✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ◆●❯❨➍◆ ì ị P ❉Ö◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✵ S hóa b i Trung tâm H c li u - i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ◆●❯❨➍◆ ❚❍➚ ì ị P ệ ❈❤✉②➯♥ ♥❣➔♥❤✿ P❍×❒◆● P❍⑩P ❚❖⑩◆ ❙❒ ❈❻P ▼❶ ❙➮✿ ✻✵✳✹✻✳✹✵ ữớ ữợ ❤å❝✿ ●❙✳❚❙❑❍✳ ◆●❯❨➍◆ ❱❿◆ ▼❾❯ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✵ S hóa b i Trung tâm H c li u - i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✐ ▼ö❝ ❧ö❝ ỵ ởt số rở ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ r ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tr ổ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷ ❑❤↔♦ s→t t➼♥❤ ❝❤➜t ❝ì số ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỗ ó ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✶ ❚➼♥❤ t ỗ ó ✷✳✷✳✷ ✣ë ❣➛♥ ✤➲✉ ✈➔ s➢♣ t❤ù tü ❝→❝ t❛♠ ❣✐→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ▼ët sè ự ỵ tr số ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶ ❈❤ù♥❣ ♠✐♥❤ sü tỗ t số ữỡ tr ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✸ ❙ü ♣❤➙♥ ❜è ♥❣❤✐➺♠ ❝õ❛ ✤❛ t❤ù❝ ✈➔ ✤↕♦ ❤➔♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹ ▼ët ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ❦❤❛✐ tr✐➸♥ ❚❛②❧♦r✲●♦♥t❝❤❛r♦✈✳ ✸✳✺ ❈❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ❇➔✐ t➟♣ ❜ê s✉♥❣ ❑➳t ❧✉➟♥ ❉❛♥❤ ♠ư❝ ❝→❝ ❝ỉ♥❣ tr➻♥❤ ❧✐➯♥ q✉❛♥ ✤➳♥ ❧✉➟♥ ✈➠♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ S hóa b i Trung tâm H c li u - i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✶✶ ✶✶ ✶✸ ✶✸ ✶✽ ✷✸ ✷✸ ✸✺ ✹✷ ✹✽ ✺✵ ✻✶ ✻✺ ✻✼ ỵ ởt số rở ỵ ỵ r ỵ ỵ tr ởt ổ ỵ q trồ tr tr✉♥❣ ❜➻♥❤ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ ❣✐↔✐ t➼❝❤ ❝ê ✤✐➸♥✳ Ù♥❣ ỵ tr ữỡ tr t ❚r✉♥❣ ❤å❝ ♣❤ê t❤ỉ♥❣ r➜t ✤❛ ❞↕♥❣ ✈➔ ♣❤♦♥❣ ♣❤ó✱ ✤➦❝ ❜✐➺t ❧➔ ❝→❝ ❞↕♥❣ t♦→♥ ✈➲ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✱ ❜✐➺♥ ❧✉➟♥ sè ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ♠ët ❦❤♦↔♥❣✱ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝✱ ①➨t ❝ü❝ trà ❝õ❛ ❤➔♠ sè✳✳✳ ❚✉② ♥❤✐➯♥✱ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ s→❝❤ ❣✐→♦ ❦❤♦❛ ❞➔♥❤ ❝❤♦ ❤å❝ s✐♥❤ ♣❤ê t❤ỉ♥❣ t❤➻ ❝→❝ ù♥❣ ❞ư♥❣ ỵ ữ ữủ tr ởt ❝→❝❤ ❤➺ t❤è♥❣ ✈➔ ✤➛② ✤õ✳ ❱ỵ✐ s✉② ♥❣❤➽ ✈➔ t ỵ tữ õ t ✈➠♥ ♥➔② ❧➔ ♥❤➡♠ ❝✉♥❣ ❝➜♣ t❤➯♠ ❝❤♦ ❝→❝ ❡♠ ❤å❝ s✐♥❤✱ ✤➦❝ ❜✐➺t ❧➔ ❝→❝ ❡♠ ❤å❝ s✐♥❤ ❦❤→✱ ❣✐ä✐✱ ❝â ♥➠♥❣ ❦❤✐➳✉ ✈➔ ②➯✉ t❤➼❝❤ ♠æ♥ t♦→♥ ♠ët t➔✐ ❧✐➺✉✱ ♥❣♦➔✐ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ❝á♥ ❝â t❤➯♠ ♠ët ❤➺ t❤è♥❣ ❝→❝ ❜➔✐ t➟♣ ♥➙♥❣ ❝❛♦✱ q✉❛ ✤â s➩ t❤➜② rã ❤ì♥ ❝→❝ ❞↕♥❣ t♦→♥ ù♥❣ ❞ư♥❣ rt ú ỵ ỵ r ởt số ỵ rở t ụ ữợ ỵ t t tỏ ỳ ❧í✐ ❣✐↔✐ ❤❛②✱ ✤ë❝ ✤→♦ ✤➦❝ t❤ị ❝❤♦ tø♥❣ ❞↕♥❣ t t tứ õ t ỵ tự s t↕♦ ♥❤ú♥❣ ❜➔✐ t♦→♥ ♠ỵ✐✳ ◆❣♦➔✐ r❛✱ ✤➙② ❝ơ♥❣ ❧➔ ♥❤ú♥❣ ❦➳t q✉↔ ♠➔ ❜↔♥ t❤➙♥ t→❝ ❣✐↔ s➩ t✐➳♣ tö❝ ❤♦➔♥ t❤✐➺♥ tr♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❣✐↔♥❣ ❞↕② t♦→♥ t✐➳♣ t❤❡♦ ð tr÷í♥❣ ♣❤ê t❤ỉ♥❣✳ ▲✉➟♥ ✈➠♥ ♥❣♦➔✐ ♠ư❝ ❧ư❝✱ ❧í✐ ♥â✐ ✤➛✉✱ ❦➳t ❧✉➟♥ ✈➔ t➔✐ t ỗ ố ữỡ ữỡ ỵ ❘♦❧❧❡ ✈➔ ♠ët sè ♠ð rë♥❣✳ S hóa b i Trung tâm H c li u - i h c Thái Ngun DeThiMau.vn http://www.lrc-tnu.edu.vn ✷ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ♥➔② ♥❤➡♠ tr ởt ỡ t ỵ ✈➲ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ❝ò♥❣ ♠ët sè ❤➺ q✉↔ q trồ ỵ tt ỡ s ✈➟♥ ❞ö♥❣ ❝❤♦ ❝→❝ ❜➔✐ t♦→♥ ù♥❣ ❞ö♥❣ ð ♥❤ú♥❣ ❝❤÷ì♥❣ s❛✉✳ ❈❤÷ì♥❣ ✷✳ ❑❤↔♦ s→t t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ sè✳ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ù♥❣ trỹ t ỵ ỵ ▲❛❣r❛♥❣❡ tr♦♥❣ ✈✐➺❝ ❦❤↔♦ s→t ❤❛✐ t➼♥❤ ❝❤➜t r➜t ❝ì ❜↔♥ ✈➔ q✉❛♥ trå♥❣ ❝õ❛ ❤➔♠ sè tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t P õ t ỗ t t ỗ ó số ữỡ ởt số ự ỵ tr♦♥❣ ✤↕✐ sè✳ ✣➙② ❧➔ ♥ë✐ ❞✉♥❣ trå♥❣ t➙♠ ❝õ❛ ú tổ ự ỵ ỵ rở tr t♦→♥ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✱ ❜✐➺♥ ❧✉➟♥ sè ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✱ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝✱ sü ♣❤➙♥ ❜è ♥❣❤✐➺♠ ❝õ❛ ✤❛ t❤ù❝ ✈➔ ✤↕♦ ❤➔♠✳ ❈→❝ ❜➔✐ t➟♣ ♠✐♥❤ ❤å❛ ✤÷đ❝ ❧ü❛ ❝❤å♥ tø ✤➲ t❤✐ ❝õ❛ ❝→❝ ❦➻ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ ◗✉è❝ ❣✐❛✱ ❝→❝ ❦➻ t❤✐ ❖❧②♠♣✐❝ ❦❤✉ ✈ü❝ ✈➔ ◗✉è❝ t➳✱ ♠ët sè ❜➔✐ t➟♣ ❞♦ t→❝ ❣✐↔ tü s→♥❣ t→❝✳ ✣è✐ ✈ỵ✐ ♠é✐ ❞↕♥❣ ❜➔✐ t➟♣ ✤➲✉ ♥➯✉ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❝ư t❤➸✱ ❝â ✤÷❛ r❛ ♥❤ú♥❣ ❜➔✐ t♦→♥ ✈ỵ✐ ❧í✐ ❣✐↔✐ ✤ë❝ ✤→♦ ✤➛② t➼♥❤ s→♥❣ t↕♦ ✈➔ ❜➜t ♥❣í✳ ❈❤÷ì♥❣ ✹✳ ❇➔✐ t➟♣ s ữỡ ợ t ởt số t t✐➯✉ ❜✐➸✉ ✤➣ ✤÷đ❝ s➢♣ ①➳♣ ✈➔ ❧ü❛ ❝❤å♥ ❦ÿ ữù ộ õ ữợ ✈➟♥ ❞ư♥❣ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ t❤✉ ✤÷đ❝ tø ❜❛ ❝❤÷ì♥❣ trữợ ♥➠♥❣ t➼♥❤ t♦→♥ ❝ư t❤➸✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ữợ sỹ ữợ ❞➙♥✱ ●❙✲❚❙❑❍ ◆❣✉②➵♥ ❱➠♥ ▼➟✉✱ t→❝ ❣✐↔ ①✐♥ ✤÷đ❝ tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ tỵ✐ ●❙ ✲ ◆❣÷í✐ ❚❤➛② r➜t ♥❣❤✐➯♠ ❦❤➢❝ ✈➔ t➟♥ t➙♠ tr♦♥❣ ổ tr tử tự qỵ ❝ơ♥❣ ♥❤÷ ❦✐♥❤ ♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ✤➲ t➔✐✳ ❚→❝ ❣✐↔ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❇❛♥ ❣✐→♠ ❤✐➺✉✱ P❤á♥❣ ✤➔♦ t↕♦ s❛✉ ✣↕✐ ❤å❝✱ ❑❤♦❛ ❚♦→♥✲❚✐♥ ❝õ❛ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ S hóa b i Trung tâm H c li u - i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✸ ❤å❝ ũ qỵ t ổ t ữợ ❝❤♦ ❧ỵ♣ ❈❛♦ ❤å❝ ❚♦→♥ ❑✷✳ ❚→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❯❇◆❉ ❚➾♥❤✱ ❙ð ●✐→♦ ❞ư❝ ✈➔ ✣➔♦ t↕♦ ❚➾♥❤ ❈❛♦ ❇➡♥❣✱ ❇❛♥ ❣✐→♠ ❤✐➺✉ ✈➔ t➟♣ t❤➸ ❝→♥ ❜ë ❣✐→♦ ✈✐➯♥ ❚r÷í♥❣ ❚❍P❚ ❉➙♥ të❝ ◆ë✐ tró ❚➾♥❤ ❈❛♦ ❇➡♥❣ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ t→❝ ❣✐↔ ❝â ❝ì ❤ë✐ ✤÷đ❝ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ✤÷đ❝ ❝↔♠ ì♥ sü q✉❛♥ t➙♠✱ ❣✐ó♣ ✤ï ♥❤✐➺t t➻♥❤ ❝õ❛ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ❈❛♦ ❤å❝ trữớ ố ợ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝✳ ✣➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✱ t→❝ ❣✐↔ ✤➣ t➟♣ tr✉♥❣ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ♠ët ❝→❝❤ ♥❣❤✐➯♠ tó❝ tr♦♥❣ s✉èt ❦❤â❛ ❤å❝✱ ❝ơ♥❣ ♥❤÷ r➜t ❝➞♥ t❤➟♥ tr♦♥❣ ❦❤➙✉ ❝❤➳ ❜↔♥ ▲❛❚❡①✳ ❚✉② ♥❤✐➯♥ ❞♦ ❝á♥ ❤↕♥ ❝❤➳ ✈➲ t❤í✐ ❣✐❛♥✱ ❦❤↔ ♥➠♥❣ ✈➔ ❤♦➔♥ ❝↔♥❤ ❣✐❛ ✤➻♥❤ ♥➯♥ tr♦♥❣ q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✱ t→❝ ❣✐↔ r➜t ữủ sỹ qỵ t ổ ỳ õ ỵ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✵✾ ♥➠♠ ✷✵✶✵✳ ◆❣÷í✐ t❤ü❝ ❤✐➺♥ ◆❣✉②➵♥ ❚❤à ❉÷ì♥❣ ❑✐➲✉ S hóa b i Trung tâm H c li u - i h c Thỏi Nguyờn DeThiMau.vn http://www.lrc-tnu.edu.vn ữỡ ỵ ❘♦❧❧❡ ✈➔ ♠ët sè ♠ð rë♥❣ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tổ ợ t ỵ ởt số rở ỵ ởt sè ❤➺ q✉↔ q✉❛♥ trå♥❣ ❝ơ♥❣ ✤÷đ❝ tr➻♥❤ ❜➔② ð ✤➙② ✤➸ t❤✉➟♥ ❧đ✐ ❝❤♦ ✈✐➺❝ ✈➟♥ ❞ư♥❣ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣ t✐➳♣ t❤❡♦✳ ỵ ỡ s ỵ ỹ ỵ ỡ t r str❛ss ✤è✐ ✈ỵ✐ ❤➔♠ ❧✐➯♥ tư❝ ❦❤➥♥❣ ✤à♥❤ r➡♥❣ ❦❤✐ f ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a, b] t❤➻ ♥â ♣❤↔✐ ✤↕t ❣✐→ trà ❧ỵ♥ ♥❤➜t ✈➔ ❣✐→ trà ♥❤ä ♥❤➜t tr õ ỵ rt ỹ trà ❝õ❛ ❤➔♠ ❦❤↔ ✈✐ ❦❤➥♥❣ ✤à♥❤ r➡♥❣ ♥➳✉ ❤➔♠ ❦❤↔ ✈✐ g(x) tr♦♥❣ (a, b) ✤↕t ❝ü❝ trà ✭❝ü❝ ✤↕✐ ❤♦➦❝ ❝ü❝ t✐➸✉✮ t↕✐ ♠ët ✤✐➸♠ tr♦♥❣ ❦❤♦↔♥❣ ✤â t t õ ỵ ỵ sỷ f tö❝ tr➯♥ ✤♦↕♥ [a; b] ✈➔ ❝â ✤↕♦ ❤➔♠ t↕✐ ♠å✐ x ∈ (a; b)✳ ◆➳✉ f (a) = f (b) t tỗ t t t ởt c (a; b) s❛♦ ❝❤♦ f ′ (c) = 0✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ f ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a; b] ♥➯♥ t ỵ rstrss f t tr ❝ü❝ ✤↕✐ ✈➔ ❣✐→ trà ❝ü❝ t✐➸✉ tr➯♥ ✤♦↕♥ [a; b]✱ tù❝ ❧➔ S hóa b i Trung tâm H c li u - i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn tỗ t x1, x2 (a; b) s❛♦ ❝❤♦ f (x1 ) = f (x) = m, f (x2 ) = max f (x) = M [a;b] [a;b] ❈â ❤❛✐ ❦❤↔ ♥➠♥❣✿ a) m = M ❑❤✐ ➜② f (x) = const tr➯♥ ✤♦↕♥ [a; b]✱ ❞♦ ✤â f ′ (x) = ✈ỵ✐ ♠å✐ x ∈ (a; b) ✈➔ c ❧➔ ✤✐➸♠ ❜➜t ❦➻ tr➯♥ ❦❤♦↔♥❣ ✤â✳ b) m < M ✳ ❑❤✐ ✤â ✈➻ ✤✐➲✉ ❦✐➺♥ f (a) = f (b) ♥➯♥ ➼t ♥❤➜t ♠ët tr♦♥❣ ❤❛✐ ✤✐➸♠ x1, x2 s➩ ❦❤ỉ♥❣ trị♥❣ ✈ỵ✐ ❝→❝ ✤➛✉ ♠ót ❝õ❛ ✤♦↕♥ [a; b]✳ ●✐↔ sû x1 (a; b) t ỵ rt t t ỵ ữủ ự t ỵ ♥â✐ ❝❤✉♥❣ s➩ ❦❤ỉ♥❣ ❝á♥ ✤ó♥❣ ♥➳✉ tr♦♥❣ ❦❤♦↔♥❣ (a; b) ❝â ✤✐➸♠ c ♠➔ t↕✐ ✤â f ′ (c) ổ tỗ t t f (x) = − x2 , x ∈ [−1; 1]✳ ❉➵ t❤➜② f (x) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥✿ f (x) ❧✐➯♥ tö❝ tr➯♥ (−1; 1) ✈➔ f (−1) = f (1)✳ ❚❛ ①➨t ✤↕♦ ❤➔♠ f ′(x) = − 3√2 x ✱ rã r➔♥❣ t↕✐ x0 = ∈ (−1; 1) ổ tỗ t số ổ t ỵ ✷✮ ✣✐➲✉ ❦✐➺♥ ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a; b] ✤è✐ ✈ỵ✐ ❤➔♠ f (x) ❝ơ♥❣ ❦❤ỉ♥❣ t❤➸ t❤❛② ❜ð✐ ✤✐➲✉ ❦✐➺♥ f (x) ❧✐➯♥ tö❝ tr♦♥❣ ❦❤♦↔♥❣ (a; b)✳ ❈❤➥♥❣ ❤↕♥✱ ①➨t ❤➔♠ 1, ♥➳✉ x = 0, f (x) = x, ♥➳✉ < x ≤ Ð ✤➙② x = ❧➔ ✤✐➸♠ ❣✐→♥ ✤♦↕♥✳ ❑❤✐ ✤â✱ ró r ổ tỗ t x0 (0, 1) f ′(x0) = ✸✮ Þ ♥❣❤➽❛ ❤➻♥❤ ❤å❝✿ ◆➳✉ ỵ ữủ t t tr ỗ t số y = f (x), x [a; b] tỗ t M (c; f (c)), c ∈ (a; b) ♠➔ t✐➳♣ t✉②➳♥ t↕✐ ✤â s♦♥❣ s♦♥❣ ✈ỵ✐ trư❝ ❤♦➔♥❤ Ox✳ ❍➺ q✉↔ ✶✳✶✳ ◆➳✉ ❤➔♠ sè f (x) ❝â ✤↕♦ ❤➔♠ tr➯♥ ❦❤♦↔♥❣ (a; b) ✈➔ ♣❤÷ì♥❣ tr➻♥❤ f (x) = ❝â n ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t t❤✉ë❝ ❦❤♦↔♥❣ (a; b) t❤➻ ♣❤÷ì♥❣ 3 S hóa b i Trung tâm H c li u - i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✻ tr➻♥❤ f ′(x) = ❝â ➼t ♥❤➜t n − ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t t❤✉ë❝ ❦❤♦↔♥❣ (a; b) ✭P❤÷ì♥❣ tr➻♥❤ f (k)(x) = ❝â ➼t ♥❤➜t n − k ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t t❤✉ë❝ ❦❤♦↔♥❣ (a; b)✱ ✈ỵ✐ k = 1, 2, , n✮✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ♣❤÷ì♥❣ tr➻♥❤ f (x) = ❝â n ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t t❤✉ë❝ ❦❤♦↔♥❣ (a; b) ✤➣ ✤÷đ❝ s➢♣ t❤ù tü x1 < x2 < · · · < xn õ ỵ n − ✤♦↕♥ [x1; x2], [x2; x3], , [xn−1; xn] t❤➻ ♣❤÷ì♥❣ tr➻♥❤ f ′(x) = ❝â ➼t ♥❤➜t n − ♥❣❤✐➺♠ t❤✉ë❝ n − ❦❤♦↔♥❣ (x1 ; x2 ), (x2 ; x3 ), , (xn−1 ; xn )✳ ●å✐ n − ♥❣❤✐➺♠ ✤â ❧➔ ξ1 , ξ2 , , ξn−1 t❤➻ t❛ ❝â f ′ (ξ1 ) = f ′ (ξ2 ) = · · · = f ′ (ξn−1 ) = ❚✐➳♣ tö❝ →♣ ỵ n (1; ξ2), , (ξn−2; ξn−1) t❤➻ ♣❤÷ì♥❣ tr➻♥❤ f ′′(x) = ❝â ➼t ♥❤➜t n − tr (a; b) tử ỵ tr s k ữợ ữỡ tr f (k)(x) = õ ➼t ♥❤➜t n − k ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t tr➯♥ ❦❤♦↔♥❣ (a; b) ❍➺ q✉↔ ✶✳✷✳ ●✐↔ sû ❤➔♠ sè f (x) ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a; b] ✈➔ ❝â ✤↕♦ ❤➔♠ tr➯♥ ❦❤♦↔♥❣ (a; b)✳ ❑❤✐ ✤â✱ ♥➳✉ ♣❤÷ì♥❣ tr➻♥❤ f ′(x) = ❝â ❦❤æ♥❣ q✉→ n − ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t tr➯♥ ❦❤♦↔♥❣ (a; b) t❤➻ ♣❤÷ì♥❣ tr➻♥❤ f (x) = ❝â ❦❤æ♥❣ q✉→ n ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t tr➯♥ ❦❤♦↔♥❣ ✤â✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ♣❤÷ì♥❣ tr➻♥❤ f (x) = ❝â ♥❤✐➲✉ ❤ì♥ n ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t tr➯♥ ❦❤♦↔♥❣ (a; b)✱ ❝❤➥♥❣ ❤↕♥ ❧➔ n + ♥❣❤✐➺♠✱ t❤➳ t❤➻ t❤❡♦ ❤➺ q✉↔ 1.1 ♣❤÷ì♥❣ tr➻♥❤ f ′(x) = ❝â ➼t ♥❤➜t n ♥❣❤✐➺♠ t❤✉ë❝ ❦❤♦↔♥❣ (a; b)✳ ✣✐➲✉ ♥➔② tr→✐ ✈ỵ✐ ❣✐↔ t❤✐➳t✳ ❱➟② ♣❤÷ì♥❣ tr➻♥❤ f (x) = ❝â ❦❤ỉ♥❣ q✉→ n ♥❣❤✐➺♠ tr➯♥ ❦❤♦↔♥❣ (a; b)✳ ❚✐➳♣ t❤❡♦✱ t❛ ①➨t ♠ët rở ỵ q số f (x) t ỗ tớ t ❝❤➜t s❛✉ ✤➙②✿ ✐✮ f (x) ①→❝ ✤à♥❤ ✈➔ ❝â ✤↕♦ ❤➔♠ ❝➜♣ n (n ≥ 1) ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a; b] ✐✐✮ f (x) ❝â ✤↕♦ ❤➔♠ ❝➜♣ n + tr♦♥❣ ❦❤♦↔♥❣ (a; b) S hóa b i Trung tâm H c li u - i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✼ ✐✐✐✮ f (a) = f ′(a) = · · · = f (n)(a) = 0, f (b) = õ tỗ t ✤✐➸♠ b1, b2, , bn+1 ♣❤➙♥ ❜✐➺t t❤✉ë❝ ❦❤♦↔♥❣ (a; b)s❛♦ ❝❤♦ f (k) (bk ) = 0, k = 1, 2, , n + ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❣✐↔ t❤✐➳t f (a) = f (b) = 0, t ỵ tỗ t b1 ∈ (a; b) s❛♦ ❝❤♦ f ′(b1) = 0, ❦➳t ❤đ♣ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ f ′(a) = 0, s✉② r tỗ t b2 (a; b1) (a; b) s❛♦ ❝❤♦ f ′′(b2) = ▲↕✐ ❦➳t ❤đ♣ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ f ′′(a) = ✈➔ t✐➳♣ tö❝ →♣ ỵ t õ f (b3) = ✈ỵ✐ b3 ∈ (a; b2 ) ⊂ (a; b) ❚✐➳♣ tử ữ ữợ tự n tỗ t bn ∈ (a; bn−1) ⊂ (a; b) s❛♦ ❝❤♦ f (n)(bn) = 0, ❦➳t ❤đ♣ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ f (n)(a) = 0, s r tỗ t bn+1 (a; bn ) ⊂ (a; b) s❛♦ ❝❤♦ f (n+1) (bn+1 ) = ữ tỗ t t b1, b2, , bn+1 tr♦♥❣ ❦❤♦↔♥❣ (a; b) s❛♦ ❝❤♦ f (k) (bk ) = 0, k = 1, 2, , n + ❈❤➼♥❤ ỳ q ỵ tr t❤➔♥❤ ♠ët ❝ỉ♥❣ ❝ư r➜t ♠↕♥❤ ✤➸ ❣✐↔✐ t♦→♥✱ ✤➦❝ t ố ợ t ữỡ tr ✈➔ ❦✐➸♠ ❝❤ù♥❣ sè ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr♦♥❣ ♠ët ❦❤♦↔♥❣ ♥➔♦ ✤â✳ ❈→❝ ù♥❣ ❞ư♥❣ ♥➔② s➩ ✤÷đ❝ tr➻♥❤ tt tr ữỡ s ỵ r ỵ t t t ởt số ỵ q t tt ợ ỵ ỵ ỵ r sỷ f ❧➔ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a; b] ✈➔ ❝â ✤↕♦ ❤➔♠ t↕✐ ♠å✐ ✤✐➸♠ tr♦♥❣ ❦❤♦↔♥❣ (a; b) ❑❤✐ õ tỗ t t t ởt c (a; b) s❛♦ ❝❤♦ f (b) − f (a) = f ′ (c)(b − a) ✭✶✳✶✮ F (x) = f (x) − λx, ✭✶✳✷✮ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ①➨t ❤➔♠ ♣❤ö S hóa b i Trung tâm H c li u - i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✽ tr♦♥❣ ✤â sè λ ✤÷đ❝ ❝❤å♥ s❛♦ ❝❤♦ F (a) = F (b)✱ tù❝ ❧➔ s❛♦ ❝❤♦ f (a) − λa = f (b) − λb ✣➸ ❝â ✤✐➲✉ ✤â ❝❤➾ ❝➛♥ ❧➜② λ= f (b) − f (a) b−a ✭✶✳✸✮ ❘ã r➔♥❣ ❤➔♠ F (x) ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a; b], ❝â ✤↕♦ ❤➔♠ tr♦♥❣ ❦❤♦↔♥❣ (a; b) ✈➔ F (a) = F (b), õ t ỵ tỗ t c (a; b) s F (c) = ❚ø ✭✶✳✷✮ t❛ ❝â F ′ (x) = f ′ (x) − λ✱ ❞♦ ✤â F ′ (c) = ⇔ f ′ (c) − λ = ⇔ f ′ (c) = λ f (a) ❚❤❛② ❣✐→ trà λ tø ✭✶✳✸✮ ✈➔♦ t❛ ❝â f ′(c) = f (b)b − , ❤❛② −a f (b) − f (a) = f ′ (c)(b − a) ❈æ♥❣ t❤ù❝ ✭✶✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ ❝ỉ♥❣ t❤ù❝ sè ❣✐❛ ❤ú✉ ❤↕♥ ▲❛❣r❛♥❣❡✳ ◆❤➟♥ ①➨t ✶✳✷✳ ✶✮ ❚❛ ✤➣ t❤✉ ✤÷đ❝ ✤à♥❤ ỵ r ữ ởt q ỵ ữ ỵ ❜✐➸✉ t❤ù❝✮ ❧↕✐ ❧➔ ♠ët tr÷í♥❣ ❤đ♣ r✐➯♥❣ ❝õ❛ ✤à♥❤ ỵ r ự ợ tt f (a) = f (b)✮✳ ✷✮ Þ ♥❣❤➽❛ ❤➻♥❤ ❤å❝✿ ◆➳✉ ❤➔♠ f (x) t ỵ r t tr ỗ t số y = f (x) tỗ t t t ởt M (c; f (c)) s t t ợ ỗ t t↕✐ ✤✐➸♠ ✤â s♦♥❣ s♦♥❣ ✈ỵ✐ ❞➙② ❝✉♥❣ AB ✱ ð ✤â A(a; f (a)) ✈➔ B(b; f (b))✳ ′ ❍➺ q✉↔ ✶✳✹✳ ●✐↔ sû f : [a; b] −→ R ❧➔ ❤➔♠ ❧✐➯♥ tö❝ ✈➔ f (x) = 0✱ ✈ỵ✐ ♠å✐ x ∈ (a; b)✳ ❑❤✐ ✤â f = const tr➯♥ ✤♦↕♥ [a; b] ❈❤ù♥❣ ♠✐♥❤✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû x0 ∈ (a; b) ❧➔ ♠ët ✤✐➸♠ ❝è ✤à♥❤ õ ỏ x tý ỵ (a; b)✳ ✣♦↕♥ t❤➥♥❣ [x0; x] ❤♦➦❝ [x; x0] ♥➡♠ trå♥ tr♦♥❣ ❦❤♦↔♥❣ (a; b)✱ ✈➻ t❤➳ f ❝â ✤↕♦ ❤➔♠ ✭✈➔ ❞♦ ✤â ♥â ❧✐➯♥ tư❝✮ ❦❤➢♣ ♥ì✐ tr➯♥ ✤♦↕♥ ỵ r t õ f (x) − f (xo ) = f ′ (c)(x − x0 ), ∀c ∈ (xo ; x) S hóa b i Trung tâm H c li u - i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✾ ◆❤÷♥❣ t❤❡♦ ❣✐↔ t❤✐➳t f ′(x) = ✈ỵ✐ ♠å✐ x ∈ (a; b) ♥➯♥ f ′(c) = ✈ỵ✐ ♠å✐ c ∈ (x0 ; x)✳ ❱➻ t❤➳ t❛ ❝â f (x) = f (x0 )✱ ✤➥♥❣ t❤ù❝ ♥➔② ❦❤➥♥❣ ✤à♥❤ r➡♥❣ ❣✐→ trà ❝õ❛ ❤➔♠ f (x) t↕✐ ✤✐➸♠ ❜➜t ❦ý x ∈ (a; b) ❧✉æ♥ ❧✉æ♥ ❜➡♥❣ ❣✐→ trà ❝õ❛ ❤➔♠ t↕✐ ♠ët ✤✐➸♠ ❝è ✤à♥❤✳ ❉♦ ✈➟②✱ f = const tr➯♥ ✤♦↕♥ [a; b] ❍➺ q✉↔ ✶✳✺✳ ◆➳✉ ❤❛✐ ❤➔♠ f (x) g(x) õ ỗ t tr➯♥ ♠ët ❦❤♦↔♥❣ t❤➻ ❝❤ó♥❣ ❝❤➾ s❛✐ ❦❤→❝ ♥❤❛✉ ❜ð✐ ❤➡♥❣ sè ❝ë♥❣✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤➟t ✈➟②✱ t❤❡♦ ❣✐↔ t❤✐➳t t❛ ❝â [f (x) − g(x)]′ = f ′ (x) − g ′ (x) = ❚❤❡♦ ❤➺ q✉↔ ✶✳✹ t❤➻ f (x) − g(x) = C (C = const) f (x) = g(x) + C ỵ ỵ sỷ f, g ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a; b] ✈➔ ❝â ✤↕♦ ❤➔♠ t↕✐ ♠å✐ ✤✐➸♠ tr♦♥❣ ❦❤♦↔♥❣ (a; b), ♥❣♦➔✐ r❛ g ′ (x) = ✈ỵ✐ ♠å✐ x ∈ (a; b) õ tỗ t t t ởt c ∈ (a; b) s❛♦ ❝❤♦ f (b) − f (a) f ′ (c) = ′ g(b) − g(a) g (c) ự rữợ ự ỵ t❛ ♥❤➟♥ ①➨t r➡♥❣ ❝æ♥❣ t❤ù❝ ✭✶✳✹✮ ❧✉æ♥ ❝â ♥❣❤➽❛✱ tù❝ ❧➔ g(b) = g(a)✳ ❚❤➟t ✈➟②✱ ♥➳✉ g(b) = g(a) t❤➻ ❤➔♠ sè g(x) t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ỵ õ tỗ t c ∈ (a; b) s❛♦ ❝❤♦ g′(c) = 0✱ ♥❤÷♥❣ ✤✐➲✉ ♥➔② tr→✐ ✈ỵ✐ ❣✐↔ t❤✐➳t g ′ (x) = 0, ∀x ∈ (a; b) ❇➙② ❣✐í t❛ ①➨t ❤➔♠ ♣❤ư F (x) = f (x) − λg(x), ✭✶✳✺✮ tr♦♥❣ ✤â sè λ ✤÷đ❝ ❝❤å♥ s❛♦ ❝❤♦ F (a) = F (b)✱ tù❝ ❧➔ f (a) − λg(a) = f (b) − λg(b) ✣➸ ❝â ✤✐➲✉ ✤â t❛ ❝❤➾ ❝➛♥ ❧➜② λ= S hóa b i Trung tâm H c li u - f (b) − f (a) g(b) − g(a) i h c Thái Nguyên DeThiMau.vn ✭✶✳✻✮ http://www.lrc-tnu.edu.vn ✶✵ ❍➔♠ F (x) t❤♦↔ ♠➣♥ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ✤à♥❤ ỵ õ c (a; b) s F ′(c) = ▼➦t ❦❤→❝ tø ✭✶✳✺✮ t❛ ❝â F ′(x) = f ′(x) − λg′(x) ♥➯♥ f ′ (c) F (c) = ⇔ f (c) − λg (c) = ⇔ λ = ′ g (c) ′ ′ ′ ✭✶✳✼✮ ❚ø ✭✶✳✻✮✈➔ ✭✶✳✼✮ t❛ t❤✉ ✤÷đ❝ f (b) − f (a) f ′ (c) = ′ g(b) − g(a) g (c) ❈ỉ♥❣ t❤ù❝ ✭✶✳✹✮ ✤÷đ❝ ❣å✐ ❧➔ ❝æ♥❣ t❤ù❝ sè ❣✐❛ ❤ú✉ ❤↕♥ ❈❛✉❝❤②✳ ◆❤➟♥ t ỵ r trữớ ủ r ỵ ợ tt g(x) = x ỵ tr ổ r t t rở ỵ r ✈ỉ ❤↕♥✳ ❈ì sð ❝õ❛ ❝→❝ ♠ð rë♥❣ ♥➔② ❧➔ ỹ ỵ r trà ❝õ❛ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a, b] ❧➜♣ ✤➛② ❝→❝ ❣✐→ trà tr♦♥❣ ✤♦↕♥ f (x), max f (x) [a,b] [a,b] ỵ sỷ ❤➔♠ sè f (x) ❧✐➯♥ tö❝ tr➯♥ [a; +∞)✱ ❝â ✤↕♦ ❤➔♠ tr♦♥❣ (a; +∞) ✈➔ lim f (x) = f (a) õ tỗ t c (a; +) s❛♦ ❝❤♦ f ′ (c) = 0✳ x→+∞ ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ f (x) = f (a) ✈ỵ✐ ♠å✐ x > a t❤➻ ❧➜② c ❧➔ ♠ët sè ❜➜t ❦ý ❧ỵ♥ ỡ a sỷ tỗ t b > a s ❝❤♦ f (b) = f (a)✱ ❝❤➥♥❣ ❤↕♥ f (b) > f (a)✳ ●å✐ µ ❧➔ ♠ët sè t❤ü❝ ❜➜t ý tở (f (a); f (b)), t ỵ tỗ t (a; b) s f () = µ✳ ❱➻ x→+∞ lim f (x) = f (a) < tỗ t d > b s f (d) < µ ❉♦ f (x) ❧✐➯♥ tư❝ tr➯♥ [a; +) t ỵ tỗ t ∈ (b; d) s❛♦ ❝❤♦ f (β) = µ = f (), õ t ỵ tỗ t c ∈ (α; β) s❛♦ ❝❤♦ f ′(c) = S hóa b i Trung tâm H c li u - i h c Thái Ngun DeThiMau.vn http://www.lrc-tnu.edu.vn ✶✶ ❈❤÷ì♥❣ ✷ ❑❤↔♦ s→t t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ sè t ỗ t ỗ ó ❝õ❛ ❤➔♠ sè ❧➔ ♥❤ú♥❣ ✈➜♥ ✤➲ ❝ì ❜↔♥ tr♦♥❣ ữỡ tr t P ỵ r õ ởt trỏ q trồ tr ự ỵ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤✳ ◆❣♦➔✐ r❛✱ tr♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ ✤➲ ❝➟♣ ✤➳♥ ❦❤→✐ ♥✐➺♠ ✤ë ❣➛♥ ✤➲✉ ✈➔ s➢♣ t❤ù tü ❝→❝ t❛♠ ❣✐→❝✱ ♠➔ ❞ü❛ ✈➔♦ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♥â t❛ ❝â ữủ rt tú ố ợ ởt số ❜➔✐ t♦→♥ ✈➲ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ t❛♠ ❣✐→❝ ✭①❡♠ ỗ ứ s❛✉✱ t❛ sû ❞ö♥❣ ❦➼ ❤✐➺✉ I(a; b) ⊂ R ❧➔ ♥❤➡♠ ♥❣➛♠ ✤à♥❤ ♠ët tr♦♥❣ ❜è♥ t➟♣ ❤ñ♣ (a; b), [a; b), (a; b] ✈➔ [a; b] ✈ỵ✐ a < b ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳ ●✐↔ sû ❤➔♠ sè f (x) ①→❝ ✤à♥❤ tr➯♥ t➟♣ I(a; b) ⊂ R ✈➔ t❤♦↔ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❱ỵ✐ ♠å✐ x1, x2 ∈ I(a; b) ✈➔ x1 < x2✱ t❛ ✤➲✉ ❝â f (x1) ≤ f (x2) t❤➻ t❛ ♥â✐ r➡♥❣ f (x) ❧➔ ♠ët ❤➔♠ ✤ì♥ ✤✐➺✉ t➠♥❣ tr➯♥ I(a; b) ✣➦❝ ❜✐➺t✱ ❦❤✐ ù♥❣ ✈ỵ✐ ♠å✐ ❝➦♣ x1, x2 ∈ I(a; b) ✈➔ x1 < x2✱ t❛ ✤➲✉ ❝â f (x1 ) < f (x2 ) t❤➻ t❛ ♥â✐ r➡♥❣ f (x) ❧➔ ♠ët ❤➔♠ ✤ì♥ ✤✐➺✉ t➠♥❣ t❤ü❝ sü S hóa b i Trung tâm H c li u - i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✶✷ tr➯♥ I(a; b) ữủ ợ x1, x2 I(a; b) ✈➔ x1 < x2✱ t❛ ✤➲✉ ❝â f (x1) ≥ f (x2 ) t❤➻ t❛ ♥â✐ r➡♥❣ f (x) ❧➔ ♠ët ❤➔♠ ✤ì♥ ✤✐➺✉ ❣✐↔♠ tr➯♥ I(a; b) ✣➦❝ ❜✐➺t✱ ❦❤✐ ù♥❣ ✈ỵ✐ ♠å✐ ❝➦♣ x1, x2 ∈ I(a; b) ✈➔ x1 < x2✱ t❛ ✤➲✉ ❝â f (x1 ) > f (x2 ) t❤➻ t❛ ♥â✐ r➡♥❣ f (x) ❧➔ ♠ët ❤➔♠ ✤ì♥ ✤✐➺✉ ❣✐↔♠ t❤ü❝ sü tr➯♥ I(a; b) ◆❤ú♥❣ ❤➔♠ ✤ì♥ ✤✐➺✉ t➠♥❣ t❤ü❝ sü tr➯♥ I(a, b) ữủ ỗ tr I(a; b) ✈➔ ❤➔♠ ✤ì♥ ✤✐➺✉ ❣✐↔♠ t❤ü❝ sü tr➯♥ I(a; b) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ I(a; b) ❚r♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ ❣✐↔✐ t➼❝❤✱ ❝❤ó♥❣ t❛ ✤➣ ❜✐➳t ✤➳♥ ❝→❝ t✐➯✉ ❝❤✉➞♥ ✤➸ ♥❤➟♥ ❜✐➳t ✤÷đ❝ ❦❤✐ ♥➔♦ t❤➻ ♠ët số trữợ tr (a; b) ❧➔ ♠ët ❤➔♠ ✤ì♥ ✤✐➺✉ tr➯♥ ❦❤♦↔♥❣ ✤â✳ ❙❛✉ ✤➙② ú t s ũ ỵ r ự ỵ ừ t ỡ số ởt ỵ rt q trồ tr ữỡ tr t ợ P ỵ số y = f (x) õ ✤↕♦ ❤➔♠ tr➯♥ ❦❤♦↔♥❣ (a; b) ✐✮ ◆➳✉ f ′(x) > ✈ỵ✐ ♠å✐ x ∈ (a; b) t❤➻ ❤➔♠ số y = f (x) ỗ tr õ ✐✐✮ ◆➳✉ f ′(x) < ✈ỵ✐ ♠å✐ x ∈ (a; b) t❤➻ ❤➔♠ sè y = f (x) ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ ❦❤♦↔♥❣ ✤â✳ ❈❤ù♥❣ ♠✐♥❤✳ ▲➜② ❤❛✐ ✤✐➸♠ x1, x2 (x1 < x2) tr➯♥ ❦❤♦↔♥❣ (a; b)✳ ❱➻ f (x) ❝â ✤↕♦ ❤➔♠ tr➯♥ ❦❤♦↔♥❣ (a; b) ♥➯♥ f (x) ❧✐➯♥ tö❝ tr➯♥ [x1 ; x2 ] ✈➔ ❝â tr (x1; x2) ỵ ▲❛❣r❛♥❣❡ ❝❤♦ ❤➔♠ sè y = f (x) tr➯♥ [x1; x2]✱ ❦❤✐ ✤â ∃c ∈ (x1 ; x2 ) s❛♦ ❝❤♦ f (x2 ) − f (x1 ) = f ′ (c)(x2 − x1 ) i) ◆➳✉ f ′ (x) > tr➯♥ ❦❤♦↔♥❣ (a; b) t❤➻ f ′ (c) > 0✱ ♠➦t ❦❤→❝ x2 −x1 > ♥➯♥ f (x2) − f (x1) > ❤❛② f (x2) > f (x1), s r f (x) ỗ tr ❦❤♦↔♥❣ (a; b)✳ S hóa b i Trung tâm H c li u - i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✶✸ ii) ◆➳✉ f ′ (x) < tr➯♥ ❦❤♦↔♥❣ (a; b) t❤➻ f ′ (c) < 0✱ ♠➦t ❦❤→❝ x2 −x1 > ♥➯♥ f (x2 ) − f (x1 ) < ❤❛② f (x2 ) < f (x1 ), s✉② r❛ ❤➔♠ f (x) ♥❣❤à❝❤ ❜✐➳♥ tr (a; b) ỵ rở ỵ sỷ số y = f (x) ❝â ✤↕♦ ❤➔♠ tr➯♥ ❦❤♦↔♥❣ (a; b)✳ ◆➳✉ f ′ (x) ≥ ✭❤♦➦❝ f ′ (x) ≤ 0✮ ✈➔ ✤➥♥❣ t❤ù❝ ❝❤➾ ①↔② r❛ t↕✐ ♠ët sè ❤ú✉ tr (a; b) t f (x) ỗ ❜✐➳♥ ✭❤♦➦❝ ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ ❦❤♦↔♥❣ ✤â✮✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤➟t ✈➟②✱ ✤➸ ✤ì♥ ❣✐↔♥ ❝→❝❤ ❧➟♣ ❧✉➟♥✱ ❣✐↔ sû r➡♥❣ f ′(x) ≥ tr➯♥ (a; b) ✈➔ f ′ (x) = t↕✐ x1 ∈ (a, b) t❤➻ ❦❤✐ õ f (x) ỗ tr tứ (a, x1 ) ✈➔ (x1 , b) ✈➔ ❧✐➯♥ tö❝ tr♦♥❣ (a, x1 ] [x1 , b) õ ụ ỗ ❜✐➳♥ tr♦♥❣ (a, x1 ] ✈➔ [x1 , b) ❚ø õ s r õ ỗ tr (a, b) ỗ ó t ỗ ó số f (x) ữủ ỗ tr➯♥ t➟♣ I(a; b) ⊂ R ♥➳✉ ✈ỵ✐ ♠å✐ x1 , x2 ∈ I(a; b) ✈➔ ✈ỵ✐ ♠å✐ ❝➦♣ sè ❞÷ì♥❣ α, β ❝â tê♥❣ α + β = 1✱ t❛ ✤➲✉ ❝â f (αx1 + βx2 ) ≤ αf (x1 ) + βf (x2 ) ✭✷✳✶✮ ◆➳✉ ❞➜✉ ✤➥♥❣ t❤ù❝ tr♦♥❣ ✭✷✳✶✮ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x1 = x2 t t õ f (x) ỗ t❤ü❝ sü ✭❝❤➦t✮ tr➯♥ I(a; b) ✐✐✮ ❍➔♠ sè f (x) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ❧ã♠ tr➯♥ t➟♣ I(a; b) ⊂ R ♥➳✉ ✈ỵ✐ ♠å✐ x1 , x2 ∈ I(a; b) ợ số ữỡ , õ tê♥❣ α + β = 1✱ t❛ ✤➲✉ ❝â f (αx1 + βx2 ) ≥ αf (x1 ) + βf (x2 ) ✭✷✳✷✮ ◆➳✉ ❞➜✉ ✤➥♥❣ t❤ù❝ tr♦♥❣ ✭✷✳✷✮ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x1 = x2 t❤➻ t❛ ♥â✐ f (x) ❧➔ ❤➔♠ ❧ã♠ t❤ü❝ sü ✭❝❤➦t✮ tr➯♥ I(a; b) S hóa b i Trung tâm H c li u - i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✶✹ ◆❤➟♥ ①➨t ✷✳✶✳ ❑❤✐ x1 < x2 t❤➻ x = x1 + x2 ợ số ữỡ , β ❝â tê♥❣ α + β = ✤➲✉ t❤✉ë❝ (x1 ; x2 ) ✈➔ α= x − x1 x2 − x ; β= x2 − x1 x2 − x1 ỵ f (x) số ❦❤↔ ✈✐ tr➯♥ I(a; b) t❤➻ f (x) ❧➔ ❤➔♠ ỗ tr I(a; b) f (x) ❧➔ ❤➔♠ ✤ì♥ ✤✐➺✉ t➠♥❣ tr➯♥ I(a; b)✳ ❈❤ù♥❣ sỷ f (x) ỗ tr I(a; b) ✤â ✈ỵ✐ x1 < x < x2 ✱ (x, x1 , x2 ∈ I(a; b))✱ t❛ ❝â ❱➻ t❤➳ x2 − x x − x1 > 0; > ✈➔ x2 − x1 x2 − x1 x2 − x x − x1 + = x2 − x1 x2 − x1 x − x1 x2 − x f (x1 ) + f (x2 ) x2 − x1 x2 − x1 f (x) − f (x1 ) f (x2 ) − f (x) ⇔ ≤ x − x1 x2 − x f (x) ≤ ✭✷✳✸✮ ❚r♦♥❣ ✭✷✳✸✮ ❝❤♦ x → x1 ✱ t❛ t❤✉ ✤÷đ❝ f (x2 ) − f (x1 ) x2 − x1 ✭✷✳✹✮ f (x2 ) − f (x1 ) ≤ f ′ (x2 ) x2 − x1 ✭✷✳✺✮ f ′ (x1 ) ≤ ❚÷ì♥❣ tü✱ tr♦♥❣ ✭✷✳✸✮ ❝❤♦ x → x2 ✱ t❛ t❤✉ ✤÷đ❝ ❚ø ✭✷✳✹✮ ✈➔ ✭✷✳✺✮✱ t❛ ♥❤➟♥ ✤÷đ❝ f ′ (x1 ) ≤ f ′ (x2 )✱ tù❝ ❤➔♠ sè f ′ (x) ❧➔ ❤➔♠ ✤ì♥ ✤✐➺✉ t➠♥❣✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû f ′ (x) ❧➔ ❤➔♠ sè ✤ì♥ ✤✐➺✉ t➠♥❣ ✈➔ x1 < x < x2 (x, x1 , x2 I(a; b)) ỵ r tỗ t x3 , x4 ✈ỵ✐ x3 ∈ (x1 ; x) ✈➔ x4 ∈ (x; x2 ) s❛♦ ❝❤♦ f (x) − f (x1 ) = f ′ (x3 ), x − x1 f (x2 ) − f (x) = f ′ (x4 ) x2 − x S hóa b i Trung tâm H c li u - i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✶✺ ❉♦ f ′ (x3 ) ≤ f ′ (x4 ) ♥➯♥ f (x) ≤ f (x2 ) − f (x) f (x) − f (x1 ) ≤ , ❤❛② t❛ ❝â x − x1 x2 − x x − x1 x2 − x f (x1 ) + f (x2 ) x2 − x1 x2 − x1 ❚ù❝ f (x) ỗ tr I(a; b) ỵ ✷✳✹✳ ◆➳✉ f (x) ❦❤↔ ✈✐ ❜➟❝ ❤❛✐ tr➯♥ I(a; b) t f (x) ỗ ó tr I(a; b) ✈➔ ❝❤➾ ❦❤✐ f ′′ (x) ≥ (f ′′ (x) ≤ 0) tr➯♥ I(a; b) ❈❤ù♥❣ ♠✐♥❤✳ ❙✉② trü❝ t tứ ỵ 2.3 s t t ỗ ó tự sè ❦❤↔ ✈✐ ❜➟❝ ❤❛✐ ❝â ✤↕♦ ❤➔♠ ❝➜♣ ✷ ❦❤æ♥❣ ✤ê✐ ❞➜✉ tr♦♥❣ I(a; b) ❍➺ q✉↔ ✷✳✶✳ ◆➳✉ số y = f (x) ỗ ó tr I(a; b) t❤➻ ♣❤÷ì♥❣ tr➻♥❤ f (x) = ❝â ❦❤æ♥❣ q✉→ ❤❛✐ ♥❣❤✐➺♠ t❤✉ë❝ I(a; b) ❈❤ù♥❣ ♠✐♥❤✳ ❚❤➟t sỷ số y = f (x) ỗ ❤♦➦❝ ❧ã♠ tr➯♥ I(a; b), tù❝ f ′′ (x) > ❤♦➦❝ f ′′ (x) < tr➯♥ I(a; b) õ số f (x) ổ ỗ ❤♦➦❝ ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ I(a; b), ♥➯♥ ♣❤÷ì♥❣ tr➻♥❤ f ′ (x) = ❝â ❦❤æ♥❣ q✉→ ✶ ♥❣❤✐➺♠ tr♦♥❣ ❦❤♦↔♥❣ I(a; b) ❉♦ ✤â t❤❡♦ ❤➺ q✉↔ 1.2 ♣❤÷ì♥❣ tr➻♥❤ f (x) = ❝â ❦❤æ♥❣ q✉→ ✷ ♥❣❤✐➺♠ tr➯♥ ❦❤♦↔♥❣ ✤â✳ ◆❤➟♥ ①➨t ✷✳✷✳ ❱ỵ✐ ❤➺ q✉↔ ♥➔②✱ ❝❤ó♥❣ t❛ ❝â t❤➯♠ ♠ët ❝ỉ♥❣ ❝ư ❤ú✉ ❤✐➺✉ ✤➸ →♣ ❞ư♥❣ ❝❤♦ ❝→❝ ❞↕♥❣ t♦→♥ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤✱ ❝❤ù♥❣ sỹ tỗ t ữỡ tr ú tổ s ợ t ữỡ tổ q ử t tr ữỡ s ỵ ✭❇➜t ✤➥♥❣ t❤ù❝ ❑❛r❛♠❛t❛✮✳ ❈❤♦ ❤❛✐ ❞➣② sè {xk , yk ∈ I(a; b), k = 1, 2, , n}✱ t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥✿ x1 ≥ x2 ≥ · · · ≥ xn , y1 ≥ y2 ≥ · · · ≥ yn S hóa b i Trung tâm H c li u - i h c Thái Nguyên DeThiMau.vn http://www.lrc-tnu.edu.vn ✶✻ ✈➔ x1 ≥ y1 , x + x2 ≥ y1 + y2 , ··· x1 + x2 + · · · + xn−1 ≥ y1 + y2 + · · · + yn−1 , x + x + · · · + x = y + y + · · · + y n n ❑❤✐ ✤â✱ ù♥❣ ợ ỗ tỹ sỹ f (x) tr I(a; b)✱ t❛ ✤➲✉ ❝â f (x1 ) + f (x2 ) + · · · + f (xn ) ≥ f (y1 ) + f (y2 ) + · · à + f (yn ) ự rữợ t t ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ f (x1 ) ≥ f (y1 ) + f ′ (y1 )(x1 − y1 ), ∀x1 , y1 ∈ I(a; b) ✭✷✳✻✮ ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x1 = y1 ❚❤➟t ✈➟②✱ t❛ ❝â ✭✷✳✻✮ ⇔ f (x1 ) − f (y1 ) ≥ f ′ (y1 )(x1 − y1 ) ✭✷✳✼✮ ❚❛ ①➨t ✸ tr÷í♥❣ ❤đ♣✳ i) ◆➳✉ x1 = y1 t❤➻ t❛ ❝â ❞➜✉ ✤➥♥❣ t❤ù❝✱ ❞♦ ✤â ✭✷✳✼✮ ✤ó♥❣✳ ii) ◆➳✉ x1 > y1 t❤➻ x1 − y1 > ♥➯♥ ✭✷✳✼✮ ⇔ f (x1 ) − f (y1 ) ≥ f ′ (y1 ) x1 − y1 ✭✷✳✽✮ f (x1 ) − f (y1 ) ≤ f ′ (y1 ) x1 − y1 ✭✷✳✾✮ ❚❤❡♦ ✤à♥❤ ỵ r t f (x1 ) f ′ (y1 ) ✈ỵ✐ y1 < x′1 < x1 ❇➜t ✤➥♥❣ t❤ù❝ ♥➔② ❧✉ỉ♥ ✤ó♥❣ ✈➻ f ′ (x) ỗ f (x) > ✭t❤❡♦ ❣✐↔ t❤✐➳t✮✱ ✈➻ t❤➳ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✻✮ ✤ó♥❣✳ iii) ◆➳✉ x1 < y1 t❤➻ x1 − y1 < ỵ r t ✭✷✳✾✮ ⇔ f ′ (x′1 ) ≤ f ′ (y1 ) ✈ỵ✐ x1 < x′1 < y1 ❇➜t ✤➥♥❣ t❤ù❝ ♥➔② ❧✉ỉ♥ ✤ó♥❣ ✈➻ f ′ (x) ❧➔ ❤➔♠ ỗ f (x) > t t❤✐➳t✮✱ ✈➻ t❤➳ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✻✮ ✤ó♥❣✳ S hóa b i Trung tâm H c li u - i h c Thái Ngun DeThiMau.vn http://www.lrc-tnu.edu.vn ✶✼ ❚÷ì♥❣ tü t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ f (xi ) ≥ f (yi ) + f ′ (yi )(xi − yi ), ∀xi , yi ∈ I(a; b), i = 1, 2, , n ◆❤÷ ✈➟② t❛ ❝â f (x1 ) ≥ f (y1 ) + f ′ (y1 )(x1 − y1 ), f (x2 ) ≥ f (y2 ) + f ′ (y2 )(x2 − y2 ), f (xn ) ≥ f (yn ) + f ′ (yn )(xn − yn ) ❉♦ ✤â i=1 n ⇔ ❳➨t n i=1 i=1 n n n f (xi ) ≥ f (xi ) − f (yi ) + i=1 n i=1 n i=1 f (yi ) ≥ i=1 f ′ (yi )(xi − yi ) f ′ (yi )(xi − yi ) ✭✷✳✶✵✮ f ′ (yi )(xi − yi ) ❙û ❞ư♥❣ ❜✐➳♥ ✤ê✐ ❆❜❡❧ ù♥❣ ✈ỵ✐ = f ′ (yi ) ✈➔ bi = (xi − yi ) t❛ ✤÷đ❝✿ n−1 n ′ i=1 f (yi )(xi − yi ) = i=1 [f ′ (yi ) − f ′ (yi+1 )][(x1 + x2 + · · · + xn−1 ) − (y1 + y2 + · · · + yn−1 )] + f ′ (yn )[(x1 + x2 + · · · + xn ) − (y1 + y2 + · · · + yn )] ❚ø ❣✐↔ t❤✐➳t t❛ ❝â f ′ (yi ) − f ′ (yi+1 ) ≥ ✭❞♦ ❤➔♠ f ′ (y) ỗ (x1 + x2 + à à · + xn−1 ) − (y1 + y2 + · · · + yn−1 ) ≥ 0, ❱➻ t❤➳ (x1 + x2 + · · · + xn ) − (y1 + y2 + · · · + yn ) = n i=1 f ′ (yi )(xi − yi ) ≥ ❚ø ✭✷✳✶✵✮ ✈➔ ✭✷✳✶✶✮ t❛ t❤✉ ✤÷đ❝ n n i−1 S hóa b i Trung tâm H c li u - f (xi ) − i−1 i h c Thái Nguyên DeThiMau.vn f (yi ) ≥ 0, http://www.lrc-tnu.edu.vn ✭✷✳✶✶✮ ... ✤➲✉ ✈➔ s➢♣ t❤ù tü ❝→❝ t❛♠ ❣✐→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ▼ët số ự ỵ tr số ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶ ❈❤ù♥❣ ♠✐♥❤ sỹ tỗ t số ữỡ tr➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳... ♥❤➟♥ ❜✐➳t ✤÷đ❝ ❦❤✐ ♥➔♦ t❤➻ ởt số trữợ tr (a; b) ❧➔ ♠ët ❤➔♠ ✤ì♥ ✤✐➺✉ tr➯♥ ❦❤♦↔♥❣ ✤â✳ ❙❛✉ ú t s ũ ỵ r ự ỵ ừ t ỡ số ởt ỵ rt q trồ tr ữỡ tr t ợ P ỵ số y = f (x) ❝â ✤↕♦ ❤➔♠ tr➯♥... ỵ ởt số rở S húa b i Trung tâm H c li u - i h c Thái Ngun DeThiMau.vn http://www.lrc-tnu.edu.vn ✷ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ♥➔② ♥❤➡♠ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ❝ì ❜↔♥ ♥❤➜t ❝→❝ ✤à♥❤ ỵ tr tr ũ ởt số q q