(Luận văn thạc sĩ) đẳng thức và bất đẳng thức trong lớp hàm logarit

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ◆●❯❨➍◆ ◆●➴❈ ◗❯❨➌◆ ✣➃◆● ❚❍Ù❈ ❱⑨ ❇❻❚ ✣➃◆● ❚❍Ù❈ ❚❘❖◆● ▲❰P ❍⑨▼ ▲❖●❆❘■❚ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✷✵ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ◆●❯❨➍◆ ◆●➴❈ ◗❯❨➌◆ ✣➃◆● ❚❍Ù❈ ❱⑨ ❇❻❚ ✣➃◆● ❚❍Ù❈ ❚❘❖◆● ▲❰P ❍⑨▼ ▲❖●❆❘■❚ ❈❤✉②➯♥ ♥❣➔♥❤✿ P❍×❒◆● P❍⑩P ❚❖⑩◆ ❙❒ ❈❻P ▼➣ sè✿ ✽ ✹✻ ✵✶ ✶✸ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ữớ ữợ ❱➠♥ ▼➟✉ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✷✵ ✐ ▼ö❝ ❧ö❝ ▼Ð ✣❺❯ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ ❧♦❣❛r✐t ✶ ✸ ✶✳✶ ▼ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ✣➦❝ tr÷♥❣ ❝õ❛ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷✳✶ ❍➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷✳✷ ❍➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✻ ✶✳✷✳✸ ❈→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✶✳✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ởt số q ợ ỗ ỗ rt ữỡ t❤ù❝ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ s✐➯✉ ✈✐➺t ❞↕♥❣ ❧♦❣❛r✐t ✶✹ ✷✳✶ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❞↕♥❣ ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷ P❤÷ì♥❣ tr➻♥❤ s✐➯✉ ✈✐➺t ❞↕♥❣ ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✸✳✶ P❤➨♣ ❝❤✉②➸♥ ✈➲ ❤➺ ✤↕✐ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✸✳✷ ❙û ❞ư♥❣ t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ❈❤÷ì♥❣ ✸✳ ❇➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❧ỵ♣ ❤➔♠ ❧♦❣❛r✐t ✸✳✶ ✸✳✷ t ữợ ữủ t tự ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✸✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✶✳✶ ❇➜t ✤➥♥❣ t❤ù❝ ❤➔♠ ❧♦❣❛r✐t ✸✳✶✳✷ P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❝❤ù❛ ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✹✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✷✳✶ ❇➔✐ t♦→♥ ❝ü❝ trà ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✷✳✷ ❇➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❞➣② sè ✈➔ ❣✐ỵ✐ ❤↕♥ ✺✻ ✸✳✷✳✸ Ù♥❣ ❞ư♥❣ ❤➔♠ ỗ rt tr ự ởt số t t ❦❤→❝ ❧✐➯♥ q✉❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑➳t ❧✉➟♥ ✻✵ ✻✻ ✶ ▼ð ✤➛✉ ❇➜t ✤➥♥❣ t❤ù❝ ❝â ✈à tr➼ ✤➦❝ ❜✐➺t q✉❛♥ trå♥❣ tr♦♥❣ t♦→♥ ❤å❝ ✈➔ ❧➔ ♠ët ❜ë ♣❤➟♥ q✉❛♥ trå♥❣ ❝õ❛ ❣✐↔✐ t➼❝❤ ✈➔ ✤↕✐ sè✳ ✣➥♥❣ t❤ù❝✱ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❧ỵ♣ ❤➔♠ ❧♦❣❛r✐t ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ♥ë✐ ❞✉♥❣ ❝ì ❜↔♥ ✈➔ q✉❛♥ trå♥❣ ❝õ❛ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ❜➟❝ tr✉♥❣ ❤å❝ ♣❤ê t❤ỉ♥❣✳ ❈❤✉②➯♥ ✤➲ ♥➡♠ tr♦♥❣ ❝❤÷ì♥❣ tr ỗ ữù ợ P ử ❝→❝ ❦ý t❤✐ ❍❙● q✉è❝ ❣✐❛ ✈➔ ❦❤✉ ✈ü❝✳ ✣➦❝ ❜✐➺t✱ tr♦♥❣ ❝→❝ ❦➻ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ t♦→♥ ❝→❝ ❝➜♣✱ ❝→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ tỵ✐ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ❧♦❣❛r✐t t❤÷í♥❣ ①✉②➯♥ ✤÷đ❝ ✤➲ ❝➟♣✳ ◆❤ú♥❣ ❞↕♥❣ t♦→♥ ♥➔② t❤÷í♥❣ ✤÷đ❝ ①❡♠ ❧➔ t❤✉ë❝ ❧♦↕✐ ❦❤â ✈➔ ✤á✐ ❤ä✐ t÷ ❞✉②✱ ❦❤↔ ♥➠♥❣ ♣❤→♥ ✤♦→♥ ❝❛♦✱ s♦♥❣ ♥â ❧↕✐ ❧✉ỉ♥ ❝â sù❝ ❤➜♣ ❞➝♥✱ t❤✉ ❤ót sü t➻♠ tá✐✱ â❝ s→♥❣ t↕♦ ❝õ❛ ❤å❝ s✐♥❤✳ ✣➸ ✤→♣ ự ỗ ữù ỗ ữù ❤å❝ s✐♥❤ ❣✐ä✐ ✈➲ ❝❤✉②➯♥ ✤➲ ❤➔♠ ❧♦❣❛r✐t✱ tæ✐ ❝❤å♥ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ✧✣➥♥❣ t❤ù❝ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❧ỵ♣ ❤➔♠ ❧♦❣❛r✐t✧✳ ❚✐➳♣ t❤❡♦✱ ❦❤↔♦ s→t ♠ët sè ❧ỵ♣ ❜➔✐ t♦→♥ tø ❝→❝ ✤➲ t❤✐ ❍❙● ◗✉è❝ ❣✐❛ t t tr ữợ ỳ trú ỗ ữỡ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥✳ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ ❧♦❣❛r✐t✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② t→❝ ❣✐↔ tr➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ ❧♦❣❛r✐t✱ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✈➔ ởt số q ợ ỗ ỗ rt ữỡ r t❤ù❝ ❧♦❣❛r✐t tr♦♥❣ ❧ỵ♣ ❤➔♠ sè ❝❤✉②➸♥ ✤ê✐ ❝→❝ ✤↕✐ ❧÷đ♥❣ tr✉♥❣ ❜➻♥❤ t❤ỉ♥❣ q✉❛ ♠ët sè ❜➔✐ t♦→♥✱ sû ❞ư♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❞↕♥❣ ❧♦❣❛r✐t✳ ❈✉è✐ ❝❤÷ì♥❣ ❞➔♥❤ ✤➸ tr➻♥❤ ❜➔② ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ s✐➯✉ ✈✐➺t ❞↕♥❣ ❧♦❣❛r✐t ❝ị♥❣ ợ tữỡ ự ữỡ t tự tr ợ rt ữỡ tr ❜➜t ✤➥♥❣ t❤ù❝ ❤➔♠ ❧♦❣❛r✐t ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❝❤ù❛ ❧♦❣❛r✐t t❤ỉ♥❣ q✉❛ ❝→❝ ✈➼ ❞ư ❝ư t❤➸✳ ◆❣♦➔✐ r❛ ❝á♥ tr➻♥❤ ❜➔② ❝→❝ ù♥❣ ❞ö♥❣ ❝õ❛ ❝→❝ ✤à♥❤ ❧➼ ✤➸ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ❝ü❝ trà rt ụ ữ t t ợ ự ỗ rt tr ự ♠✐♥❤ ♠ët ❧ỵ♣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❦✐♥❤ ✤✐➸♥✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ữợ sỹ ữợ ●✐→♦ s÷✱ ❚✐➳♥ s➽ ❦❤♦❛ ❤å❝ ◆❣✉②➵♥ ❱➠♥ ▼➟✉✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤è✐ ợ ữớ t t ữợ tr ✤↕t ❦✐➳♥ t❤ù❝✱ ❦✐♥❤ ♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ t tợ ổ tr trữớ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ ❣✐↔♥❣ ❞↕②✱ ❣✐ó♣ ✤ï ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ t→❝ ❣✐↔ tr sốt tớ t t rữớ ỗ tớ tỉ✐ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ ❣✐❛ ✤➻♥❤ ỗ ổ ú ù ✤ë♥❣ ✈✐➯♥ tæ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✵✸ ♥➠♠ ✷✵✷✵✳ ❚→❝ ❣✐↔ ◆❣✉②➵♥ ◆❣å❝ ◗✉②➳♥ ✸ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ ❧♦❣❛r✐t ▼ư❝ ✤➼❝❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ tr➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ ❧♦❣❛r✐t❀ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✈➔ ♠ët số q ợ ỗ ỗ rt t q ữỡ ữủ t❤❛♠ ❦❤↔♦ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪✳ ✶✳✶ ▼ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ ❧♦❣❛r✐t ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ a > 0, a = f (x) = loga x ❈❤♦ ❤➔♠ sè ❧♦❣❛r✐t ✳ ❑❤✐ ✤â ❤➔♠ sè ✤÷đ❝ a✳ log x x ❚ø ✤à♥❤ ♥❣❤➽❛ ♥➔② t❛ s✉② r❛✿ loga a = 1✱ loga = 0✱ x = a a ✱ x = loga a ✳ ❣å✐ ❧➔ ❝ì sè ❚r♦♥❣ ❝→❝ ♣❤➛♥ t✐➳♣ t❤❡♦✱ t❛ ❣✐↔ sû ◆❤➟♥ ①➨t ✶✳✶✳ D = (0; +∞) ✐✮ ❍➔♠ sè ❧♦❣❛r✐t ❝â t➟♣ ①→❝ ✤à♥❤ ✐✐✮ ❍➔♠ sè f (x) = loga x f (x) = loga x ✳ ✭❚➼♥❤ ✤ì♥ ✤✐➺✉✮ ln a > a>1 x ln a a > 1✳ ♥➯♥ s✉② r❛ f (x) = loga x < a < 1✳ t❤➻ ✲ ❚r÷í♥❣ ❤đ♣ ✷✿ ❤ì♥ ♥ú❛ ❚❛ ❦❤↔♦ s→t t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❤➔♠ sè f (x) = (loga x) = ❱➟②✱ ❦❤✐ I = R✳ x > 0✱ tr♦♥❣ ✷ tr÷í♥❣ ❤đ♣✳ ✲ ❚r÷í♥❣ ❤đ♣ ✶✿ ❑❤✐ ✤â✱ ✈➔ t➟♣ ❣✐→ trà ❧✐➯♥ tö❝ ✈➔ ❝â ✤↕♦ ❤➔♠ ✈ỵ✐ ♠å✐ f (x) = ❚➼♥❤ ❝❤➜t ✶✳✶ < a = 1✳ > 0, ∀x > x ln a ỗ tr ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② loga x f (x) < 0, ∀x ∈ D✳ ❱➟②✱ ❦❤✐ 0 ỗ ó t số t❛ ❝â f (x) = (loga x) = f (x) = ✲ ◆➳✉ ✲ ◆➳✉ −1 x2 ln a , x ln a a > tù❝ ln a > t❤➻ y < s✉② r❛ ❤➔♠ sè ❧ã♠ tr➯♥ (0; +∞)✳ < a < tù❝ ln a < t❤➻ y > s✉② r số ỗ tr (0; +) t ❱ỵ✐ ♠å✐ a > 0, a = ✈➔ ❚➼♥❤ ❝❤➜t ✶✳✹✳ ❱ỵ✐ ♠å✐ a > 0✱ a = ✈➔ x1 , x2 ∈ (0; +∞)✱ t❛ ❝â x1 loga (x1 x2 ) = loga x1 + loga x2 , loga = loga x1 − loga x2 x2 loga xα = αloga x, loga x = ❚➼♥❤ ❝❤➜t ✶✳✺✳ ❱ỵ✐ ♠å✐ x > 0✳ ❱ỵ✐ ♠å✐ < a = 1, b = ✈➔ x > 0✱ < a = 1, < c = loga x = ❚➼♥❤ ❝❤➜t ✶✳✼✳ ❍➔♠ sè ❚➼♥❤ ❝❤➜t ✶✳✽✳ ❱ỵ✐ ♠å✐ α ❜➜t ❦ý✱ t❛ ❝â loga xα = α logaα x = logaα xα α loga b logb c = loga c, loga b = ❚➼♥❤ ❝❤➜t ✶✳✻✳ ❱ỵ✐ t❛ ❝â logb a ✈➔ x > 0✱ t❛ ❝â logc x logc a f (x) = loga x (0 < a = 1) ❝â ✤↕♦ ❤➔♠ t↕✐ ♠å✐ ◆➳✉ ❤➔♠ sè u = u(x) ❝â ✤↕♦ ❤➔♠ ✤✐➸♠ x ∈ (0; +∞) ✈➔ (loga x) = x ln a tr➯♥ ❦❤♦↔♥❣ J ∈ R t❤➻ ❤➔♠ sè y = loga u(x)✱ (0 < a = 1) ❝â ✤↕♦ ❤➔♠ tr➯♥ u (x) J ✈➔ (loga u(x)) = u(x) ln a ✐✮ ❑❤✐ ✐✐✮ ❑❤✐ a>1 t❤➻ a > 0✱ a = ✈➔ x1 , x2 ∈ (0; +∞)✱ loga x1 < loga x2 ⇔ x1 < x2 0 1) ❍➔♠ sè M tr➯♥ ♥➳✉ f (x) ✤÷đ❝ ❣å✐ M ⊂ D(f ) ✈➔ ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉  ∀x ∈ M s✉② r❛ a±1 x ∈ M f (ax) = f (x), ∀x ∈ M ❱➼ ❞ö ✶✳✶✳ f (x) = sin(2π log2 x)✳ ❑❤✐ ✤â f (x) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ + + ±1 + ❦ý ✷ tr➯♥ R ✳ ❚❤➟t ✈➟②✱ t❛ ❝â ∀x ∈ R t❤➻ x ∈ R ✈➔ ❳➨t ♥❤➙♥ t➼♥❤ ❝❤✉ f (2x) = sin(2π log2 (2x)) = sin(2π(1 + log2 x)) = sin(2π log2 x) = f (x) ❚➼♥❤ ❝❤➜t ✶✳✾✳ ◆➳✉ ❦ý t÷ì♥❣ ù♥❣ ❧➔ a f (x) ✈➔ b ✈➔ tr➯♥ g(x) M ✈➔ ❧➔ ❤❛✐ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ln |a| m = , m, n ∈ N∗ ln |b| n t❤➻ F (x) = f (x) + g(x) ✈➔ G(x) = f (x).g(x) ❧➔ ❝→❝ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ tr➯♥ M ✳ ❈❤ù♥❣ ♠✐♥❤✳ ln |a| m n m = s✉② r❛ |a| = |b| ✳ ln |b| n ❝õ❛ F (x) ✈➔ G(x)✳ ❚❤➟t ✈➟②✱ t❛ ❚ø ❣✐↔ t❤✐➳t T := a2n = b2m ❧➔ ❝❤✉ ❦ý ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❝â F (T x) = f (a2n x) + g(b2m x) = f (x) + g(x) = F (x), ∀x ∈ M ; G(T x) = f (a2n x)g(b2m x) = f (x)g(x) = G(x), ∀x ∈ M ∀x ∈ M, T ±1 x ∈ M ✳ t➼♥❤ tr➯♥ M ✳ ❍ì♥ ♥ú❛✱ ♥❤➙♥ ❚➼♥❤ ❝❤➜t ✶✳✶✵✳ tr➯♥ tr➯♥ R t❤➻ R+ ✳ ❉♦ ✤â✱ F (x), G(x) ❧➔ ❝→❝ ❤➔♠ t✉➛♥ ❤♦➔♥ f (x) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ❝ë♥❣ t➼♥❤ ❝❤✉ ❦ý a✱ a > g(t) = f (ln t)✱ ✭t > 0✮ ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦ý ea ◆➳✉ ✻ f (x) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦ý a ✭a > 1✮ g(t) = f (et ) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ❝ë♥❣ t➼♥❤ ❝❤✉ ❦ý ln a tr➯♥ R✳ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ R+ t❤➻ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû f (x) ❧➔ ❤➔♠ t✉➛♥ tr➯♥ R✳ ❳➨t g(t) = f (ln t)✱ ✭t > 0✮✳ ❤♦➔♥ ❝ë♥❣ t➼♥❤ ❝❤✉ ❦ý tr➯♥ a✱ a > ❚❛ ❝â g(ea t) = f (ln(ea t)) = f (ln ea + ln t) = f (a + ln t) = f (ln t) = g(t), ∀t ∈ R+ ❱➟② g(t) ◆❣÷đ❝ ✭0 ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦ý ❧↕✐✱ ❣✐↔ sû f (x) ❧➔ < a = 1✮ tr➯♥ R+ ✳ t ❳➨t g(t) = f (e ), ∀t ∈ R✳ ❤➔♠ t✉➛♥ ea tr➯♥ ❤♦➔♥ R+ ✳ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦ý a ❚❛ ❝â g(t + ln a) = f (et+ln a ) = f (et eln a ) = f (aet ) = f (et ) = g(t), ∀t ∈ R ❱➟② g(t) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ❝ë♥❣ t➼♥❤ ❝❤✉ ❦ý ln a tr➯♥ R✳ ✶✳✷✳✷ ❍➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ❝❤✉ ❦ý f (x) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ a (a > 1) tr➯♥ M ♥➳✉ M ⊂ D(f ) ✈➔  ∀x ∈ M s✉② r❛ a±1 x ∈ M f (ax) = −f (x), ∀x ∈ M ❱➼ ❞ö ✶✳✷✳ ❍➔♠ sè f (x) = cos(π log2 x)✳ ❑❤✐ ✤â f (x) ❧➔ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ + ❦ý tr➯♥ R ✳ ❳➨t ♥❤➙♥ t➼♥❤ ❝❤✉ ❚❤➟t ✈➟②✱ t❛ ❝â ∀x ∈ R+ t❤➻ f (2x) = cos(π log2 (2x)) = cos(π+π log2 x) = − cos(π log2 x) = −f (x), ∀x ∈ R+ ❱➼ ❞ö ✶✳✸✳ √ [sin(2π log2 ( 2x)) − sin(2π log2 x)]✳ √ f (x) ❧➔ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦ý tr➯♥ R+ ✳ √ + ±1 + ❚❤➟t ✈➟②✱ t❛ ❝â ∀x ∈ R t❤➻ ( 2) x ∈ R ✈➔ √ √ f ( 2x) = [sin(2π log2 (2x)) − sin(2π log2 ( 2x))] ❳➨t f (x) = ❑❤✐ ✤â ✼ √ = [sin(2π(1 + log2 x)) − sin(2π log2 ( 2x))] √ = [sin(2π log2 x) − sin(2π log2 ( 2x))] = −f (x) ❚➼♥❤ ❝❤➜t ✶✳✶✶✳ ▼å✐ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ tr➯♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ tr➯♥ ❈❤ù♥❣ ♠✐♥❤✳ M ✤➲✉ ❧➔ ❤➔♠ M✳ ❚❤❡♦ ❣✐↔ tt tỗ t b > s x M t❤➻ b±1 ∈ M ✈➔ f (bx) = −f (x), ∀x ∈ M ❙✉② r❛✱ ∀x ∈ M t❤➻ b±1 ∈ M ✈➔ f (b2 x) = f (b(bx)) = −f (bx) = −(−f (x)) = f (x), ∀x ∈ M ◆❤÷ ✈➟②✱ f (x) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦ý ❚➼♥❤ ❝❤➜t ✶✳✶✷✳ f (x) tr➯♥ M ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ b2 tr➯♥ M✳ ❧➔ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦ý f (x) b ✭b > 1✮ ❝â ❞↕♥❣✿ f (x) = (g(bx) − g(x)), tr♦♥❣ ✤â✱ g(x) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦ý b2 tr➯♥ M✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ●✐↔ sû f ❧➔ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦ý b tr➯♥ M ✳ ❑❤✐ ✤â g(x) = −f (x) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦ý b2 tr➯♥ M ✈➔ ✭✐✐✮ 1 (g(bx) − g(x)) = (−f (bx) − (−f (x))) 2 = (−(−f (x)) + f (x)) = f (x), ∀x ∈ M ◆❣÷đ❝ ❧↕✐✱ f (x) = (g(bx) − g(x)), t❤➻ 1 f (bx) = (g(b2 x) − g(bx)) = (g(x) − g(bx)) 2 = − (g(bx) − g(x)) = −f (x), ∀x ∈ M ∀x ∈ M tr➯♥ M ✳ ❍ì♥ ♥ú❛✱ ♥❤➙♥ t➼♥❤ t❤➻ b±1 x ∈ M ✳ ❉♦ ✤â✱ f (x) ❧➔ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ✺✸ ❙✉② r❛ S≥3 2 3 ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❝õ❛ S ❱➼ ❞ö ✸✳✷✵✳ x=y=z= 3 ❧➔ =3 ✳ ❱➟② ❣✐→ trà ♥❤ä ♥❤➜t ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ √ + x ln(x + + x2 ) ≥ √ + x2 , ∀x, y ∈ R ▲í✐ ❣✐↔✐ ✳ ❳➨t ❤➔♠ sè f (t) = ln(t + ❘ã r➔♥❣ f (t) > ✈ỵ✐ t>0 √ + t2 ) > 0, ∀x ∈ R+ f (t) = ❦❤✐ t = 0✳ ❑❤✐ x √ ln(t + + t2 )dt > ✈➔ ✤â✱ ✈ỵ✐ 0 + x2 + > t❤➻ ln(t + ♥➯♥ ❦❤✐ t x + t2 ) − x < 0✱ √ + t2 ) = − ln(−t + √ + t2 ) < 0, t❛ ❝â ln(t + √ + t2 )dx < x ❱ỵ✐ x = ❜➜t ✤➥♥❣ t❤ù❝ trð t❤➔♥❤ ✤➥♥❣ t❤ù❝✳ ❙✉② r❛ ự ữợ ởt số ❞ư ✈➲ ♣❤÷ì♥❣ ♣❤→♣ t➼❝❤ ♣❤➙♥ tr♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❧♦❣❛r✐t✳ ❚❛ t❤÷í♥❣ ❞ị♥❣ ❦➳t q✉↔ s❛✉ ✤➙②✳ ✺✹ ✣à♥❤ ỵ số y = f (x) tư❝✱ ❦❤ỉ♥❣ ➙♠✱ ✤ì♥ ✤✐➺✉ t➠♥❣ tr➯♥ [0, c) ✈ỵ✐ c > 0✳ ●å✐ f −1 (x) ❧➔ ❤➔♠ ♥❣÷đ❝ ❝õ❛ ♥â✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ a ∈ [0, c) ✈➔ b ∈ [f (0), f (c)) t❛ ❝â a b f −1 (x)dx ≥ ab f (x)dx + f (0) ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ b = f (a)✳ ❱➼ ❞ö ✸✳✷✶✳ f (x) tử ỗ tr [0; b] a ∈ [0; b]✳ ❈❤♦ ❤➔♠ sè ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ a b f (x)dx ≥ a b f (x)dx ▲í✐ t tự tr tữỡ ữỡ ợ a (b − a) b f (x)dx ≥ a f (x)dx ❉♦ f (x) ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ a [0; a] ✈➔ [a; b] ♥➯♥ a (b − a) a f (x)dx ≥ (b − a) f (a)dx = (b − a)a.f (a) b b f (a)dx ≥ a =a a ❱➟② ❜➜t ✤➥♥❣ t❤ù❝ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❱➼ ❞ö ✸✳✷✷✳ ❈❤♦ < a < b✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ a−b a a−b < ln < a b b ▲í✐ ❣✐↔✐ ✳ ❚❛ ❝â ♥❤➟♥ ①➨t✿ ✈ỵ✐ ♠å✐ a ❤❛② ❱➟② a ∈ (a, b) t❤➻ b b dx > a a 1 > > a x b dx > x b b−a > ln |x| a b a f (x)dx a > ♥➯♥ b dx, a b−a b b−a b b−a > ln > a a b ✺✺ ❱➼ ❞ö ✸✳✷✸✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ f (x) = (3 + ln 2)x − 2x+1 − ln 2.x2 , x ∈ [0; 2] ▲í✐ ❣✐↔✐ ✳ ❚❛ ❝â g(t) = 2t + t ❧➔ ❤➔♠ ❧✐➯♥ tử ỗ tr [0; 2] t ❞ö ✸✳✷✶✱ t❛ ❝â x t (2t + t)dt (2 + t)dt ≤ x 0 x t t2 2 2t ⇔2 + + ≤x ln 2 ln 2 4x x 2x+1 + x2 − ≤ + 2x − ⇔ ln ln ln ln x+1 ⇔2 + x ln − ≤ 4x + 2x ln − x t ⇔ (3 + ln 2)x − 2x+1 − ln 2.x2 ≥ −2 ❱➟② ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❱➼ ❞ö ✸✳✷✹✳ f (x) = √ [0, 2]✱ −2 ❦❤✐ x = 0, x = 2✳ √ √ √ √ 2x5 − x[4 − ln(1 + 2)] − ln(1 + x), x ∈ [0, 2] ✳ √ ❜➡♥❣ ❚➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❤➔♠ sè ▲í✐ ❣✐↔✐ ❚❛ t❤➜② f (x) g(t) = −t4 + t+1 ❧➔ ❤➔♠ ❧✐➯♥ tư❝ ✈➔ ♥❣❤à❝❤ ❜✐➳♥ ✈ỵ✐ ♠å✐ ❞♦ ✤â √ x √ dt ≥ x −t + t+1 dt −t4 + t+1 0 √ √ √ x5 ⇔ − + ln(1 + x) ≥ x − + ln(1 + 2) 5 √ √ √ √ ⇔ − 2x + ln(x + 1) ≥ [−4 + ln(1 + 2)]x ❙✉② r❛ √ ❱➟② ❣✐→ trà √ √ √ 2x5 − x[4 − ln(1 + 2)] − ln(1 + x) ≤ √ ❧ỵ♥ ♥❤➜t ❝õ❛ f (x) ❜➡♥❣ ✵ ❦❤✐ x = ❤♦➦❝ x = 2✳ ❱➼ ❞ö ✸✳✷✺✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ e A = e b + b(ln b − 1), < b ≤ e t∈ ✺✻ ▲í✐ ❣✐↔✐ e ✳ ✣➦t b = a✱ s✉② r❛ ab = e ✈➔ ✤ì♥ ✤✐➺✉ t➠♥❣ tr➯♥ a ≥ 1✳ ✈➔ (0, +∞) ❝â ❍➔♠ sè f (0) = f (x) = ex ❧✐➯♥ tư❝✱ ❦❤ỉ♥❣ ➙♠ ✈➔ ❝â ❤➔♠ ♥❣÷đ❝ ❧➔ y = ln x✱ t ỵ t õ a b x ln xdx ≥ ab e dx + a b ⇔ ex + (a ln x − x) ≥ ab ⇔ ea − + b ln b − b + ≥ ab ⇔ ea + b(ln b − 1) ≥ ab ❤❛② e e b + b(ln b − 1) ≥ e ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ b = f (a) ❱➟② ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ b = ea ✱ ❦❤✐ ✤â t❛ A ❜➡♥❣ e ❦❤✐ b = e✳ ❤❛② ❝â a = 1, b = e✳ ✸✳✷✳✷ ❇➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❞➣② sè ✈➔ ❣✐ỵ✐ ❤↕♥ ❚r♦♥❣ ♣❤➛♥ ♥➔② t s sỷ t ởt số ỵ s ỵ ỵ sü ❤ë✐ tư ❝õ❛ ❞➣② ✤ì♥ ✤✐➺✉ ❜à ❝❤➦♥✮ {an }+∞ n=1 ✤ì♥ ✤✐➺✉ t➠♥❣ ✈➔ ❜à ❝❤➦♥ tr➯♥ t❤➻ ❝â ❣✐ỵ✐ ❤↕♥ lim an = sup an n→+∞ ✷✳ ◆➳✉ ❞➣② {an }+∞ n=1 ✤ì♥ ✤✐➺✉ ❣✐↔♠ ✈➔ ữợ t õ ợ lim an = inf an n+ ỵ số {an}+ n=1 ỵ ợ ❤↕♥ ❝õ❛ ❞➣② sè✮ +∞ + {bn }n=1 ✱ {cn }+∞ n=1 ✳ ◆➳✉ ✈ỵ✐ ♠å✐ n ∈ N ♠➔ an ≤ bn ≤ cn ✈➔ lim an = lim cn = L (L ∈ R) n→+∞ n→+∞ t❤➻ lim bn = L ❱➼ ❞ö ✸✳✷✻✳ n→+∞ ❈❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ln(x + 1) ≤ x, ∀x ≥ ✺✼ ▲í✐ ❣✐↔✐ ✳ f (x) = ln(x + 1) − x, ∀x ≥ ❳➨t ❤➔♠ sè f (x) = ❚❛ ❝â −x −1= ≤0 x+1 x+1 f (x) ≤ f (0) = ln − = ❚ø ✤➙② ❞➵ ❞➔♥❣ s✉② r❛ ln(x + 1) ≤ x, ∀x ≥ ✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = ♥➯♥ ln(x + 1) < x, ∀x > ♥➯♥ ✤➙② ❧➔ ❤➔♠ ♥❣❤à❝❤ ❜✐➳♥✳ ❙✉② r❛ ❇➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ✈➼ ❞ư tr➯♥ ✤÷đ❝ →♣ ❞ư♥❣ ❝❤♦ ❝→❝ ✈➼ ❞ư s❛✉ ♥➔②✳ ❱➼ ❞ö ✸✳✷✼✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❞➣② an = + 1 + · · · + − ln n, n ∈ N n ❝â ❣✐ỵ✐ ❤↕♥ ❤ú✉ ❤↕♥✳ ▲í✐ ❣✐↔✐ ✳ ❚❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ x < ln(x + 1) < x, ∀x > x+1 ✭✸✳✶✼✮ ❙û ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✭✸✳✶✼✮✱ t❛ ✤÷đ❝ an+1 − an = ❉♦ ✤â ❞➣② (an ) 1 − ln(n + 1) + ln n = − ln + n+1 n+1 n ❧➔ ❞➣② ❣✐↔♠✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❞➣② (an ) < ❜à ữợ ỷ t tự ✭✸✳✶✼✮✱ t❛ ❝â n+1 = ln n n an > ln(1 + 1) + ln + + ln + = ln + · · · + ln + C✳ ✣➦t ✤â✱ t❛ ❝â 1+ ❙è C − ln n n+1 > > n 1+n (an ) ❝â an − C = γn , n ∈ N✳ ❑❤✐ ✤â✱ t❤❡♦ ♥❣✉②➯♥ ỵ rstrss ợ õ n ❣✐ỵ✐ ❤↕♥ ❤ú✉ ❤↕♥✳ ❚❛ ❦➼ ❍✐➸♥ ♥❤✐➯♥ γn → ∞✳ ❚ø 1 + · · · + = ln n + C + γn n ✤÷đ❝ ❣å✐ ❧➔ ❤➡♥❣ sè ❒❧❡✳ ❱➼ ❞ö ✸✳✷✽ ✭✣➲ t❤✐ ❍❙● t➾♥❤ ◆✐♥❤ ❇➻♥❤✱ ✈á♥❣ ✷ ♥➠♠ ❤å❝ ✷✵✶✶ ✲ ✷✵✶✷✮ ❈❤ù♥❣ ♠✐♥❤ ❞➣② n {xn } ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝ un = k=1 k − ln n ✳ ❝â ❣✐ỵ✐ ✺✽ ❤↕♥ ❤ú✉ ❤↕♥✳ ▲í✐ ❣✐↔✐ ✳ ❚❛ ①➨t ❤✐➺✉ s❛✉ − ln(n + 1) + ln n n+1 n+1 = − ln n+1 n 1 = − ln + < n+1 n un+1 − un = ❙✉② r❛ ❞➣② ✤➣ ❝❤♦ ✤ì♥ ✤✐➺✉ ❣✐↔♠✳ ▼➦t ❦❤→❝✱ t❛ ❝ơ♥❣ ❝â n un = k=1 n − ln n > k n ln k=1 + − ln n k [ln(k + 1) − ln k] − ln n = ln = k=1 n+1 >0 n ♥➯♥ ❞➣② ♥➔② ❜à ❝❤➦♥ ữợ ữợ ❝â ❣✐ỵ✐ ❤↕♥ ❤ú✉ ❤↕♥✳ ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❱➼ ❞ö ✸✳✷✾✳ ❈❤♦ sè t❤ü❝ ❛ ✈➔ ❞➣② sè t❤ü❝ {xn } ①→❝ ✤à♥❤ ❜ð✐ x1 = a, xn+1 = ln(3 + cos xn + sin xn ) − 2020, ∀n = 0, 1, 2, ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❞➣② sè {xn } ❝â ❣✐ỵ✐ ❤↕♥ ❤ú✉ ❤↕♥ ❦❤✐ n t✐➳♥ ✤➳♥ ❞÷ì♥❣ ✈ỉ ❝ị♥❣✳ ▲í✐ ❣✐↔✐ ✳ ✣➦t f (x) = ln(3 + cos x + sin x) − 2020✱ t❛ ❝â cos x − sin x + sin x + cos x √ √ |cos x − sin x| ≤ 2✱ |sin x + cos x| ≤ √ √ = q < |f (x)| ≤ 3− f (x) = ❚❛ sû ❞ö♥❣ ✤→♥❤ ❣✐→ g(x) = f (x) − x✱ ❦❤✐ ✤â g(x) ❧➔ ❤➔♠ sè ①→❝ ✤à♥❤ R✳ ❍ì♥ ♥ú❛ g (x) = f (x) − < 0, ∀x ∈ R✳ ❚❛ ❧↕✐ ❝â ❳➨t ❤➔♠ sè tr➯♥ g(0) =f (0) = ln − 2020 < 0, t❛ s✉② r❛ ✈➔ ❧✐➯♥ tö❝ ✺✾ g(−2020) = ln(3 + cos(−2020) + sin(−2020)) = ln(3 + cos2020 − sin 2020) > g(0).g(−2020) < 0✳ ❚â♠ ❧↕✐ g(x) ❧➔ số tử ỗ tr R✱ ✈➔ g(0).g(−2020) < 0✳ ❉♦ ✤â ♣❤÷ì♥❣ tr➻♥❤ g(x) = 0✱ ❤❛② ♣❤÷ì♥❣ tr➻♥❤ f (x) = x ❝â ♠ët ♥❣❤✐➺♠ ❞✉② ♥❤➜t tr➯♥ ❦❤♦↔♥❣ (−2020; 0)✳ ❚ù❝ ❧➔ tỗ t t l (2020; 0) s f (l) = l ỵ r ❝❤♦ x, y ∈ R✱ ❞♦ ❤➔♠ f (x) ❧✐➯♥ tử tr R tỗ t z tở R s ❝❤♦ ❙✉② r❛ f (x) − f (y) = f (z)(x − y) ❙✉② r❛ |f (x) − f (y)| ≤ q |x − y| , ∀x, y ∈ R ❚❛ ❝â |xn+1 − l| = |f (xn ) − f (l)| = |f (x)| |xn − l| ≤ q |xn − l| , ∀n = 1, 2, ❉♦ ✤â ≤ |xn − l| ≤ q |xn−1 − l| ≤ ≤ q n−1 |x1 − l| = q n−1 |a − l| ❉♦ ❝â q n−1 n + ỵ ❦❤✐ n → +∞✱ ❞➣② ✤➣ ❝❤♦ ❣✐ỵ✐ ❤↕♥ ❧➔ l ✳ ❙✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❱➼ ❞ö ✸✳✸✵ a1 , x2 , ✭❱▼❖ ✶✾✾✽✮ ♥❤÷ s❛✉✿ x1 = a xn+1 ✳ ❈❤♦ a≥1 ❧➔ ♠ët sè t❤ü❝ ✈➔ ✤à♥❤ ♥❣❤➽❛ ❞➣② ✈➔ xn (x2n + 3) = + log 3x2n + ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❞➣② tr➯♥ ❝â ❣✐ỵ✐ ❤↕♥ ❤ú✉ ❤↕♥ ✈➔ t➼♥❤ ❣✐ỵ✐ ❤↕♥ ✤â✳ ▲í✐ ❣✐↔✐ ✳ ❱ỵ✐ x ≥ 1✱ t❛ ❝â x(x2 + 3) (x − 1) ≥ ⇔ x + 3x ≥ 3x + ⇔ ≥ 3x2 + 3 ❙✉② r❛ x(x2 + 3) + log ≥ 1, 3x2 + t❤❡♦ ♠ët ❝→❝❤ q✉② ♥↕♣ t❛ ❝â ♠å✐ ♣❤➛♥ tû ❝õ❛ ❞➣② ✤➲✉ ❧ỵ♥ ❤ì♥ ❤♦➦❝ ❜➡♥❣ ✶ ✈➔ ≤ x ⇒ x2 + ≤ 3x2 + ⇒ x(x2 + 3) ≥x 3x2 + ✻✵ x(x2 + 3) ⇒ + log ≥ + log x ≤ x 3x2 + ✭✸✳✶✽✮ ❱➙② ỡ ữợ tỗ t ợ ỳ ợ x t❤ä❛ ♠➣♥ x = + log ♣❤↔✐ ①↔② r❛ ❞➜✉ ❜➡♥❣✱ ♥❣❤➽❛ ❧➔ x = 1✳ x(x2 + 3) 3x2 + ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ✭✸✳✶✽✮ ❱➟② ❣✐ỵ✐ t ỗ ❤➔♠ ❧♦❣❛r✐t tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❚r♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ t❤➻ ❧ỵ♣ ❜➜t ✤➥♥❣ t❤ù❝ ❦✐♥❤ ✤✐➸♥ ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣✱ ❧➔ ❝ì sð ✤➸ ❝❤ù♥❣ ♠✐♥❤ r➜t ♥❤✐➲✉ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❦❤→❝✳ ❙❛✉ ✤➙② ởt ợ t tự ữủ ự t ữỡ ỗ ỵ b1 , b2 , , bn √ n ✳ ✭❇➜t ✤➥♥❣ t❤ù❝ ▼✐♥❝♦♣①❦✐✮ ❈❤♦ ❤❛✐ ❞➣② sè t❤ä❛ ♠➣♥ n a1 a2 an + ❈❤ù♥❣ ♠✐♥❤✳ > 0, bi > 0, i = 1, n✳ b1 b2 bn ≤ ❳➨t ❤➔♠ sè ex f (x) = >0 (1 + ex )2 f (x) = ln(1 + e )✳ ✈ỵ✐ ♠å✐ ln + e b2 bn a2 +···+ln an n x ∈ R✳ ❙✉② r❛ xi = ln ln + ≤ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ bi b1 a1 ex f (x) = + ex ❚❛ ❝â f (x) ỗ tr s r R t õ + · · · + ln + bn an n ❙✉② r❛  ln + n b1 b2 bn a1 a2 an  ≤ ln  n (a1 + b1 )(a2 + b2 ) (an + bn )  a1 a2 an ❉♦ ✤â 1+ n b1 b2 bn ≤ a1 a2 an n (a1 + b1 )(a2 + b2 ) (an + bn ) a1 a2 an ❙✉② r❛ √ n a1 a2 an + n b1 b2 bn ≤ n ✈➔ (a1 + b1 )(a2 + b2 ) (an + bn ) x ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❏❡♥s❡♥ ✈ỵ✐ b ln a1 +ln n a1 , a2 , , an (a1 + b1 )(a2 + b2 ) (an + bn ) ✻✶ ❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ln b1 b2 bn b1 b2 bn = ln = = ln ⇔ = = = a1 a2 an a1 a2 an ❱➟② t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✣à♥❤ ỵ ợ số ổ t ý a, b ✈➔ ✭❇➜t ✤➥♥❣ t❤ù❝ ❨♦✉♥❣✮ p > 0, q > s❛♦ ❝❤♦ 1 + = ❚❛ ❝â p q ap bq ab ≤ + p q ❈❤ù♥❣ ♠✐♥❤✳ ❇➜t ✤➥♥❣ t❤ù❝ ❤✐➸♥ ♥❤✐➯♥ ✤ó♥❣ ❦❤✐ a = ❤♦➦❝ b = 0✳ ●✐↔ sû a > 0, b > 0✳ ❳➨t ❤➔♠ sè f (x) = ex s✉② r❛ f (x) = ex > ✈ỵ✐ ♠å✐ x ∈ R✳ ❙✉② r❛ f (x) ỗ tr R õ f 1 ln ap + ln bq p q 1 ≤ f (ln ap ) + f (ln bq ) p q 1 f (ln a + ln b) ≤ f (p ln a) + f (q ln b) p q ❤❛② 1 eln ab ≤ ep ln a + eq ln b p q p q a b ⇔ ab ≤ + p q ❱➟② t❛ ❝â ✤✐➲✉ ự ỵ x1, x2, , xn > 0✳ ❚❛ ①➨t ❝→❝ ✤↕✐ ❧÷đ♥❣ s❛✉ ma = mq = x1 + x2 + · · · + xn ; n n x21 + x22 + · · · + a2n ; n mg = √ n mh = x1 x2 xn ; n 1 + + ··· + x1 x2 xn , tr♦♥❣ ✤â ma , mg , mq , mh t÷ì♥❣ ù♥❣ ❣å✐ ❧➔ tr✉♥❣ ❜➻♥❤ ❝ë♥❣✱ tr✉♥❣ ❜➻♥❤ ♥❤➙♥✱ tr✉♥❣ ❜➻♥❤ t♦➔♥ ♣❤÷ì♥❣ ✈➔ tr✉♥❣ ❜➻♥❤ ✤✐➲✉ ❤á❛ ❝õ❛ ❝→❝ sè x1 , x2 , , xn ✳ ❑❤✐ ✤â✱ t❛ ❝â mh ≤ mg ≤ ma ≤ mq ✻✷ ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ❤➔♠ sè f (x) = x2 tr➯♥ R✳ ❚❛ ❝â f (x) > 0, ∀x✳ ❉♦ ✤â f (x) ỗ t tự s f tr➯♥ t♦➔♥ trö❝ sè✳ ⑩♣ ❞ö♥❣ λ1 = λ2 = = λn = x1 + x2 + · · · + xn n ❑❤✐ ✤â x1 + x2 + · · · + xn n ≤ n ✮✱ n t❛ ❝â n f (x) i=1 x21 + x22 + · · · + x2n ≤ n ❍❛② x21 + x22 + · · · + x2n ⇔ ma ≤ mq n x1 + x2 + · · · + xn ≤ n f (x) = − ln x✱ ✈ỵ✐ x > 0✳ 1 ❚❛ ❝â f (x) = − s✉② r❛ f (x) = ✱ ✈ỵ✐ ♠å✐ x > 0✳ x x2 ❦❤✐ x > 0✳ ❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ❏❡♥s❡♥ t❛ ❝â ✭✸✳✶✾✮ ❳➨t ❤➔♠ sè f x1 + x2 + · · · + xn n ≤ f n x1 +f x2 ❱➟② f (x) + ··· + f ỗ xn r 1 + + ··· + 1 1 x x2 xn − ln ≤ − ln + ln + · · · + ln n n x1 x2 xn ❙✉② r❛ ln n 1 + + ··· + x1 x2 xn n ≤ ln y = ln x ✈ỵ✐ x > 0✱ tø ✭✸✳✷✵✮ √ ≤ n x1 x2 xn ⇒ mh ≤ mg t ỗ số n 1 + + ··· + x1 x2 xn x1 x2 xn n ✭✸✳✷✵✮ s✉② r❛ ✭✸✳✷✶✮ ❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤② t❤➻ mg ≤ ma ✭✸✳✷✷✮ ✻✸ mh ≤ mg ≤ ma ≤ mq ✳ ❚ø ✭✸✳✶✾✮✱ ✭✸✳✷✶✮ ✈➔ ✭✸✳✷✷✮ t❛ ❝â ❙✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ◆❣♦➔✐ r❛✱ t❛ ❝á♥ →♣ ❞ö♥❣ ❤➔♠ ỗ ự t tự số ❝â ❝→❝ ✈➼ ❞ư s❛✉✳ ❱➼ ❞ư ✸✳✸✶ ✈ỵ✐ ♠å✐ x ∈ R✱ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ t❛ ❝â 12 ▲í✐ ❣✐↔✐ ✳ ✭✣➲ t❤✐ t✉②➸♥ s✐♥❤ ✣↕✐ ❤å❝ ❦❤è✐ ❇ ✲ ✷✵✵✺✮ x 15 + x + 20 ❑❤✐ ✤â f (x) x ≥ 3x + x + x ✳ ❳➨t ❤➔♠ sè f (x) = − ln x✳ a1 + a2 − ln ❚❛ ❝❤å♥ 12 a1 =  x ; a2 = 12  − ln   x 12 x ❉♦ ✤â + ≤− ❚❛ ❝â x 15 ✳ ❑❤✐ ✤â✱ t❛ ❝â 15 x 15 x 12 ≥ x 12   ≤ − ln  (0; +∞)✳ √ ln a1 + ln a2 = − ln a1 a2 2 + ỗ tr x 15 15 x x ❙✉② r❛ 12 x x + 15 15 x x ≥ 2.3x ✭✸✳✷✸✮ ❚r÷ì♥❣ tü t❛ ❝â 12 x + 20 x ≥ 2.4 ; + 20 x ≥ 2.5x ✭✸✳✷✹✮ ❈ë♥❣ ✈➳ t❤❡♦ ✈➳ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ✭✸✳✷✸✮ ✈➔ ✭✸✳✷✸✮✱ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❱➼ ❞ö ✸✳✸✷ ✳ ✭❈❛♥❛❞❛ ▼❖ ✷✵✵✷✮ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ a3 b3 c3 + + ≥ a + b + c, ∀a, b, c > bc ca ab ✻✹ ▲í✐ ❣✐↔✐ ✳ ❳➨t ❤➔♠ sè f (x) = − ln x✳ ❑❤✐ ✤â f (x) ❧➔ ỗ tr (0; +) õ a1 + a2 + a3 − ln √ ln a1 + ln a2 + ln a3 = − ln a1 a2 a3 a3 a1 = ; a2 = b; a3 = c✳ ❚❛ ❝â bc   a  bc + b + c   ≤ − ln − ln    ❚❛ ❝❤å♥ ❙✉② r❛ ≤− ✭✸✳✷✺✮ a3 bc bc a3 + b + c ≥ 3a✳ bc ❚÷ì♥❣ tü t❛ ❝â c3 b3 + a + c ≥ 3b; + a + b ≥ 3c ac ab ✭✸✳✷✻✮ ❚ø ✭✸✳✷✺✮ ✈➔ ✭✸✳✷✻✮✱ s✉② r❛ a3 b c3 + + ≥ a + b + c bc ca ab ❱➼ ❞ö ✸✳✸✸✳ ❈❤♦ < a < 1, < b < aa + bb ≥ ▲í✐ ❣✐↔✐ ✈➔ √ a + b = 1✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✳ ❳➨t ❤➔♠ sè (0, 1)✳ f (x) = xx , < x < 1✳ ❘ã r➔♥❣ f (x) ❧➔ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ ❙✉② r❛ f (x) = + ln x ⇒ f (x) = f (x)(1 + ln x) f (x) ❉♦ ✤â ❤❛② f (x) = f (x)(1 + ln x) + f (x) x f (x) = f (x)(1 + ln x)2 + f (x) x ❙✉② r❛ f (x) = xx (1 + ln x)2 + x ✭✸✳✷✼✮ ✻✺ ❚ø ✭✸✳✷✼✮ s✉② r❛ f (x) > 0, < x < f (x) õ ỗ tr (0, 1)✳ ▼➦t ❦❤→❝✱ t❛ ❝â aa + bb = aa + (1 − a)1−a = f (a) + f (1 − a) ⑩♣ ❞ư♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❏❡♥s❡♥ ✈ỵ✐ ỗ f (a) + f (1 a) f a+1−a ❚❤❡♦ ✭✸✳✷✽✮✱ t❛ ❝â aa + bb ≥ f (x) =f √ tr➯♥ = ✭✸✳✷✽✮ (0, 1)✱ t❛ ❝â 1 = 2 ❱➟② t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❱➼ ❞ö ✸✳✸✹✳ ❈❤♦ a1 , a2 , , an > 0✳ a1 a1 aa22 aann ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ a1 + a1 + · · · + a1 n ≥ a1 +a2 +···+an ▲í✐ ❣✐↔✐ ✳ ❳➨t ❤➔♠ sè ✈ỵ✐ ♠å✐ f (x) = x ln x✱ x > 0✳ ❙✉② r❛ f (x) t❛ ❝â f (x) = ln x + ỗ tr s r f (x) = (0, +∞)✳ ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❏❡♥s❡♥ ❝❤♦ ❤❛✐ ❜ë sè a1 , a2 , , an ✈➔ n sè ✤÷đ❝ f a1 + a2 + · · · + an n ≤ > 0✱ x n t❛ (f (a1 ) + f (a2 ) + · · · + f (an )) n ❍❛② a1 + a2 + · · · + an a1 + a2 + · · · + an ln ≤ (a1 ln a1 + · · · + an ln an ) n n n ✣✐➲✉ ♥➔② t÷ì♥❣ ữỡ ợ a1 + a2 + à à à + an ln n ❱➟② a1 + a2 + · · · + an n ❇➜t ✤➥♥❣ t❤ù❝ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ a1 +a2 +···+an a1 ≤ ln(a1 aa22 aann ) a1 +a2 +···+an a1 ≤ a1 aa22 aann ✻✻ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✏✣➥♥❣ t❤ù❝ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❧ỵ♣ ❤➔♠ ❧♦❣❛r✐t✑ ✤➣ tr➻♥❤ ❜➔② ♥❤ú♥❣ ✈➜♥ ✤➲ s❛✉✿ ✶✳ ❍➺ t❤è♥❣ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ ❧♦❣❛r✐t❀ ❤➔♠ t✉➛♥ ❤♦➔♥ ✈➔ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✈➔ ♠ët sè ✤à♥❤ ❧➼ ❧✐➯♥ q✉❛♥ ✤➳♥ ❧ỵ♣ ỗ ỗ rt t tr ❜➔② ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ t❤✐➳t ❧➟♣ ❝→❝ ✤➥♥❣ t❤ù❝ t❤ỉ♥❣ q✉❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✈➔ ①➨t ♠ët sè ♣❤÷ì♥❣ tr➻♥❤ s✐➯✉ ✈✐➺t ❞↕♥❣ ❧♦❣❛r✐t ❧✐➯♥ q✉❛♥✳ ✸✳ ❈✉è✐ ❝ò♥❣✱ ❧✉➟♥ tr t ữợ ữủ ữỡ ♣❤→♣ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❝❤ù❛ ❤➔♠ ❧♦❣❛r✐t✳ ❳➨t ♠ët sè ❞↕♥❣ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ ❧♦❣❛r✐t ♥❤÷ ❜➔✐ t♦→♥ ❝ü❝ trà✱ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❞➣② số ợ õ ự ỗ rt ❚✉② ♥❤✐➯♥✱ ✈ỵ✐ t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❦❤↔ ♥➠♥❣ ỏ ợ ữ r ✤÷đ❝ ♠ët sè ❜➔✐ t♦→♥ ✤➥♥❣ t❤ù❝ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❧ỵ♣ ❤➔♠ ❧♦❣❛r✐t ✈➔ →♣ ❞ư♥❣ ✈➔♦ ♠ët sè ❜➔✐ t♦→♥ ✈➲ ❞➣② sè ✈➔ ❣✐ỵ✐ ❤↕♥✳ ▲✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✱ t→❝ ❣✐↔ r➜t ữủ ỵ õ õ t tø ❝→❝ t❤➛② ❣✐→♦✱ ❝æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ✻✼ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ ❱➠♥ ▼➟✉ ✭✶✾✾✼✮✱ ❬✷❪ ◆❣✉②➵♥ ❱➠♥ ▼➟✉ ✭✷✵✵✻✮✱ P❤÷ì♥❣ tr➻♥❤ ❤➔♠✱ ◆❳❇ ●✐→♦ ❞ư❝✳ ❇➜t ✤➥♥❣ t❤ù❝✱ ỵ ◆❣✉②➵♥ ❱➠♥ ▼➟✉ ✭✶✾✾✸✮✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤✱ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ◆❳❇ ●✐→♦ ❞ư❝✳ ❬✹❪ ✣à♥❤ ❚❤à ◆❛♠ ✭✷✵✶✶✮✱ ♣❤÷ì♥❣ tr➻♥❤ s✐➯✉ ✈✐➺t✱ ▲✉➟♥ ✈➠♥ ❚❤↕❝ sÿ✱ ✣↕✐ ❤å❝ ✣➔ ◆➤♥❣✳ ❬✺❪ ❚↕♣ ❝❤➼ ❚❍&❚❚ ✭✷✵✵✼✮✱ ❱✐➺t ◆❛♠ ✭✶✾✾✵✲✷✵✵✻✮✱ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ❈→❝ ❜➔✐ t❤✐ ❖❧②♠♣✐❝ ❚♦→♥ tr✉♥❣ ❤å❝ ♣❤ê t❤æ♥❣ ◆❳❇ ●✐→♦ ❞ö❝✳ ❚✐➳♥❣ ❆♥❤ ❬✻❪ ❚❡♦❞♦r❛✲▲✐❧✐❛♥❛✱ ❚✳ ❘❛❞✉❧❡s❝✉✱ ❘❛❞✉❧❡s❝✉✱ ❱✳❉✳ ❘❛❞✉❧❡s❝✉✱ ❚✳ ❆♥✲ ❞r❡❡s❝✉ ✭✷✵✵✾✮✱ r❡❛❧ ❛①✐s✳ Pr♦❜❧❡♠s ✐♥ ❘❡❛❧ ❆♥❛❧②s✐s✿ ❆❞✈❛♥❝❡❞ ❈❛❧❝✉❧✉s ♦♥ t❤❡ ❙♣r✐♥❣❡r✳