ⇐ x = t2 + ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ✭✶✮ trð t❤➔♥❤✿ m(t2 + 3) − t ≤ m + ⇔ m ≤ t+1 = f (t) (2) t2 + √ −t2 − 2t + ✿f (t) = (t + 1)2 = ⇔ t = −1 ± ❈â ▲➟♣ ❜↔♥❣ ❜✐➳♥ t❤✐➯♥✿ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ✭✶✮ ❝â ♥❣❤✐➺♠ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ✭✷✮ ❝â ♥❣❤✐➺♠ t t❤✉ë❝ ❬✵✱+∞]✳ ❚ù❝ ❧➔ ✿ m ≤ max f (t) = f (1 + [0,+∞] ✳ √ √ 3) = 3+1 ❱➼ ❞ư ✼ ❚➻♠ ❦ ✤➸ ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ❝â ♥❣❤✐➺♠ ✿ |x − 1|3 − 3x − k < (1) log2 x2 + log2 (x − 1)3 ≤ (2) ▲í✐ ❣✐↔✐✿ ✣✐➲✉ ❦✐➺♥ ✿ x > 1✳ ❑❤✐ x > t❤➻ ✭✷✮⇔ log2 x + log2 (x − 1) ≤ ⇔ x(x − 1) ≤ ⇔ ✻✼ x2 − x − ≤ ⇔ −1 ≤ x ≤ ❱➻ x > ⇐ < x ≤ 2✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ✭✶✮⇔ (x − 1)3 − 3x < k✳ ✣➦t f (x) = (x − 1)3 − 3x ❝â f (x) = 3(x − 1)2 − = 3x(x − 2) ❱ỵ✐ < x ≤ ⇐ f (x) ≤ ❙✉② r❛ ❤➔♠ ❢✭①✮ ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ ✭✶✱✷❪✳ ❑❤✐ ✤â min(1,2]f (x) = f (2) = −5✳ ✣➸ ❤➺ ❝â ♥❣❤✐➺♠ t❤➻ k > −5 ❇➔✐ t➟♣ t❤❛♠ ❦❤↔♦ ❜➔✐ t➟♣ ✶ ❚➻♠ ♠ ✤➸ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ❝â ♥❣❤✐➺♠✿ √ √ √ √ x x + x + 12 = m( − x + − x) √ ✣→♣ sè m ∈ [2( 15 − √ 12), 12] ❇➔✐ t➟♣ ✷ ✭❑❤è✐ ❆ ✲✷✵✵✼✮ ❚➻♠ ♠ ✤➸ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ❝â ♥❣❤✐➺♠ t❤ù❝✿ √ x − + m x + = x2 − √ ✣→♣ sè −1 < m ≤ 31 ❇➔✐ t➟♣ ✸ ❚➻♠ t ữỡ tr ú ợ tở ❘✿ m4x + (m − 1)2x+2 + m − > ✣→♣ sè m > 1✳ ✻✽ ❑➳t ❧✉➟♥ ❙❛✉ t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ t↕✐ ❦❤♦❛ ❚♦→♥ ✲ ❈ì ✲ ❚✐♥ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ❚ü ♥❤✐➯♥✱ ✣❍◗● ❍➔ ◆ë✐✳ ✣÷đ❝ ❝→❝ t❤➛② ❝ỉ trü❝ t✐➳♣ ❣✐↔♥❣ ữợ t P ❙❛♥❣✱ ❡♠ ✤➣ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ✈ỵ✐ ✤➲ t➔✐ ✧P❤÷ì♥❣ ♣❤→♣ ❤➔♠ ✈➔ ù♥❣ ❞ư♥❣✧✳ ▲✉➟♥ ✈➠♥ ✤➣ ✤↕t ✤÷đ❝ ♠ët sè ❦➳t q✉↔✿ ✶✳ ▲✉➟♥ ✈➠♥ ✤➣ ❦❤❛✐ t❤→❝ ✤÷đ❝ ❝→❝ ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❤➔♠ ✈➔♦ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ❤å❝ ♣❤ê t❤ỉ♥❣ ❦❤→ ❤✐➺✉ q✉↔ ✈➔ ❧í✐ ❣✐↔✐ ✤➭♣✱ t↕♦ ✤÷đ❝ ♥✐➲♠ ✤❛♠ ♠➯ t➻♠ tá✐ ✈➔ s→♥❣ t↕♦ ❤å❝ t➟♣ t♦→♥ ❝õ❛ ❤å❝ s✐♥❤✳ ✷✳ ▲✉➟♥ ✈➠♥ ✤➣ ❤➺ t❤è♥❣ ❤â❛ ✈➔ ♣❤➙♥ ❧♦↕✐ ✤÷đ❝ ❝→❝ ❞↕♥❣ t♦→♥ ❝ì ❜↔♥ ợ ữỡ ❣✐↔✐ ♣❤♦♥❣ ♣❤ó ❦➧♠ t❤❡♦ ❝→❝ ❜➔✐ t➟♣ t❤❛♠ ❦❤↔♦ ✤÷đ❝ tr➼❝❤ tø ❝→❝ ❦➻ t❤✐ ❣✐ä✐ t♦→♥ q✉è❝ ❣✐❛✱ t❤✐ ♦❧②♠♣✐❝ t♦→♥ q✉è❝ t➳✱ t❤✐ ✤↕✐ ❤å❝✱ ✈➻ ✈➟② ❜↔♥ ❧✉➟♥ ✈➠♥ ❝â t❤➸ ❧➔♠ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤♦ ❤å❝ s✐♥❤ ❝→❝ ❧ỵ♣ ❝❤✉②➯♥ t♦→♥ ♣❤ê t❤ỉ♥❣ ✈➔ s✐♥❤ ✈✐➯♥ ♥➠♠ ♥❤➜t ❝→❝ tr÷í♥❣ ❦❤♦❛ ❤å❝ ❝ì ❜↔♥✳ t ữủ ữợ ự s→♥❣ t↕♦ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❤➔♠✳ ✹✳ ❍✐➺♥ ♥❛② ♣❤÷ì♥❣ ♣❤→♣ ❤➔♠ ❝á♥ ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ ❦❤→❝ ♥ú❛ ❝➛♥ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉✳ ✻✾ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ✈✐➺t ❬✶✳❪ ❚æ ❱➠♥ ❇❛♥✱ ●✐↔✐ t➼❝❤ ♥❤ú♥❣ ❜➔✐ t➟♣ ♥➙♥❣ ❝❛♦✱ ◆❳❇ ●✐→♦ ❉ö❝✱ ✷✵✵✺✳ ❬✷✳❪ ❚r➛♥ ✣ù❝ ▲♦♥❣✱ ◆❣✉②➵♥ ✣➻♥❤ ❙❛♥❣✱ ❍♦➔♥❣ ◗✉è❝ ❚♦➔♥✱ ●✐→♦ tr➻♥❤ ❣✐↔✐ t➼❝❤✱ ❇➔✐ t➟♣ ❣✐↔✐ t➼❝❤ ■✱ ■■✱ ◆❳❇ ✣❍◗● ❍➔ ◆ë✐✱ ✷✵✵✼✳ ❬✸✳❪ ◆❣✉②➵♥ ❱➠♥ ▼➟✉✱ ởt số t ỗ ữù s ❣✐ä✐ tr✉♥❣ ❤å❝ ♣❤ê t❤ỉ♥❣✱ ◆❳❇ ●✐→♦ ❉ư❝✱ ✷✵✶✵✳ ❬✹✳❪ ◆❣✉②➵♥ ❱➠♥ ▼➟✉✱ ❉➣② sè ✈➔ →♣ ❞ö♥❣✱ ✣❛ t❤ù❝ ✈➔ →♣ ❞ö♥❣✱ ◆❳❇ ●✐→♦ ❉ö❝✱ ✷✵✵✹✳ ❬✺✳❪ ✣♦➔♥ ◗✉ý♥❤✱ ❚r➛♥ ◆❛♠ ❉ơ♥❣✱ ◆❣✉②➵♥ ❱ơ ▲÷ì♥❣✱ ✣➦♥❣ ❍ị♥❣ ❚❤➢♥❣✱ ❚➔✐ ❧✐➺✉ ❝❤✉②➯♥ ✤➲ t♦→♥ ✤↕✐ sè ✈➔ ❣✐↔✐ t➼❝❤ ✶✶✱ ◆❳❇ ●✐→♦ ❉ö❝✱ ✷✵✶✵✳ ❬✻✳❪ ❚↕♣ ❝❤➼ t♦→♥ ❤å❝ t✉ê✐ tr➫✱ ❈→❝ ❜➔✐ t❤✐ ♦❧②♠♣✐❝ t♦→♥ tr✉♥❣ ❤å❝ ♣❤ê t❤æ♥❣ ❱✐➺t ◆❛♠✱ ◆❳❇ ●✐→♦ ❉ư❝✱ ✷✵✵✼✳ ❬✼❪✳ P❤ị♥❣ ✣ù❝ ❚❤➔♥❤✱ ▲✉➟♥ ✈➠♥ ✿ Ù♥❣ ❞ö♥❣ ✤↕♦ ❤➔♠ ✤➸ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ♣❤ê t❤æ♥❣✱ ✷✵✶✶✳ ❚✐➳♥❣ ❛♥❤ ❬✽✳❪ ❲✳❏✳❑❛❝❦♦r ✱ ▼✳❚✳◆♦✇❛r❦✱ Pr♦❜❧❡♠ ✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ❛♥❛❧②s✐s ■✱ ❘❡❛❧ ♥✉♠❜❡r✱ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s✱ ❆▼❙✱ ✷✵✵✵✳ ❬✶✶❪✳ ❲✳❏✳❑❛❝❦♦r✱ ▼✳❚✳◆♦✇❛❦✱ Pr♦❜❧❡♠ ✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ❛♥❛❧②s✐s ■■✱ ❘❡❛❧ ♥✉♠❜❡r✱ ❈♦♥✲t✐♥✉✐t② ❛♥❞ ❞✐❢❢❡r❡♥t✐❛t✐♦♥✱ ❆▼❙✱ ✷✵✵✶✳ ✼✵ ✼✶ ... t2 ✈ỵ✐ t1 ∈ (13, 1003) ✈➔ t2 ∈ (1023, 2 013) s❛♦ ❝❤♦✿ f (t1 )(1003 − 13) = f (1003) − f (13) = 1003x − 13x f (t2 )(2 013 − 1023) = f (2 013) − f (1023) = 2013x − 1023x ❙✉② r❛ ✿f (t1).990 = f (t2).990... s❛✉✿ 2013x + 13x = 1023x + 1003x ▲í✐ ❣✐↔✐✿ P❤÷ì♥❣ tr➻♥❤ t÷ì♥❣ ✤÷ì♥❣ ✿ 2013x − 1023x = 1003x − 13x✳ ❳➨t ❤➔♠ f (t) = tx t > ✱ ❧➔ ❤➔♠ ❦❤↔ ✈✐ tr➯♥ R+✱ t❤ä❛ ♠➣♥ tt ỵ r tỗ t t1, t2 ✈ỵ✐ t1 ∈ (13, 1003)... ●✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ✿ ✣→♣ sè x4 − 13x2 + 18x − = √ √ −3 ± ± x={ , } 2 ❇➔✐ t➟♣ ✷ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ✿ 2x4 − 10x3 + 11x2 + x − = ✷✽ ✣→♣ sè √ √ ± ± 13 ; } x={ 2 t ỗ t số s❛✉ ❝â trö❝