Phương pháp giải phương trình bất phương trình chứa logarit và các bài toán liên quan_2

Header Page of 128 ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ị Pì PP P❍×❒◆● ❚❘➐◆❍ ❇❻❚ P❍×❒◆● ❚❘➐◆❍ ❈❍Ù❆ ▲❖●❆❘■❚ ❱⑨ ❈⑩❈ ❇⑨■ ❚❖⑩◆ ▲■➊◆ ◗❯❆◆ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙ß ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ✲ ◆❿▼ ✷✵✶✹ Footer Page of 128 Header Page of 128 ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ị Pì PP P❍×❒◆● ❚❘➐◆❍ ❇❻❚ P❍×❒◆● ❚❘➐◆❍ ❈❍Ù❆ ▲❖●❆❘■❚ ❱⑨ ❈⑩❈ ❇⑨■ ❚❖⑩◆ ▲■➊◆ ◗❯❆◆ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙ß ❚❖⑩◆ ❍➴❈ ❈❤✉②➯♥ Pì PP P số ữớ ữợ ◆●❯❨➊◆ ✲ ◆❿▼ ✷✵✶✹ Footer Page of 128 Header Page of 128 ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✸ ✶ ❚➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ sè ❧♦❣❛r✐t ✈➔ ❝→❝ ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✺ ✶✳✶ ❚➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ sè ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ❈→❝ ỵ trủ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸ ▲ỵ♣ ❤➔♠ t✉➛♥ ❤♦➔♥ ✈➔ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✳ ✳ ✳ ✳ ✶✵ ✶✳✸✳✶ ▲ỵ♣ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✸✳✷ ▲ỵ♣ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷ P❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t ✶✸ ✷✳✶ ✷✳✷ ✷✳✸ P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✶✳✶ P❤÷ì♥❣ ♣❤→♣ ♠ơ ❤â❛ ✈➔ ✤÷❛ ✈➲ ❝ò♥❣ ❝ì sè ✳ ✳ ✳ ✳ ✶✸ ✷✳✶✳✷ P❤÷ì♥❣ ♣❤→♣ ✤➦t ➞♥ ♣❤ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✶✳✸ P❤÷ì♥❣ ♣❤→♣ ❤➡♥❣ sè ❜✐➳♥ t❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✶✳✹ P❤÷ì♥❣ ♣❤→♣ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✶✳✺ Ù♥❣ ❞ö♥❣ ✤à♥❤ ỵ r ỵ ✷✳✶✳✻ P❤÷ì♥❣ ♣❤→♣ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✶✳✼ P❤÷ì♥❣ ♣❤→♣ ✤→♥❤ ❣✐→ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✷✳✶ P❤÷ì♥❣ ♣❤→♣ ♠ơ ❤â❛ ✈➔ ✤÷❛ ✈➲ ❝ò♥❣ ❝ì sè ✳ ✳ ✳ ✳ ✸✻ ✷✳✷✳✷ P❤÷ì♥❣ ♣❤→♣ ✤➦t ➞♥ ♣❤ư ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✷✳✸ P❤÷ì♥❣ ♣❤→♣ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✷✳✹ P❤÷ì♥❣ ♣❤→♣ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✷✳✺ P❤÷ì♥❣ ♣❤→♣ ✤→♥❤ ❣✐→ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♠ët sè ❤➺ ❝❤ù❛ ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✐ Footer Page of 128 Header Page of 128 ✷✳✸✳✶ P❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ✤ê✐ t÷ì♥❣ ✤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✷✳✸✳✷ P❤÷ì♥❣ ♣❤→♣ ✤➦t ➞♥ ♣❤ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✳✸✳✸ P❤÷ì♥❣ ♣❤→♣ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✷✳✸✳✹ P❤÷ì♥❣ ♣❤→♣ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✷✳✸✳✺ P❤÷ì♥❣ ♣❤→♣ ✤→♥❤ ❣✐→ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✸ ❈→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ sè ❧♦❣❛r✐t ✸✳✶ Pữỡ tr t ữỡ tr tr ợ ❧♦❣❛r✐t ✸✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Pữỡ tr tr ợ rt ✳ ✳ ✳ ✳ ✺✻ ✸✳✶✳✷ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ tr♦♥❣ ❧ỵ♣ ❤➔♠ ❧♦❣❛r✐t ✳ ✳ ✳ ✻✹ ❈→❝ ❜➔✐ t♦→♥ ✈➲ ❞➣② sè ✈➔ ❣✐ỵ✐ ❤↕♥ ❞➣② sè s✐♥❤ ❜ð✐ ❤➔♠ ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✻✼ ✼✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐ Footer Page of 128 ✺✻ ✼✻ Header Page of 128 ▼ð ✤➛✉ P❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ♥ë✐ ❞✉♥❣ ❝ì ❜↔♥ ✈➔ q✉❛♥ trå♥❣ ❝õ❛ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ❜➟❝ tr✉♥❣ ❤å❝ ♣❤ê t❤ỉ♥❣✳ ✣➦❝ ❜✐➺t ❧➔ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t ❧➔ ♥❤ú♥❣ ♥ë✐ ❞✉♥❣ ❤❛② ✈➔ ❦❤â ố ợ s ú tữớ t tr ❝→❝ ✤➲ t❤✐ t✉②➸♥ s✐♥❤ ✤↕✐ ❤å❝✱ ❝❛♦ ✤➥♥❣ ✈➔ ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐✳ ❱✐➺❝ ❣✐↔♥❣ ❞↕② ❤➔♠ sè rt ữủ ữ ữỡ tr ợ tr ✤â ♣❤➛♥ ❦✐➳♥ t❤ù❝ ✈➲ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t ❝❤✐➳♠ ✈❛✐ trá trå♥❣ t➙♠✳ ❚✉② ♥❤✐➯♥ ❞♦ t❤í✐ ❣✐❛♥ ❤↕♥ ❤➭♣ ❝õ❛ ❝❤÷ì♥❣ tr➻♥❤ ♣❤ê t❤ỉ♥❣ ♥➯♥ tr♦♥❣ s→❝❤ ❣✐→♦ ❦❤♦❛ ❦❤ỉ♥❣ ♥➯✉ ✤÷đ❝ ✤➛② ✤õ ✈➔ ❝❤✐ t✐➳t t➜t ❝↔ ❝→❝ ❞↕♥❣ ❜➔✐ t♦→♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t ✈➔ ❝→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥✳ ❱➻ ✈➟② ❤å❝ s✐♥❤ t❤÷í♥❣ ❣➦♣ ♥❤✐➲✉ ❦❤â ❦❤➠♥ ❦❤✐ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ♥➙♥❣ ❝❛♦ ✈➲ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t tr♦♥❣ ❝→❝ ✤➲ t❤✐ ✤↕✐ ❤å❝✱ ❝❛♦ ✤➥♥❣ ✈➔ ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐✳ ▼➦❝ ❞ò ✤➣ ❝â ♥❤✐➲✉ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✈➲ rt ợ ữ ữ õ ✤➲ r✐➯♥❣ ❦❤↔♦ s→t ✈➲ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t ♠ët ❝→❝❤ ❤➺ t❤è♥❣✳ ✣➦❝ ❜✐➺t✱ ♥❤✐➲✉ ❞↕♥❣ t♦→♥ ✈➲ ✤↕✐ sè ✈➔ ❧♦❣❛r✐t ❝â q✉❛♥ ❤➺ ❝❤➦t ợ ổ t t rớ ữủ t♦→♥ ❝❤ù❛ ❧♦❣❛r✐t ❝➛♥ ❝â sü trđ ❣✐ó♣ ❝õ❛ ✤↕✐ sè✱ ❣✐↔✐ t➼❝❤ ✈➔ ♥❣÷đ❝ ❧↕✐✳ ❉♦ ✤â✱ ✤➸ ✤→♣ ù♥❣ ♥❤✉ ❝➛✉ ✈➲ ❣✐↔♥❣ ❞↕②✱ ❤å❝ t➟♣ ✈➔ ❣â♣ ♣❤➛♥ ♥❤ä ❜➨ ✈➔♦ sü ♥❣❤✐➺♣ ❣✐→♦ ❞ö❝✱ ❧✉➟♥ ✈➠♥ ✧P❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t ✈➔ ❝→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥✧ ♥❤➡♠ ❤➺ t❤è♥❣ ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t ❦➳t ✸ Footer Page of 128 Header Page of 128 ❤đ♣ ✈ỵ✐ ❦✐➳♥ t❤ù❝ ✤↕✐ sè✱ ❣✐↔✐ t➼❝❤ ✤➸ tê♥❣ ❤ñ♣✱ ❝❤å♥ ❧å❝ ✈➔ ♣❤➙♥ ❧♦↕✐ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t ✈➔ ①➙② ❞ü♥❣ ♠ët sè ❧ỵ♣ ❜➔✐ t♦→♥ ợ ữủ ữỡ ữỡ ❚➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ sè ❧♦❣❛r✐t ✈➔ ❝→❝ ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥✳ ✲ ◆❤➢❝ ❧↕✐ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ số rt ỵ trủ ▲ỵ♣ ❤➔♠ t✉➛♥ ❤♦➔♥ ✈➔ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤✳ ❈❤÷ì♥❣ ✷✳ P❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t✳ ✲ ❚r➻♥❤ ❜➔② ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t✳ ✲ ❚r➻♥❤ ❜➔② ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ❧♦❣❛r✐t✳ ✲ ❚r➻♥❤ ❜➔② ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♠ët sè ❤➺ ❝❤ù❛ ❧♦❣❛r✐t✳ ❈❤÷ì♥❣ ✸✳ ❈→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ sè ❧♦❣❛r✐t✳ ✲ ữỡ tr t ữỡ tr tr ợ ❤➔♠ ❧♦❣❛r✐t✳ ✲ ◆➯✉ ❝→❝ ❜➔✐ t♦→♥ ✈➲ ❞➣② sè ✈➔ ❣✐ỵ✐ ❤↕♥ ❞➣② sè s✐♥❤ ❜ð✐ ❤➔♠ ❧♦❣❛r✐t✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤è✐ ợ sữ s ữớ t trỹ t ữợ t ❧✐➺✉ ✈➔ tr✉②➲♥ ✤↕t ♥❤ú♥❣ ❦✐♥❤ ♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ ❝❤♦ tỉ✐✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛②✱ ❝ỉ ❣✐→♦ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✲ ❚✐♥✱ ♣❤á♥❣ ✣➔♦ t↕♦ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❚r÷í♥❣ P ỗ ú ✤ï t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ ❤♦➔♥ t❤➔♥❤ ❜↔♥ ❧✉➟♥ ỵ ự Footer Page of 128 Header Page of 128 ❈❤÷ì♥❣ ✶ ❚➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ sè ❧♦❣❛r✐t ✈➔ ❝→❝ ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✶✳✶ ❚➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ sè ❧♦❣❛r✐t f (x) = loga x, < a = ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ sè ❧♦❣❛r✐t ❝ì sè a✳ ◆❤➟♥ ①➨t r➡♥❣ t➟♣ ①→❝ ✤à♥❤ D = (0; +∞) ✈➔ t➟♣ ❣✐→ trà I = R✳ ❚r♦♥❣ ❝→❝ ♣❤➛♥ t✐➳♣ t❤❡♦✱ t❛ ❣✐↔ sû < a = 1✳ ◆❤➟♥ ①➨t r➡♥❣ ❤➔♠ sè f (x) = loga x ❧✐➯♥ tư❝ ✈➔ ❝â ✤↕♦ ❤➔♠ ✈ỵ✐ ♠å✐ x > 0✱ ❤ì♥ ♥ú❛ f (x) = x ln a ❚❛ ❦❤↔♦ s→t t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❤➔♠ sè f (x) = loga x tr♦♥❣ ✷ tr÷í♥❣ ❤đ♣✳ ✲ ❚r÷í♥❣ ❤đ♣ ✶✿ a > 1✳ ❑❤✐ ✤â✱ ln a > ♥➯♥ s✉② r❛ f (x) = > 0, ∀x > x ln a ❱➟②✱ ❦❤✐ a > t❤➻ f (x) = loga x ❧➔ ❤➔♠ ỗ tr õ f (1) = 0, f (a) = ✈➔ lim loga x = −∞; lim loga x = +∞ + ❍➔♠ sè x→+∞ x→0 ❚❛ ❝â ❜↔♥❣ ❜✐➳♥ t❤✐➯♥ s❛✉✿ x 1 y = loga x −∞ ✺ Footer Page of 128 a +∞ +∞ Header Page of 128 < a < 1✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② f (x) < 0, ∀x ∈ D ✳ ❱➟②✱ ❦❤✐ < a < t❤➻ f (x) = loga x ❧➔ ❤➔♠ ✲ ❚r÷í♥❣ ❤đ♣ ✷✿ sè ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ ❉✳ ❚❛ ❝â ❜↔♥❣ ❜✐➳♥ t❤✐➯♥ s❛✉✿ x +∞ a +∞ y = loga x −∞ ❚➼♥❤ ❝❤➜t ✶✳✶✳ f (x) = logax ✈➔ ♥❣❤à❝❤ ỗ tr D = R+ ❦❤✐ a>1 < a < ❚➼♥❤ ❝❤➜t ✶✳✷✳ ❱ỵ✐ ♠å✐ ❚➼♥❤ ❝❤➜t ✶✳✸✳ ❱ỵ✐ ♠å✐ a > 0✱ a = ✈➔ x1 , x2 ∈ (0; +∞)✱ x1 loga (x1 x2 ) = loga x1 + loga x2 ✱ loga = loga x1 − loga x2 x2 a > 0✱ a = ✈➔ x > 0✳ ❱ỵ✐ α t❛ ❝â ❜➜t ❦ý✱ t❛ ❝â loga xα = αloga x ❚➼♥❤ ❝❤➜t ✶✳✹✳ ❱ỵ✐ ♠å✐ < a = 1, < c = loga x = ❚➼♥❤ ❝❤➜t ✶✳✺✳ ❍➔♠ sè ❚➼♥❤ ❝❤➜t ✶✳✻✳ ❱ỵ✐ ♠å✐ ✈➔ x > 0✱ t❛ ❝â logc x logc x f (x) = loga x (0 < a = 1) ❝â ✤↕♦ ❤➔♠ t↕✐ ♠å✐ ✤✐➸♠ x ∈ (0; +∞) ✈➔ (loga x) = ◆➳✉ ❤➔♠ sè u = u(x) ❝â ✤↕♦ ❤➔♠ x ln a tr➯♥ ❦❤♦↔♥❣ J ∈ R t❤➻ ❤➔♠ sè y = loga u(x)✱ (0 < a = 1) ❝â ✤↕♦ ❤➔♠ u (x) tr➯♥ ❏ ✈➔ (loga u(x)) = u(x) ln a • ❑❤✐ a>1 • ❑❤✐ 0 x2 ✻ Footer Page of 128 ✈➔ t❛ ❝â Header Page of 128 ✶✳✷ ❈→❝ ✤à♥❤ ỵ trủ c (a; b) t tỗ t y = f (x) ❧✐➯♥ ❝❤♦ f (c) = 0✳ ◆➳✉ ❤➔♠ sè s❛♦ ◆➳✉ ❤➔♠ sè tö❝ tr➯♥ [a; b] ✈➔ f (a).f (b) < y = f (x) ❧✐➯♥ tö❝ tr➯♥ [a; b]✱ f (a) = A, f (b) = B t❤➻ ❤➔♠ sè ♥❤➟♥ ♠å✐ ❣✐→ trà tr✉♥❣ ❣✐❛♥ ❣✐ú❛ ❆ ✈➔ ❇✳ ❍➺ q✉↔ ✶✳✶✳ ◆➳✉ ❤➔♠ sè y = f (x) ❧✐➯♥ tö❝ tr➯♥ [a; b] t❤➻ ♥â ♥❤➟♥ ♠å✐ ❣✐→ trà tr✉♥❣ ❣✐❛♥ ❣✐ú❛ ❣✐→ trà ❧ỵ♥ ♥❤➜t ✈➔ ❣✐→ trà ♥❤ä ♥❤➜t✳ ✣à♥❤ ❧➼ ✶✳✸ [a; b]✱ ✳ ❝â ✤↕♦ ❤➔♠ tr➯♥ ❦❤♦↔♥❣ s❛♦ ❝❤♦ f : [a; b] → R t❤ä❛ ♠➣♥ ❢ (a; b) f (a) = f (b) t tỗ ❤➔♠ sè ❧✐➯♥ tö❝ tr➯♥ t↕✐ c ∈ (a; b) f (c) = 0✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ f (x) ❧✐➯♥ tö❝ tr➯♥ [a; b] ♥➯♥ t❤❡♦ ✤à♥❤ ❧➼ ❲❡✐❡rstr❛ss f (x) ♥❤➟♥ ❣✐→ trà [a; b]✳ tr➯♥ [a; b]✱ ❧ỵ♥ ♥❤➜t ▼ ✈➔ ❣✐→ trà ♥❤ä ♥❤➜t ♠ tr➯♥ M = m t❛ ❝â f (x) ❧➔ ❤➔♠ ❤➡♥❣ ❞♦ ✤â ✈ỵ✐ ♠å✐ c ∈ (a; b) ❧✉ỉ♥ ❝â f (c) = 0✳ ✲ ❑❤✐ M > m✱ ✈➻ f (a) = f (b) tỗ t c (a; b) s❛♦ ❝❤♦ f (c) = m ❤♦➦❝ f (c) = M ✱ t❤❡♦ ❜ê ✤➲ ❋❡r♠❛t s✉② r❛ f (c) = 0✳ ✲ ❑❤✐ ❍➺ q✉↔ ✶✳✷✳ ◆➳✉ ❤➔♠ sè f (x) ❝â ♥ ♥❣❤✐➺♠ ✭♥ ❧➔ sè ♥❣✉②➯♥ ữỡ ợ ỡ tr t n1 (a; b) tr➯♥ ❍➺ q✉↔ ✶✳✸✳ ❍➺ q✉↔ ✶✳✹✳ f (x) ❝â ✤↕♦ ❤➔♠ f (x) ❝â ♥❤✐➲✉ ♥❤➜t ◆➳✉ ❤➔♠ sè ✈æ ♥❣❤✐➺♠ tr➯♥ (a; b) (a; b) ✈➔ f (x) (a; b) t❤➻ f (x) ❝â ➼t ❝â ✤↕♦ ❤➔♠ tr➯♥ ❦❤♦↔♥❣ t❤➻ ◆➳✉ ❤➔♠ sè f (x) ✶ ♥❣❤✐➺♠ n+1 ✣à♥❤ ❧➼ ✶✳✹ tr➯♥ ✤♦↕♥ ♥❣❤✐➺♠ tr➯♥ ✳ ✭▲❛❣r❛♥❣❡✮ [a; b]✱ (a; b)✳ f : [a; b] → R t❤ä❛ ♠➣♥ (a; b)✱ ❦❤✐ ✤â ∃c ∈ (a; b) : ❈❤♦ ❤➔♠ sè ❦❤↔ ✈✐ tr➯♥ ❦❤♦↔♥❣ f (c) = f (b) − f (a) b−a ✼ Footer Page of 128 (a; b) ✈➔ f (x) (a; b) t❤➻ f (x) ❝â ❝â ✤↕♦ ❤➔♠ tr➯♥ ❦❤♦↔♥❣ ❝â ♥❤✐➲✉ ♥❤➜t ♥ ♥❣❤✐➺♠ ✭♥ ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣✮ tr➯♥ ♥❤✐➲✉ ♥❤➜t (a; b) ✈➔ f (x) tr➯♥ (a; b)✳ tr➯♥ ❦❤♦↔♥❣ ❢ ❧✐➯♥ tö❝ Header Page 10 of 128 ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ❤➔♠ sè F (x) = f (x) − f (b) − f (a) x b−a ❚❛ ❝â F (x) ❧➔ ❤➔♠ ❧✐➯♥ F (a) = F (b)✳ tö❝ tr➯♥ ✤♦↕♥ [a; b] ✱ ❝â ✤↕♦ ❤➔♠ tr➯♥ ❦❤♦↔♥❣ (a; b) ✈➔ c ∈ (a; b) s❛♦ ❝❤♦ F (c) = 0✳ f (b) − f (a) f (b) − f (a) F (x) = f (x) − ✱ s✉② r❛ f (c) = ✳ b−a b−a tỗ t q ◆➳✉ F (x) = ✈ỵ✐ ♠å✐ ① t❤✉ë❝ ❦❤♦↔♥❣ (a; b) t❤➻ F (x) ❜➡♥❣ ❤➡♥❣ sè tr➯♥ ❦❤♦↔♥❣ ✤â✳ ✣à♥❤ ❧➼ ✶✳✺✳ ✲ ◆➳✉ ✲ ◆➳✉ f (x) ❝â ✤↕♦ ❤➔♠ tr➯♥ ❦❤♦↔♥❣ (a; b)✳ f (x) > 0, x (a; b) t f (x) ỗ tr➯♥ (a; b)✳ f (x) < 0, ∀x ∈ (a; b) t❤➻ f (x) ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ (a; b)✳ ✣à♥❤ ❧➼ ✶✳✻ ❦ý ❈❤♦ ❤➔♠ sè ✳ ✭❇➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲❙❝❤✇❛r③✮ a1 , a2 , , an ✈➔ b1 , b2 , , bn ✳ ❈❤♦ ❤❛✐ ❝➦♣ ❞➣② sè ❜➜t ❑❤✐ ✤â (a1 b1 + a2 b2 + + an bn )2 ≤ (a21 + a22 + + a2n )(b21 + b22 + + b2n ) ❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❈❤ù♥❣ ♠✐♥❤✳ ∃k ✤➸ = kbi , ∀i ∈ (1, 2, , n) ❳➨t t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ f (x) = (a21 +a22 + +a2n )x2 −2(a1 b1 +a2 b2 + +an bn )x+(b21 +b22 + +b2n ) ✲ ◆➳✉ a21 + a22 + + a2n = ⇔ a1 = a2 = = an = ❜➜t ✤➥♥❣ t❤ù❝ ❤✐➸♥ ♥❤✐➯♥ ✤ó♥❣✳ ✲ ◆➳✉ a21 + a22 + + a2n > t t f (x) ữợ ❞↕♥❣ f (x) = (a1 x − b1 )2 + (a2 x − b2 )2 + + (an x − bn )2 ≥ 0, ∀x ∈ R ❚❤❡♦ ✤à♥❤ ỵ t tự t = (a1 b1 + a2 b2 + + an bn )2 − (a21 + a22 + + a2n )(b21 + b22 + + b2n ) ≤ ⇔ (a1 b1 + a2 b2 + + an bn )2 ≤ (a21 + a22 + + a2n )(b21 + b22 + + b2n ) ✽ Footer Page 10 of 128