✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ▲➊ ❚❍➚ ▼■◆❍ ❆◆❍ ▼❐❚ ❙➮ ❇⑨■ ❚❖⑩◆ ❚✃◆● ❍ĐP ❱➋ ❍⑨▼ ❙➱ ❈❤✉②➯♥ ♥❣➔♥❤✿ P❤÷ì♥❣ P❤→♣ ❚♦→♥ ❙ì ❈➜♣ ▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✶✸ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ữớ ữợ t❤→♥❣ ✾ ♥➠♠ ✷✵✶✹ ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❍➔♠ sè ❧✐➯♥ tö❝ ✈➔ ❦❤↔ ✈✐ ✶✳✶ ✶✳✷ ✶✳✸ ●✐ỵ✐ ❤↕♥ ❝õ❛ ❤➔♠ sè ♠ët ❜✐➳♥ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✸ ✶✳✶✳✶ ❈→❝ ✤à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✷ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❣✐ỵ✐ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ❙ü ❧✐➯♥ tö❝ ❝õ❛ ❤➔♠ ♠ët ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷✳✶ ❈→❝ ✤à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷✳✷ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ ✳ ỵ ❤➔♠ ❦❤↔ ✈✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ❘♦❧❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷ ▼ët sè ❜➔✐ t♦→♥ tê♥❣ ❤ñ♣ ✈➲ ❤➔♠ sè ✷✳✶ ✷ ✶✶ ✷✳✸ ❇➔✐ t♦→♥ tê♥❣ ❤ñ♣ ✈➲ ❤➔♠ ❜➟❝ ❤❛✐ tr➯♥ ❜➟❝ ♥❤➜t ax2 + bx + c ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ y= dx + e x2 − mx + ✷✳✶✳✶ ❇➔✐ t♦→♥✿ ❈❤♦ ❤➔♠ sè y = ✳ ✳ x−1 x2 − mx + ❇➔✐ t♦→♥ tê♥❣ ❤ñ♣ y = (∗) ✳ ✳ ✳ ✳ x−1 ❇➔✐ t♦→♥ tê♥❣ ❤ñ♣ ✈➲ ❤➔♠ y = ax3 + bx2 + cx + d ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✹ ▼ët sè ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✷ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✶✶ ✳ ✳ ✳ ✳ ✳ ✶✶ ✳ ✳ ✳ ✳ ✳ ✷✺ ✹✶ ✹✷ ✶ ▼Ð ✣❺❯ ❍➔♠ sè ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ♣❤➛♥ ❝ì ❜↔♥ ✈➔ trå♥❣ t➙♠ ❝õ❛ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ❚r✉♥❣ ❤å❝ ♣❤ê t❤æ♥❣✳ ❚r♦♥❣ ✤➲ t❤✐ ✤↕✐ ❤å❝✱ ❝❛♦ ✤➥♥❣ ✈➔ ❝→❝ ❦ý t❤✐ ❖❧②♠♣✐❝ ❧✉æ♥ ❧✉æ♥ ❝â t số ỵ tt sè ❧✐➯♥ tư❝ ✈➔ ❦❤↔ ✈✐ ✤÷đ❝ sû ❞ư♥❣ r➜t rë♥❣ r➣✐ tr♦♥❣ ❝→❝ ❜➔✐ t➟♣ ❝ơ♥❣ ♥❤÷ ❝→❝ s→❝❤ ✈✐➳t ✈➲ ❤➔♠ sè✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ tr ởt số ỵ q trồ ❝õ❛ ❤➔♠ ❦❤↔ ✈✐✱ ❧✐➯♥ tö❝ tø ✤â →♣ ❞ö♥❣ ❣✐↔✐ ♠ët sè ❜➔✐ t➟♣ tê♥❣ ❤ñ♣ ✈➲ ❤➔♠ sè✳ ▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ✈➔ ❣✐↔✐ ❜➔✐ t♦→♥ tê♥❣ ❤ñ♣ số tr t ỗ tớ ✤÷❛ r❛ ❝→❝ ❜➔✐ t♦→♥ tê♥❣ ❤đ♣ ✈➲ ❤➔♠ sè ❜➟❝ ❜❛✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ỵ ỡ ợ sỹ tử ởt ỵ ữỡ ỗ P ✶ tr➻♥❤ ❜➔② ❜➔✐ t♦→♥ tê♥❣ ❤ñ♣ ✈➲ ❤➔♠ sè ❜➟❝ ❤❛✐ tr➯♥ ❜➟❝ ♥❤➙t ✈ỵ✐ ❧í✐ ❣✐↔✐ ❝❤✐ t✐➳t✳ P❤➛♥ ✷ tr➻♥❤ ❜➔② ❝→❝ ❜➔✐ t♦→♥ tê♥❣ ❤ñ♣ ❤➔♠ ❜➟❝ ✸✳ ◗✉❛ ✤➙②✱ tỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ s s tợ ữớ ữớ ữợ ❝❛♦ ❤å❝ ❝õ❛ ♠➻♥❤✱ ❚❙✳ ◆❣✉②➵♥ ▼✐♥❤ ❑❤♦❛ ✲ tr÷í♥❣ ✣↕✐ ❤å❝ ✣✐➺♥ ▲ü❝✳ ❚❤➛② ✤➣ ❞➔♥❤ ♥❤✐➲✉ t❤í✐ ❣✐❛♥ t t ữợ qt ỳ t❤➢❝ ♠➢❝ ❝❤♦ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ tæ✐ ❧➔♠ ❧✉➟♥ ✈➠♥✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧í✐ ❝↔♠ ì♥ t tợ ổ tr ỗ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽✱ ❝→❝ ❚❤➛② ❈ỉ ❣✐↔♥❣ ❞↕② ❧ỵ♣ ỗ ✤➣ t↕♦ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ♥❤➜t ✤➸ tỉ✐ ❝â t❤➸ ❤♦➔♥ t❤✐➺♥ ❦❤â❛ ❤å❝ ❝ơ♥❣ ♥❤÷ ❧✉➟♥ ✈➠♥ ❝õ❛ ♠➻♥❤✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✾ ♥➠♠ ✷✵✶✹✳ ❍å❝ ✈✐➯♥ ▲➯ ❚❤à ▼✐♥❤ ❆♥❤ ✷ ❈❤÷ì♥❣ ✶ ❍➔♠ sè ❧✐➯♥ tư❝ ✈➔ ❦❤↔ ✈✐ ✶✳✶ ●✐ỵ✐ ❤↕♥ ❝õ❛ ❤➔♠ sè ♠ët ❜✐➳♥ sè ✶✳✶✳✶ ❈→❝ ✤à♥❤ ♥❣❤➽❛ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ố ữủ ợ số y = f (x) ❦❤✐ x → x0 ♥➳✉ ❤➔♠ sè y = f (x) ①→❝ ✤à♥❤ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ x0 ✭❝â t❤➸ ❦❤æ♥❣ ①→❝ ✤à♥❤ t↕✐ x0 ✮✿ ∀ε > 0, ∃δ = δ(ε) : < |x − x0 | < δ ⇒ |f (x) − A| < ε✳ ❱➼ ❞ö ✶✳✶✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ x→1 lim (2x + 3) = 5✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â |(2x + 3) − 5| < ε ⇔ 2|x − 1| < ε ⇔ |x − 1| < 2ε ε ⇒ ∀x : |x − 1| < δ ⇒ |(2x + 3) − 5| < ε✳ ❉♦ ✤â t❤❡♦ ✤à♥❤ ♥❣❤➽❛ t❛ ❝â lim (2x + 3) = 5✳ ❱➟② ∀ε > 0, ∃δ(ε) = x→1 ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ❍➔♠ y = f (x) ①→❝ ✤à♥❤ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ x0 ✭❝â t❤➸ ❦❤æ♥❣ ①→❝ ✤à♥❤ t↕✐ x0 ✮ ❣å✐ ❧➔ ❝â ❣✐ỵ✐ ❤↕♥ ❆ ❦❤✐ x → x0 ♥➳✉ ✤è✐ ✈ỵ✐ ♠å✐ ❞➣② xn , xn = x0 ❤ë✐ tö ✤➳♥ x0 t❤➻ ❞➣② ❝→❝ ❣✐→ trà ❝õ❛ ❤➔♠ t÷ì♥❣ ù♥❣ f (x1 ); f (x2 ); f (x3 ) ; f (xn ) ❤ë✐ tö ✤➳♥ ❆✳ ❱➼ ❞ö ✶✳✷✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ x→0 lim x.sin = 0✳ x ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ♥❤➟♥ t❤➜② ❤➔♠ f (x) = x.sin x1 ❦❤æ♥❣ ①→❝ ✤à♥❤ t↕✐ x0 = ♥❤÷♥❣ ①→❝ ✤à♥❤ t↕✐ ❧➙♥ ❝➟♥ x0 = 0✳ ▲➜② ❞➣② xn ❜➜t ❦➻ tr♦♥❣ ❦❤♦↔♥❣ ( ✸ −π π ; ) 4 s❛♦ ❝❤♦ lim xn = 0✳ ❚❛ ❝â✿ n→∞ ≤ |f (xn )| = |xn sin | ≤ |xn |✳ xn ❱➻ lim xn = → lim |xn | = ⇒ lim f (xn ) = 0✳ n→∞ n→∞ n→∞ ❱➟② t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ✷ t❛ ❝â✿ lim x.sin = 0✳ x→0 x ✳ ❱➼ ❞ö ✶✳✸✳ ự r ổ tỗ t x1 lim sin x1 ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❧➜② ❤❛✐ ❞➣② xn = + nπ ❀① ✄n = + ✳ (4n + 1)π ❚❛ ❝â lim xn = 1; lim ① ✄n = 1✳ ❉➣② ❝→❝ ❣✐→ trà t÷ì♥❣ ù♥❣ ❝õ❛ ❤➔♠ ❧➔ n→∞ n→∞ = sinnπ = 0✱ f (xn ) = sin 1+ −1 nπ π = sin( + 2nπ) = 1✳ f (① ✄n ) = sin 2 1+ (4n + 1)π ⇒ lim f (xn ) = 0; lim f (① ✄ n ) = 1✳ n→∞ n→∞ ❱➟② t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ổ tỗ t lim sin x1 x1 ①➨t ✶✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✶ ✈➔ ✤à♥❤ ♥❣❤➽❛ ✷ ❧➔ t÷ì♥❣ ✤÷ì♥❣✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ❍➔♠ sè y = f (xn) ①→❝ ✤à♥❤ ❧➙♥ ❝➟♥ ❜➯♥ ♣❤↔✐ x0✳ ❙è ❆ ữủ ợ số x x0 ỵ A = lim x→(x0 +0) f (x) = f (x0 + 0) ♥➳✉ ∀ε > 0, ∃δ = δ(ε) > 0✱∀x : < x − x0 < δ ⇒ |f (x) − A| < ε✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ❍➔♠ y = f (x) ①→❝ ✤à♥❤ t↕✐ ❧➙♥ ❝➟♥ ❜➯♥ tr→✐ x0 ✭❝â t❤➸ ❦❤æ♥❣ ①→❝ ✤à♥❤ t↕✐ x0 ✮✳ ❙è ❆ ❣å✐ ❧➔ ❣✐ỵ✐ ❤↕♥ tr→✐ ❝õ❛ ❤➔♠ f (x) ❦❤✐ x x0 ỵ A = lim x(x0 0) f (x) = f (x0 − 0) ♥➳✉ ∀ε > 0✱ ∃δ = δ(ε) > : ∀x ∈ < x0 − x < δ ⇒ |f (x) − A| < ε✳ ✹ ❱➼ ❞ư ✶✳✹✳ ❚➻♠ ❣✐ỵ✐1 ❤↕♥ ♠ët ♣❤➼❛ ❝õ❛ ❤➔♠ sè✿ f (x) = 2014 + 1 + 5x − , x → 1✳ ●✐↔✐✳ ❚❛ ❝â✿ −1 x → +∞ ❦❤✐ x → − 0✳ ❉♦ ✤â 1 + 5x − → 0✳ ❱➟②f (1 − 0) = 2014 ❦❤✐ x → + 0✳ ❚❛ ❝â 1 → −∞✱ ❞♦ ✤â x − → 0✳ ❱➟② f (1 + 0) = 2015✳ 1−x ✶✳✶✳✷ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❣✐ỵ✐ ❤↕♥ ❚➼♥❤ ❝❤➜t ✶✳✶✳ ◆➳✉ x→x lim f (x) = A✱ ❆ ❧➔ ♠ët sè ❤ú✉ ❤↕♥ ❦❤✐ ✤â ❤➔♠ f (x) ❧➔ ❜à ❝❤➦♥ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ♥➔♦ ✤â V (x0)✱ tù❝ ❧➔ ∃ ♠ët sè M > s❛♦ ❝❤♦✿ |f (x)| ≤ M, ∀x ∈ V (x0), x = x0✳ ❈❤ù♥❣ ỵ tỗ t ♠ët ❧➙♥ ❝➟♥ V (x0) s❛♦ ❝❤♦✿ > |f (x) − A| ≥ |f (x)| − |A|✳ ⇒ |f (x)| < + |A| ✈➟② + |A| ✤â♥❣ ✈❛✐ trá ❝õ❛ ▼✳ ❚➼♥❤ ❝❤➜t ✶ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❚➼♥❤ ❝❤➜t ✶✳✷✳ ◆➳✉ x→x lim f (x) = A, A = ❧➔ sè ❤ú✉ ❤↕♥✱ ❦❤✐ ✤â ❝â ♠ët ❧➙♥ ❝➟♥ V (x0) ✤➸ s❛♦ ❝❤♦ |f (x)| > |A| , ∀x ∈ V (x0 ), x = x0 ✳ ❚➼♥❤ ❝❤➜t ✶✳✸✳ ◆➳✉ x→x lim f1 (x) = A1 , lim f2 (x) = A2 ✈➔ ❝â ♠ët ❧➙♥ ❝➟♥ x→x t❤➻ A1 ≤ A2✳ V (x0 ) : f1 (x) ≤ f2 (x), ∀x ∈ V (x0 ), x = x0 ❚➼♥❤ ❝❤➜t ✶✳✹✳ ◆➳✉ x→x lim f1 (x) = A, lim f2 (x) = A ✈➔ f1 (x) ≤ ϕ(x) ≤ x→x t❤➻ x→x lim ϕ(x) = A✳ f2 (x), ∀x ∈ V (x0 ), x = x0 0 ❚➼♥❤ ❝❤➜t ✶✳✺✳ ✭❚✐➯✉ ❝❤✉➞♥ ❈❛✉❝❤②✮ ❈➛♥ ✈➔ ✤õ ✤➸ ∃ x→x lim f (x) ❤ú✉ ❤↕♥ ❧➔ ❤➔♠ y = f (x) ①→❝ ✤à♥❤ ð ♠ët ❧➙♥ ❝➟♥ ❝õ❛ x0 ✭❝â t❤➸ trø r❛ x0✮ ✈➔ ∀ε > ∃ ❧➙♥ ❝➟♥ V (x0 ) s❛♦ ❝❤♦✿ |f (x ) − f (x”)| < ε✱ ∀x , x” ∈ V (x0 ); x , x” = x0 ✳ ✺ ❚➼♥❤ ❝❤➜t ✶✳✻✳ ❈❤♦ x→x lim f (x) x→x0 ✤â✿ x→x lim [f (x) ± g(x)] = A ± B; lim [f (x).g(x)] = A.B x→x ❤ú✉ ❤↕♥✳❑❤✐ ✈➔ ♥➳✉ B = t❤➻ = A, lim g(x) = B; A, B 0 f (x) A lim = ✳ x→x0 g(x) B sinx ❱➼ ❞ö ✶✳✺✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ x→0 lim = 1✳ x ❈❤ù♥❣ ♠✐♥❤✳ ❍➔♠ f (x) = sinx ❦❤æ♥❣ ①→❝ ✤à♥❤ t↕✐ ✤✐➸♠ x0 = ♥❤÷♥❣ x π ①→❝ ✤à♥❤ t↕✐ ❧➙♥ ❝➟♥ ❝õ❛ ♥â ❝❤➥♥❣ ❤↕♥ V (x0 ) = x : < |x| < ✳ π ❚r÷í♥❣ ❤đ♣ ✶✿ < x < ✱ tø ❤➻♥❤ ✈➩ t❛ ❝â✿ S AOM < SquatAOM < S AOT 1 ⇔ OA.M H < OAAM < OA.AT ✭✶✳✶✮ 2 sinx ⇔ M H < AM < AT ⇔ sinx < x < tanx ⇔ < < ✳ x cosx −π π ❚r÷í♥❣ ❤đ♣ ✷✿ < x < 0✱ ✤➦t x = −t ⇒ < t < ✳ 2 sin(−t) sint π ❱➻ cosx = cos(−t) = cost; sinx = = ✈➔ ❞♦ < t < ♥➯♥ x −t t −π ✭✶✳✶✮ ✈➝♥ ✤ó♥❣ ❦❤✐ < x < 0✳ sinx ❉♦ lim = ⇒ lim = 1✳ x→0 cosx x→0 x ✻ ✶✳✷ ❙ü ❧✐➯♥ tö❝ ❝õ❛ ❤➔♠ ♠ët ❜✐➳♥ ✶✳✷✳✶ ❈→❝ ✤à♥❤ ♥❣❤➽❛ ✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ❍➔♠ ❢✭①✮ ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tư❝ t↕✐ ✤✐➸♠ x0 ♥➳✉ ♥â t❤ä❛ ♠➣♥ ❤❛✐ ✤✐➲✉ ❦✐➺♥✿ ✐✮ ❢✭①✮ ①→❝ ✤à♥❤ t↕✐ x0 ✈➔ ❧➙♥ ❝➟♥✳ ✐✐✮ lim = f (x0 )✳ ✣✐➸♠ x0 ❦❤✐ ✤â ❣å✐ ❧➔ ✤✐➸♠ ❧✐➯♥ tö❝ ❝õ❛ ② ❂ ❢✭①✮✳ x→x0 ✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ❍➔♠ ❢✭①✮ ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tư❝ tr→✐ ✭❤♦➦❝ ♣❤↔✐✮ t↕✐ ✤✐➸♠ x0 ♥➳✉ ♥â t❤ä❛ ♠➣♥ ❤❛✐ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✐✮ ❢✭①✮ ①→❝ ✤à♥❤ t↕✐ x0 ✈➔ ❧➙♥ ❝➟♥ tr→✐ ❤♦➦❝ ♣❤↔✐ ❝õ❛ ✤✐➸♠ x0 ✳ ✐✐✮ f (x0 − 0) = f (x0 ) ❤♦➦❝ f (x0 + 0) = f (x0 )✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✼✳ ❍➔♠ ❢✭①✮ ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a, b] ♥➳✉ ♥â ❧✐➯♥ tö❝ t↕✐ ∀x ∈ (a, b) ✈➔ ❧✐➯♥ tö❝ ♣❤↔✐ t↕✐ ① ❂ ❛✱ ❧✐➯♥ tö❝ tr→✐ t↕✐ ① ❂ ❜✳ ✶✳✷✳✷ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ ❚➼♥❤ ❝❤➜t ✶✳✼✳ ❈❤♦ ❢✭①✮ ❧➔ ❤➔♠ sè ❧✐➯♥ tö❝ tr➯♥ ❬❛✱❜❪ ✈➔ f (a).f (b) < 0✳ ❑❤✐ ✤â ∃c ∈ (a, b) : f (c) = 0✳ ❈❤ù♥❣ ♠✐♥❤✳ ❑❤æ♥❣ ❣✐↔♠ t➼♥❤ tê♥❣ q✉→t t❛ ❣✐↔ t❤✐➳t ❢✭❛✮ ❁ ✵❀ ❢✭❜✮ ❃ ✵✳ ✣➦t α0 = a, β0 = b ⇔ f (α0 ) < 0; f (β0 ) > 0✳ α0 + β0 ✣➦t u0 = ✱ ♥➳✉ f (u0 ) = t❤➻ c = u0 ✱ ♥➳✉ f (u0 < 0) t❤➻ ✤➦t α1 = u0 , β1 = β0 ❝á♥ ♥➳✉ f (u0 > 0) t❤➻ ✤➦t α1 = α0 , β1 = u0 ✳ ❚❛ ❧↕✐ ①➨t [α1 , β1 ] ✈➔ ❝â f (α1 ).f (β1 ) < 0✳ α1 + β1 ✈➔ q✉→ tr➻♥❤ t✐➳♣ ❞✐➵♥ ✈ỵ✐ t❤✉➟t t♦→♥ ❧➦♣ ❧↕✐✳◆❤÷ ❚✐➳♣ tư❝ ✤➦t u1 = αn + βn ✈➟② t❛ s➩ ♥❤➟♥ ✤÷đ❝ [αn , β(n)], un = ✳ ◆➳✉ f (un ) = t❤➻ c = un ✈➔ ❝ ❝❤➾ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❢✭①✮ ❂ ✵✳ ◆➳✉ f (un ) < t❤➻ t❛ ✤➦t αn+1 = un , βn+1 = βn ❀ ❝á♥ ♥➳✉ f (un > 0) t❤➻ ✤➦t αn+1 = αn , βn+1 = un ✳ ✼ ❚✐➳♣ tö❝ q✉→ tr➻♥❤ ♥➔② r❛ ✈ỉ ❤↕♥ t❛ ♥❤➟♥ ✤÷đ❝ ✷ ❞➣② sè αn , βn ❝ị♥❣ ❤ë✐ tư ✈➔ ❝â ợ ứ t ữủ ❢✭❝✮ ❂ ✵ ✈➔ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❚➼♥❤ ❝❤➜t ✶✳✽✳ ✭❲❡✐❡rstr❛ss ✶✮ ◆➳✉ ❤➔♠ sè ❢✭①✮ ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ ❬❛✱ ❜❪ t❤➻ ♥â s➩ ❜à ❝❤➦♥ tr➯♥ ✤♦↕♥ ➜②✳ ❚➼♥❤ ❝❤➜t ✶✳✾✳ ✭ ❲❡✐❡rstr❛ss ✷✮ ◆➳✉ ❤➔♠ sè f (x) ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ ❬❛✱❜❪ t❤➻ ♥â ✤↕t ❣✐→ trà ❧ỵ♥ ♥❤➜t✱ ❣✐→ trà ♥❤ä ♥❤➜t tr➯♥ ✤♦↕♥ ➜②✳ ❚➼♥❤ ❝❤➜t ✶✳✶✵✳ ◆➳✉ ❤➔♠ sè ❢✭①✮ ❧✐➯♥ tư❝ tr➯♥ ✤♦↕♥ ❬❛✱ ❜❪ ✈➔ µ ∈ [m, M ] m = f (x), M = maxf (x) t❤➻ ∃ξ ∈ (a, b) : f (ξ) = µ✳ ỵ ỵ t ỵ sỷ y = f (x) ①→❝ ✤à♥❤ tr♦♥❣ ❦❤♦↔♥❣ ✭❛✱ ❜✮✳ ◆➳✉ f (x) ✤↕t ❝ü❝ trà t↕✐ ♠ët ✤✐➸♠ c ∈ (a, b) t tỗ t ỳ ❤↕♥ f (c) t❤➻ f (c) = 0✳ ✶✳✸✳✷ ✣à♥❤ ỵ ỵ số y = f (x) ①→❝ ✤à♥❤ ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a; b] ✈➔ ❦❤↔ ✈✐ tr♦♥❣ ❦❤♦↔♥❣ ✭❛❀❜✮ ❣✐↔ sû f (a) = f (b) õ tỗ t c (a; b) s❛♦ ❝❤♦ f (c) = 0✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ❧✐➯♥ tö❝ ⇒ ∃M = max f (x)✱ m = minf (x)✳ ❑❤✐ ✤â ❝â ❤❛✐ ❦❤↔ ♥➠♥❣ ①↔② r❛✿ ❤♦➦❝ ❝↔ ✷ ❣✐→ trà ▼✱ ♠ ✤↕t t↕✐ ✷ ♠ót ❛✱❜ tù❝ ❧➔✿ f (a) = f (b) = m = M ⇒ f (x) = C(const)✱ ∀x ∈ (a; b) ⇒ f (n) = 0✱ ∀x ∈ (a; b) ⇒ f (x) = 0, ∀x ∈ (a; b) ❤♦➦❝ ❝â ♠ët ❣✐→ trà ✤↕t t c (a; b) t ỵ t t❛ ❝â f (c) = 0✳ ❱➼ ❞ö ✶✳✻✳ ❈❤♦ f (x) = (x − 1)(x − 2) (x − 2014)✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ f (x) = ❝â ✤ó♥❣ ✷✵✶✸ ♥❣❤✐➺♠✳ ✽ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â f1 = f2 = = f2014 ỵ ❘♦❧❧❡ ❝❤♦ ❝→❝ ✤♦↕♥ [1; 2]; [2; 3]; ; [2013; 2014] ⇒ ∃c1 ∈ [1; 2]; c2 ∈ [2; 3]; ; c2013 ∈ [2013; 2014] s❛♦ ❝❤♦ f (c1 ) = 0, f (c2 ) = 0, , f (c2013 ) = 0✳ ❚❛ ❧↕✐ ❝â ❞♦ ❢✭①✮ ❧➔ ✤❛ t❤ù❝ ❜➟❝ ✷✵✶✹ ⇒ f (x) ❧➔ ✤❛ t❤ù❝ ❜➟❝ ✷✵✶✸✳ ⇒ f (x) = ❝â ✤ó♥❣ ✷✵✶✸ ♥❣❤✐➺♠ C1 ; C2 ; C3 ; ; C2013 ✳ ❱➼ ❞ư ✶✳✼✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ f (x) = x2 − xsinx − cosx = 0✳ Π ●✐↔✐✳ ❱➻ f −Π > 0, f (0) < 0, f > ❞♦ ✤â t❤❡♦ t➼♥❤ ❝❤➜t ❝õ❛ 2 ❤➔♠ ❧✐➯♥ tö❝ ❢✭①✮ ❂ ✵ ❝â ➼t ♥❤➜t ✷ ♥❣❤✐➺♠✳ ◆➳✉ ❢✭①✮ ❂ ✵ ❝â ✈ỉ sè ♥❣❤✐➺♠ ❧ỵ♥ ❤ì♥ t t ỵ f , (x) = ❝â ➼t ♥❤➜t ✷ ♥❣❤✐➺♠ ♥❤÷♥❣ ✈➻ f , (x) = 2x − sinx − xcosx + sinx = x(2 − cosx) = ❝❤➾ ❝â ✶ ♥❣❤✐➺♠ ① ❂ ✵✳ ❉♦ ✤â ❢✭①✮ ❂ ✵ ❝❤➾ õ ú ỵ r ỵ ✶✳✸✳ ❈❤♦ ❤➔♠ sè y = f (x) ①→❝ ✤à♥❤✱ ❧✐➯♥ tö❝ tr➯♥ ❬❛✱ ❜❪ ✈➔ ❦❤↔ ✈✐ tr♦♥❣ ✭❛✱ õ tỗ t t t ởt c ∈ (a, b) s❛♦ ❝❤♦ f (b) − f (a) ✳ f (c) = b−a − f (a) ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ❤➔♠ ❜ê trñ h(x) = f (x) − f (a) − (x − a) f (b)b − a ∀x ∈ [a, b]✳ ❚❤➜② r➡♥❣ ❤➔♠ ❤✭①✮ t❤ä❛ ♠➣♥ ✤à♥❤ ỵ c (a, b) h (c) = 0✳ f (b) − f (a) f (b) − f (a) ⇒ h (c) = f (c) − = → ❱➻ h (x) = f (x) − b−a b−a f (b) − f (a) f (c) = , c ∈ (a, b)✳ b−a ❱➼ ❞ö ✶✳✽✳ ❈❤♦ < b < a✱ ❝❤ù♥❣ ♠✐♥❤✿ a −a b < ln ab < a −b b ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ❤➔♠ sè f (x) = lnx tr➯♥[a, b]✿ ❢✭①✮ ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [b, a], f (x) = 1, ∀x ∈ (b, a) õ t ỵ r c (b, a) s❛♦ ✾ v(x).u, (x) − u(x)v , (x) ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â y (x) = ⇒ y , (x0 ) = v (x) u, (x0 ) , , ⇒ v(x0 ).u (x0 ) = u(x0 ).v (x0 ), v(x0 ) = ⇒ y(x0 ) = , v (x0 , ●✐↔✐ ❝➙✉ ✺✸✿ u(x) = x2 − mx + 1✱ v(x) = x − ⇒ v , (x) = = 0, ∀x 2x1 − m y , (x1 ) = 0, y , (x2 ) = 0✳ ⑩♣ ❞ö♥❣ ❜ê ✤➲ ⇒ |y(x1 ) − y(x2 )| = | − 2x2 − m | = 2|x1 − x2 | ❱ỵ✐ ❣✐→ trà ♥➔♦ ❝õ❛ ♠ t❤➻ ❤➔♠ sè ✭✯✮ ❝â ❝ü❝ ✤↕✐✱ ❝ü❝ t✐➸✉ s❛♦ ❝❤♦ ❈➙✉ ✺✹✳ |yCD − yCT | = 8✳ ●✐↔✐✳ ❍➔♠ sè ✭✯✮ ❝â ❝ü❝ ✤↕✐✱ ❝ü❝ t✐➸✉ y, = ✈➔ ✤ê✐ ❞➜✉ ⇔ f (x) = x2 − 2x + m − = ✭✐✐✮ ❝â ✷ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t ❦❤→❝ ✶ ⇒ ✤✐➲✉ , >0 2−m>0 ❦✐➺♥ ⇔ ⇔ m < 2✳ f (1) = m=2 ❑❤✐ ✤â |yCD − yCT | = ⇔ |y(x1 ) − y(x2 )| = ⇔ 2|x1 − x2 | = ⇔ (x1 − x2 )2 = 16 ⇔ (x1 + x2 )2 − 4x1 x2 = 16 tr♦♥❣ ✤â x1 , x2 ❧➔ ✷ ♥❣❤✐➺♠ ❝õ❛ ✭✐✐✮✳ ❚❤❡♦ ❱✐➨t t❛ ❝â✿ − 4(m − 1) = 16 ⇔ −4m = ⇔ m = ỗ t❤à ✭✯✮ ❝➢t ❖① t↕✐ ✷ ✤✐➸♠ ❝â ❤♦➔♥❤ ✤ë x1, x2 s❛♦ ❝❤♦ t✐➳♣ t✉②➳♥ t↕✐ x1 ✈✉æ♥❣ ❣â❝ ợ t t t x2 ỗ t ❝➢t ❖① t↕✐ ✷ ✤✐➸♠ x1, x2 ⇔ x2 − mx + = ❝â ✷ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t ❦❤→❝ ✶ ⇒ ✤✐➲✉ ❦✐➺♥ m=2 ⇔ m=2 ⇔ |m| > >0 m2 − > ❚✐➳♣ t✉②➳♥ t↕✐ x1 ✈✉ỉ♥❣ ❣â❝ ✈ỵ✐ t✐➳♣ t✉②➳♥ t↕✐ x2 ⇔ y , (x1 ).y , (x2 ) = −1 ❀ ❝â ✤↕♦ ❤➔♠ t↕✐ x0✱ ❣✐→ trà ❝õ❛ ❤➔♠ t↕✐ ❇ê ✤➲ ✷✳✷✳ ❈❤♦ ❤➔♠ sè y = u(x) v(x) x0 ❧➔ u, (x0 ) y(x0 ) = ⇒ y (x0 ) = ) v(x0 , v(x).u, (x) − u(x).v , (x) ❈❤ù♥❣ ♠✐♥❤✳ y (x) = v (x) u, (x0 ) u(x0 ) v , (x0 ) u, (x0 ) , ⇒ y (x0 ) = − = v(x0 ) v(x0 ) v(x0 v(x0 ) , ⑩♣ ❞ư♥❣ ❜ê ✤➲✿②➯✉ ❝➛✉ ❜➔✐ t♦→♥ t÷ì♥❣ ✤÷ì♥❣ ❚ø ✤➙② ❞ò♥❣ ❱✐➨t t❛ ❣✐↔✐ r❛ ♠✳ ✷✽ 2x1 − m 2x2 − m =1 x1 − x2 − ❈➙✉ ✺✻✳ ❱ỵ✐ ❣✐→ trà ♥➔♦ ♠ t❤➻ ỗ t t ố ự ❚❛ ❝â y = x + − m + 2x−−m1 ❱➟② y = x + − m ❧➔ t✐➺♠ ❝➟♥ ①✐➯♥✱① ✲ ✶ ❧➔ t✐➺♠ ❝➙♥ ✤ù♥❣ ⇒ ❣✐❛♦ ❝õ❛ ✷ t✐➺♠ ❝➟♥ ❏✭✶✱✷ ✲ ♠✮ ❧➔ t➙♠ ố ự ỗ t t ố ự ỗ t m = ⇒ m = −1✳ ❈➙✉ ✺✼✳ ❱ỵ✐ ❣✐→ trà t t ỗ t t↕♦ ✈ỵ✐ ✷ trư❝ t♦❛ ✤ë ♠ët t❛♠ ❣✐→❝ ❝â ❞✐➺♥ t➼❝❤ ❜➡♥❣ ✷✳ ●✐↔✐✳ ✰ ❚✐➺♠ ❝➟♥ ①✐➯♥ ❣✐❛♦ ✈ỵ✐ ❖① t↕✐ ❊✭♠ ✲ ✶✱✵✮ ✰ ❚✐➺♠ ❝➟♥ ①✐➯♥ ❣✐❛♦ ✈ỵ✐ ❖② t↕✐ ❋✭✵✱✶ ✲ ♠✮ 1 ❉✐➺♥ t➼❝❤ t❛♠ ❣✐→❝✿ S = |xE |.|yF | = (m − 1)2 = 2 m=3 ⇒ (m − 1)2 = ⇒ m − = ±2 ⇒ m = −1 ❱ỵ✐ ❣✐→ trà ♥➔♦ ♠ t❤➻ t✐➺♠ ỗ t t ú ợ ✤÷í♥❣ trá♥ t➙♠ ❖✭✵✱✵✮ ❜→♥ ❦➼♥❤ R = 2✳ ❈➙✉ ✺✽✳ ●✐↔✐✳ ✰ ❚✐➺♠ ❝➟♥ ①✐➯♥ y = x + − m ⇔ x − y + − m = √ ✰ ❚✐➺♠ ❝➟♥ ①✐➯♥ t✐➳♣ ①ó❝ ợ ữớ trỏ t R = √ |1 − m| √ ⇔ kc(O, T CX) = ⇔ √ = ⇔ |m − 1| = 2 √ m=3 ⇔m−1=± 2⇔ m = −1 √ ❱ỵ✐ ❣✐→ trà ♥➔♦ ♠ t❤➻ ❤➔♠ sè ✭✯✮ t❤ä❛ ♠➣♥ |y| ≥ 2, ∀x = ❈➙✉ ✺✾✳ ●✐↔✐✳ ❨➯✉ ❝➛✉ ❜➔✐ t♦→♥ t÷ì♥❣ ✤÷ì♥❣ ⇔ t➻♠ ♠ ✤➸ ♣❤÷ì♥❣ tr➻♥❤ x − mx + = y ✭✐✐✐✮ ✈æ ♥❣❤✐➺♠ ∀y : −2 < y < x−1 ⇔ x2 − (y + m)x + + y = ✈æ ♥❣❤✐➺♠ ∀y : −2 < y < ⇔ < 0, ∀y : −2 < y < ⇔ (y + m)2 − 4(1 + y) < 0, ∀y ∈ (−2, 2) ⇔ f (y) = y + 2(m − 2)y + m2 − < 0, ∀y ∈ (−2, 2) ✰ ❚❍✶✿ , ≤ ⇒ f (y) ≥ 0, ∀y ❦❤æ♥❣ t❤ä❛ ♠➣♥ ✰ ❚❍✷✿ , > ⇒ f (y) ❝â ✷ ♥❣❤✐➺♠ y1 < y2 ✈➔ ❝â sü ♣❤➙♥ ❞➜✉ ✷✾ ✣➸ ❢✭②✮ ❁ ✵ ∀y ∈ (−2, 2) t❛ ♣❤↔✐ ❝â y1 < −2 < < y2 ⇒ ✤✐➲✉ ❦✐➺♥ 1.f (−2) < m2 − 4m + < (m − 2)2 + < ⇔ ⇔ 1.f (2) < m2 + 4m − < m2 + 4m + < √ √ ⇔ m2 + 4m + < 0✳ ⇔ −2 − < m < −2 + ợ tr t ỗ t số ỗ tr [2, +) ỗ t số ỗ tr [2, +) ⇔ y,(x) ≥ 0, ∀x > ⇔ f (x) = x2 − 2x + m − ≥ 0, ∀x > ✰❚❍✶✿ , = − m ≤ ⇔ m ≥ ⇒ f (x) ≥ 0, ∀x ⇒ f (x) ≥ 0, ∀x > 2✳ ✰❚❍✷✿ , > 0, f (x) ❝â ✷ ♥❣❤✐➺♠ x1 < x2 ✈➔ ❝â sü ♣❤➙♥ ❞➜✉ ✣➸ f (x) ≥ 0, ∀x > t❛ ♣❤↔✐ ❝â x1 < x2 ≤ ⇒ ✤✐➲✉ ❦✐➺♥ m0 ⇔ 2>0 S>0 −k − > P >0 ợ tr t ữớ t t ỗ t t t õ ữỡ ợ ❣✐→ trà ♥➔♦ ❝õ❛ ❛ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ❝â ✹ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t |x3 − 3x2 + 2| = a✳ ❈➙✉ ✶✽✳ ❈❤♦ ✤÷í♥❣ trá♥ Cα : x2 + y2 − 2αx − 4αy + 5α2 − = 0✳ ❱ỵ✐ ❣✐→ trà ♥➔♦ ❝õ❛ α t❤➻ ❝ü❝ ✤↕✐✱❝ü❝ t ỗ t ữớ trá♥ Cα ✳ ❈➙✉ ✶✾✳ ❚➻♠ tr➯♥ ✤÷í♥❣ t❤➥♥❣ y = −2 ♥❤ú♥❣ ✤✐➸♠ ♠➔ tø ✤â ❦➫ ✤÷đ❝ ✸ t t ợ ỗ t sỷ ✲✷✮ t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉ ❜➔✐ t♦→♥✳ ✣÷í♥❣ t❤➥♥❣ ✭❞✮ q✉❛ ❆✭❛✱ ✲✷✮ ❝â ❤➺ sè ❣â❝ ❦ ❧➔✿ y = k(x − a) + 2✳ ✸✷ k(x − a) − = x3 − 3x2 + ✭❞✮ ❧➔ t✐➳♣ t✉②➳♥ ⇔ ❤➺ s❛✉ ❝â ♥❣❤✐➺♠ k = 3x2 − 6x ⇔ (3x2 − 6x)(x − a) − = x3 − 3x2 + 2✳ ⇔ 2x3 − 3x2 a − 3x2 + 6xa − = 0✳ ⇔ (x − 2)(2x2 + (1 − 3a)x + 2) = 0✳ x=2 ⇔ g(x) = 2x2 + (1 − 3a)x + = (∗) ✣➸ q✉❛ ❆ ❦➫ ✤÷đ❝ t t ợ ỗ t t õ ♥❣❤✐➺♠ ♣❤➙♥ a < −1 >0 9a − 6a − 15 > ❜✐➺t ❦❤→❝ ✷ ⇔ ⇔ ⇔ a> g(2) = 12 − 6a = a=2 = a > ⇔ a < −1 ❚➻♠ t➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ tr➯♥ ❖① ♠➔ tø õ ữủ t t ợ ỗ t❤à ✭✶✮✳ ●✐↔✐✳ ●✐↔ sû ❆✭❛✱ ✵✮ t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉ ❜➔✐ t♦→♥✳ ✣÷í♥❣ t❤➥♥❣ ✭❞✮ q✉❛ ❆✭❛✱ ✵✮ ❝â ❤➺ sè ❣â❝ ❦ ❧➔✿ y = k(x − a)✳ k(x − a) = x3 − 3x2 + ✭❞✮❧➔ t✐➳♣ t✉②➳♥ ⇔ ❤➺ s❛✉ ❝â ♥❣❤✐➺♠ k = 3x2 − 6x ⇔ (3x2 − 6x)(x − a) = x3 − 3x2 + ⇔ 2x3 − 3x2 a − 3x2 + 6xa − = 0✳ ✭✐✮ ✣➸ q✉❛ ữủ t t ợ ỗ t t ✭✐✮ ❝â ✸ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t t÷ì♥❣ ✤÷ì♥❣ ①➨t ❤➔♠ sè 2x3 − 3x2 (1 + a) + 6xa − 2✳ ❚❛ ❝â y , = 6x2 − 6(1 + a)x + 6a = 0✳ ⇔ y , = x2 − (1 + a)x + a = x1 = a, y1 = −a3 + 3a2 − ⇔ x2 = 1, y2 = 3a − ✣➸ ✭✐✮ ❝â ✸ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t ⇒ ✤✐➲✉ ❦✐➺♥ ⇔ a=1 (−a3 + 3a2 − 2).(3a − 3) < ⇔ ✸✸ a=1 y1 y2 < a=1 (−a3 + 3a2 − 2).(3a − 3) < a=1 ⇔ −3(a − 1)2 (a − √ − 1).(a + √ − 1) < ⇔ a> √ 3+1 √ a 0✳ 3m 3m ❱ỵ✐ ❣✐→ trà ♥➔♦ ❝õ❛ số ỗ tr [2, +) ❈➙✉ ✷✾✳ ❱ỵ✐ ❣✐→ trà ♥➔♦ ❝õ❛ ♠ ❤➔♠ sè ỗ tr [, 0) ợ tr ♥➔♦ ❝õ❛ ♠ ❤➔♠ sè ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ [0, 1]✳ ợ tr ỗ t ✭✯✯✯✮ ❝â ❝ü❝ ✤↕✐✱ ❝ü❝ t✐➸✉✳ ❱✐➳t ♣❤÷ì♥❣ tr➻♥❤ ✤÷í♥❣ t❤➥♥❣ q✉❛ ✷ ✤✐➸♠ ❝ü❝ ✤↕✐✱ ❝ü❝ t✐➸✉✳ ❈➙✉ ✸✷✳ ❱ỵ✐ ❣✐→ trà ♥➔♦ ❝õ❛ ♠ ❤➔♠ sè ✭✯✯✯✮ ❝â ❈✣✱❈❚✿ xCD , xCT > 0✳ ❈➙✉ ✸✸✳ ❱ỵ✐ ❣✐→ trà ♥➔♦ ❝õ❛ ♠ ❤➔♠ sè ✭✯✯✯✮ ❝â ❈✣✱ ❈❚ ữớ t q ỹ ỹ t t ợ ✷ trö❝ t♦❛ ✤ë ♠ët t❛♠ ❣✐→❝ ❝â ❞✐➺♥ t➼❝❤ ❜➡♥❣ ✷✵✶✹✳ ❈➙✉ ✸✹✳ ❱ỵ✐ ❣✐→ trà ♥➔♦ ❝õ❛ ♠ ❤➔♠ sè ✭✯✯✯✮ ❝â ❈✣✱ ❈❚ ✈➔ ❝→❝ ✤✐➸♠ ❝ü❝ trà ❝→❝❤ ✤➲✉ ❣è❝ tå❛ ✤ë ❖✭✵✱✵✮✳ ●✐↔✐✳ y = x3 − 3mx2 + 3(m − 1)x + y , = 3x2 − 6mx + 3(m − 1) y , = ⇔ x2 − 2mx + m − = 0✳ ✣➸ ❤➔♠ sè ❝â ❝ü❝ ✤↕✐✱ ❝ü❝ t✐➸✉ t❤➻✿ , > ⇔ m2 − m + > t❤ä❛ ♠➣♥✳ P❤÷ì♥❣ tr➻♥❤ ✤÷í♥❣ t❤➥♥❣ q✉❛ ❝ü❝ ✤↕✐✱ ❝ü❝ t✐➸✉✿ y = (2m − − 2m2 )x + (2 + m2 − m)✳ ●å✐ A(x1 , y1 ); B(x2 , y2 ) ❧➔ tå❛ ✤ë ❝õ❛ ❤❛✐ ✤✐➸♠ ❝ü❝ ✤↕✐✱ ❝ü❝ t✐➸✉✳ ❨➯✉ ❝➛✉ ❜➔✐ t♦→♥ t÷ì♥❣ ✤÷ì♥❣ OA = OB ⇔ OA2 = OB ⇔ x21 = x22 ⇔ (x1 − x2 )(x1 + x2 ) = ⇔ (x1 − x2 ).2m = ❚❛ ❝â✿ (x1 − x2 )2 = (x1 + x2 )2 − 4x1 x2 ✳ √ ⇔ x1 − x2 = m2 − m + > ⇒ 2m = ⇒ m = 0✳ ✸✻ ❱➟② ✈ỵ✐ ♠ ❂ ✵ t❤➻ ❝→❝ ✤✐➸♠ ❝ü❝ ✤↕✐ ✈➔ ❝ü❝ t✐➸✉ ❝→❝❤ ✤➲✉ ❖✭✵✱✵✮✳ ❈➙✉ ✸✺✳ ❱ỵ✐ ❣✐→ trà ỗ t õ ỹ trà ✈➔ ♠ët tr♦♥❣ ❤❛✐ ✤✐➸♠ t❤✉ë❝ ❖①✳ ❈➙✉ ✸✻✳ ❚➻♠ ♠ ✤➸ ❈✣✱ ❈❚ ❝õ❛ ❤➔♠ sè ✭✯✯✯✮ ♥➡♠ tr ữớ t y = 2x ỗ t õ ổ t ú ợ ởt ữớ ố ✤à♥❤ ❦❤ỉ♥❣❄ ❈➙✉ ✸✽✳ ❱ỵ✐ ❣✐→ trà ♥➔♦ ❝õ❛ ♠ t❤➻ ❤➔♠ sè ✭✯✯✯✮ ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ ✶ ✤♦↕♥ ❝â ✤ë ❞➔✐ ❜➡♥❣ ✶✳ ❈➙✉ ✸✾✳ ❚➻♠ ♠ ✤➸ ❤➔♠ sè ✭✯✯✯✮ ❝â ❝ü❝ trà t↕✐ x1, x2 : x1 < −1 < x2✳ ●✐↔✐✳ y = x3 − 3mx2 + 3(m − 1)x + 2✳ y, = f (x) = 3x2 − 6mx + 3(m − 1)✳ ❨➯✉ ❝➛✉ ❜➔✐ t♦→♥ t÷ì♥❣ ✤÷ì♥❣ y , ❝â ✷ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t t❤ä❛ ♠➣♥ x1 < −1 < x2 ⇔ a.f (−1) < ⇔ 3.3m < ⇒ m < ỗ t ❖① t↕✐ ✸ ✤✐➸♠ ♣❤➙♥ ❜✐➺t✳ ❈➙✉ ✹✶✳ ❱ỵ✐ ❣✐→ tr t ỗ t t t↕✐ ✸ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❝â ❤♦➔♥❤ ✤ë ❧ỵ♥ ❤ì♥ ✵✳ ❈➙✉ ✹✷✳ ❱ỵ✐ ❣✐→ trà ♥➔♦ ❝õ❛ ♠ t❤➻ ỗ t t t t ❝â ❤♦➔♥❤ ✤ë ♥❤ä ❤ì♥ ✵✳ ❈➙✉ ✹✸✳ ❱ỵ✐ ❣✐→ tr t ỗ t t t↕✐ ✷ ✤✐➸♠ ❝â ❤♦➔♥❤ ✤ë ❧ỵ♥ ❤ì♥ ✵✱ ♠ët ✤✐➸♠ ❝â ❤♦➔♥❤ ✤ë ♥❤ä ❤ì♥ ✵✳ ❈➙✉ ✹✹✳ ❱ỵ✐ tr t ỗ t t ❖① t↕✐ ✷ ✤✐➸♠ ❝â ❤♦➔♥❤ ✤ë ♥❤ä ❤ì♥ ✵✱✶ ✤✐➸♠ ❝â ❤♦➔♥❤ ✤ë ❧ỵ♥ ❤ì♥ ✵✳ ❈➙✉ ✹✺✳ ❱ỵ✐ tr t ỗ t t ❖① t↕✐ ✸ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❝â ❤♦➔♥❤ ✤ë ♥❤ä ❤ì♥ ✷✳ ❈➙✉ ✹✻✳ ❱ỵ✐ ❣✐→ trà ♥➔♦ ❝õ❛ ♠ t ỗ t t t ✶ ✤✐➸♠✳ ●✐↔✐✳ P❤÷ì♥❣ tr➻♥❤ ❤♦➔♥❤ ✤ë ❣✐❛♦ ✤✐➸♠ ❝õ❛ ✭✯✯✯✮ ✈➔ ❖① ❧➔✿ x3 − 3mx2 + 3(m − 1)x + = 0✳ x=1 ⇔ x2 + (1 − 3m)x − = (2) ✣➸ ✭✯✯✯✮ ❣✐❛♦ ✈ỵ✐ ❖① t↕✐ ✶ ✤✐➸♠ ✈➔ ❝❤➾ ✶ ✤✐➸♠ ⇔ ✭✷✮ ❝â ✶ ♥❣❤✐➺♠ x = ❤♦➦❝ ✈æ ♥❣❤✐➺♠✳ ❚❍✶✿ ✭✷✮ ✈æ ♥❣❤✐➺♠ < ⇔ 9m2 − 6m + < ♠➔ t❛ ❝â 9m2 − 6m + = (3m − 1)2 + > 0✳ ❉♦ ✤â ❦❤æ♥❣ ❝â ❣✐→ trà ❝õ❛ ♠ t❤ä❛ ♠➣♥✳ ✸✼ 9m2 − 6m + = =0 ❚❍✷✿ ✭✷✮ ❝â ✶ ♥❣❤✐➺♠ x = ⇔ ⇔ 3m − = 3m − = 2 2 ♠➔ t❛ ❝â 9m − 6m + = (3m − 1) + > 0✳ ❉♦ ✤â ❦❤æ♥❣ ❝â ❣✐→ trà ❝õ❛ ♠ t❤ä❛ ♠➣♥✳ ❈➙✉ ✹✼✳ ❱ỵ✐ ❣✐→ trà t ỗ t t t ✶ ✤✐➸♠ ✈➔ t✐➳♣ ①ó❝ ✈ỵ✐ ❖① t↕✐ ✶ ✤✐➸♠ ❦❤→❝✳ ❈➙✉ ✹✽✳ ❱ỵ✐ ❣✐→ trà ♥➔♦ ❝õ❛ ♠ t❤➻ ỗ t t t t ❝â ❤♦➔♥❤ ✤ë x1 , x2 , x3 : x21 + x22 + x23 > 10✳ ●✐↔✐✳ y = x3 − 3mx2 + 3(m − 1)x + 2✳ P❤÷ì♥❣ tr➻♥❤ ❤♦➔♥❤ ✤ë ❣✐❛♦ ✤✐➸♠ ❝õ❛ ✭✯✯✯✮ ✈➔ ❖① ❧➔✿ x3 − 3mx2 + 3(m − 1)x + = x=1 ⇔ x2 + (1 − 3m)x − = (2) ỗ t t t ✤✐➸♠ ♣❤➙♥ ❜✐➺t t❤➻ ✭✷✮ ❝â ✷ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t >0 9m2 − 6m + > ❦❤→❝ ✶ ⇔ ⇔ ⇔ m = (∗) −3m = −3m = ❚❤❡♦ ❜➔✐ r❛ t❛ ❝â✿ x21 + x22 + x23 > 10 x21 + x22 + > 10 ⇔ x21 + x22 > (x1 + x2 )2 − 2x1√x2 > ⇔ 9m2 − 6m − > 1− m< 3√ ⇒ + m> √ − m< 3√ ❑➳t ❤ñ♣ ✤✐➲✉ ❦✐➺♥ ✭✯✮ t❛ ✤÷đ❝ + m> tr ỳ ỗ t❤à ✭✯✯✯✮ ❦❤ỉ♥❣ ❜❛♦ ❣✐í ✤✐ q✉❛✳ ❈➙✉ ✹✾✳ ❈➙✉ ợ tr t ỗ t ✭✯✯✯✮ ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ ♠ët ✤♦↕♥ ❝â ✤ë ❞➔✐ ❜➡♥❣ ✶✳ ✸✽ ✷✳✹ ▼ët sè ✤➲ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ ❈➙✉ ✶✳ ❈❤♦ ❤➔♠ sè y = 13 + (m 1)x2 + (4 3m)x + õ ỗ t❤à ❧➔ (Cm )✱ ♠ ❧➔ t❤❛♠ sè✳ ❚➻♠ ❝→❝ ❣✐→ trà ❝õ❛ ♠ ✤➸ tr➯♥ (Cm ) ❝â ❞✉② ♥❤➜t ♠ët ✤✐➸♠ ❝â ❤♦➔♥❤ ✤ë ➙♠ ♠➔ t✐➳♣ t✉②➳♥ (Cm ) t õ ổ õ ợ ữớ t❤➥♥❣ ✭❞✮✿ x + 2y = 0✳ ✭✣➲ t❤✐ ❝❤å♥ ❤å❝ s✐♥❤ ❣✐ä✐ ❧ỵ♣ ✶✷ ✈á♥❣ ✶ ❜↔♥❣ ❆ t➾♥❤ ▲♦♥❣ ❆♥✳✮ ●✐↔✐✳ ❚❛ ❝â y, = mx2 + 2(m − 1)x + − 3m✳ ❚✐➳♣ t✉②➳♥ ✈ỵ✐ (Cm ❝â ❤➺ sè ❣â❝ ❜➡♥❣ ✷ ♥➯♥ t❛ ❝â✿ mx2 + 2(m − 1)x + − 3m = (∗) ❨➯✉ ❝➛✉ ❜➔✐ t♦→♥ t÷ì♥❣ ✤÷ì♥❣ ✭✯✮ ❝â ✤ó♥❣ ✶ ♥❣❤✐➺♠ ➙♠ x=1 ⇔ (x − 1)(mx + 3m − 2) = ⇔ mx = − 3m ❱ỵ✐ ♠ ❂ ✵ ⇒ 0.x = ✭❧♦↕✐✮✳ m ✭✷✮ ❝â ✷ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t x = ⇔ ⇔ m = + 2m + − = ❑❤✐ ✤â ✸ ❣✐❛♦ ✤✐➸♠ ❧➔ A(1, 2m − 2); B(x1 , 2mx1 − 2); C(x2 , 2mx2 − 2)✳ ❚❛ ❝â✿ S OBC = BC.d tr♦♥❣ ✤â d = d(0; ) = √ + 4m2 2 2 BC = (x1 − x2 ) + (2mx1 − 2mx2 ) = (x1 + x2 ) − 4x1 x2 (4m2 + 1) ⇒ BC = [(2m + 1)2 + 8] (4m2 + 1) (2m + 1)2 + √ √ √ ❱➟② S = 17 ⇔ 4m2 + 4m + = 17 m=1 ⇔ m = −2 2x + ❈❤♦ ❤➔♠ sè y = õ ỗ t ữớ t ❧➔ x+2 y = −2x + m✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✭❞✮ ❝➢t ✭❈✮ t↕✐ ✷ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❆✱ ❇ ✈ỵ✐ ⇒S= ❈➙✉ ✸✳ ♠å✐ sè t❤ü❝ ♠✳ ●å✐ k1 , k2 ❧➛♥ ❧÷đt ❧➔ ❤➺ sè ❣â❝ ❝õ❛ t✐➳♣ t✉②➳♥ ❝õ❛ ✭❈✮ t↕✐ ❆ ✈➔ ❇✳ ❚➻♠ ♠ ✤➸ P = (k1 )2013 + k2 )2013 ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t✳ ✭✣➲ t❤✐ ❝❤å♥ ❤å❝ s✐♥❤ ❣✐ä✐ ❧ỵ♣ ✶✷ t➾♥❤ ❍↔✐ ❉÷ì♥❣✳✮ ●✐↔✐✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ ❤♦➔♥❤ ✤ë ❣✐❛♦ ✤✐➸♠ ❝õ❛ ✭❈✮ ✈➔ ✭❞✮✿ 2x + = −2x + m ⇔ x+2 ❳➨t ♣❤÷ì♥❣ tr➻♥❤ ✭✯✮ t❛ ❝â✿ x = −2 2x2 + (6 − m)x + − 2m = (∗) > 0, ∀m ∈ R ✈➔ x = −2 ❦❤æ♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✯✮ ♥➯♥ ✭❞✮ ❧✉æ♥ ❝➢t ✭❈✮ t↕✐ ✷ ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❆✱ ❇ ✈ỵ✐ ∀m✳ 1 ❍➺ sè ❣â❝ ❝õ❛ t✐➳♣ t✉②➳♥ t↕✐ ❆ ✈➔ ❇ ❧➔✿ k1 = ; k = (x1 + 2)2 (x2 + 2)2 tr♦♥❣ ✤â x1 , x2 ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✯✮✳ 1 = ❚❛ t❤➜② k1 k2 = (x1 + 2)2 (x2 + 2)2 (x1 x2 + 2(x1 x2 ) + 4)2 = 4(k1 > 0, k2 ) > 0✳ P = (k1 )2013 + k2 )2013 ≥ (k1 k2 )2013 = 22014 ✱ ❞♦ ✤â ▼✐♥ P ❂ ✷✵✶✹ ✤↕t 1 ✤÷đ❝ ❦❤✐ k1 = k2 ⇔ = k = ⇔ (x1 + 2)2 = (x2 + 2)2 2 (x1 + 2) (x2 + 2) ⇔ x1 + x2 = −4 ⇔ m = −2✳ ❱➟② ♠ ❂ ✲✷ ❧➔ ❣✐→ trà ❝➛♥ t➻♠✳ ✹✵ ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❣✐ỵ✐ ❤↕♥✱ t➼♥❤ ❧✐➯♥ tư❝✱ t➼♥❤ ❦❤↔ ✈✐ ❝õ❛ ❤➔♠ ♠ët ❜✐➳♥✳ ❚r➯♥ ❝ì sð ✤â t→❝ ❣✐↔ ✤➣ ①➙② ❞ü♥❣ ✈➔ ❣✐↔✐ ❜➔✐ t♦→♥ tê♥❣ ❤ñ♣ ✈➲ tự tr t ỗ tớ ✤÷❛ r❛ ❜➔✐ t♦→♥ tê♥❣ ❤đ♣ ✈➲ ❤➔♠ ❜➟❝ ❜❛✳ ❈→❝ ❜➔✐ t♦→♥ tê♥❣ ❤đ♣ ♥➔② ❣✐ó♣ ❣✐→♦ ✈✐➯♥ ✈➔ ❤å❝ s✐♥❤ ♣❤ê t❤æ♥❣ ❝â ❝→✐ ♥❤➻♥ ❜❛♦ q✉→t ①✉♥❣ q✉❛♥❤ ❧ỵ♣ ❝→❝ ❤➔♠ sè ✤❛ t❤ù❝ ✈➔ ♣❤➙♥ t❤ù❝ ✤÷đ❝ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝❤õ ②➳✉ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ ♣❤ê t❤æ♥❣✳ ✹✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❙✳ ▼✳ ◆✐❦♦❧s❦②✱ ❆ ❝♦✉rs❡ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s✱ ❱♦❧ ✶✱ ▼✐r ♣✉❜❧✐s❤❡rs✱ ✶✾✽✶✳ ❬✷❪ ✣➲ t❤✐ t✉②➸♥ s✐♥❤ ✤↕✐ ❤å❝ ✭✶✾✼✵ ✲ ✷✵✶✹✮✳ ❬✸❪ P✳ ❊✳ ✣❛♥❦♦✱ ❆✳ ❨✳ P♦r♦♣✱ ❚✳ ❧❛✳ ❈♦❣✐❡♣♥❤✐❝♦✈❛✱ ❇➔✐ t➟♣ t♦→♥ ❤å❝ ❝❛♦ ❝➜♣✱ ◆❤➔ ①✉➜t ❜↔♥ ✧▼✐r✧ ▼❛①❝♦✈❛✳ ❬✹❪ ◆❣✉②➵♥ ❈❛♠✱ ●✐↔✐ t♦→♥ ✤↕♦ ❤➔♠ ✈➔ ❦❤↔♦ s→t ❤➔♠ sè✱ ◆❤➔ ①✉➜t ❜↔♥ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✱ ✶✾✾✾✳ ❬✺❪ ◆❣✉②➵♥ ❱➠♥ ◗ó②✱ ◆❣✉②➵♥ ❚✐➳♥ ❉ô♥❣✱ ◆❣✉②➵♥ ❱✐➺t ❍➔✱ ❑❤↔♦ sè✳ ✹✷ s→t ❤➔♠ ... ❧✉æ♥ ❝â ❝→❝ ❜➔✐ t➟♣ ✈➲ số ỵ tt số tử ❦❤↔ ✈✐ ✤÷đ❝ sû ❞ư♥❣ r➜t rë♥❣ r➣✐ tr♦♥❣ ❝→❝ ❜➔✐ t➟♣ ❝ơ♥❣ ♥❤÷ ❝→❝ s→❝❤ ✈✐➳t ✈➲ ❤➔♠ sè✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ởt số ỵ q trồ ❧✐➯♥ tö❝... < m2 + 4m + < √ √ ⇔ m2 + 4m + < 0✳ ⇔ −2 − < m < −2 + 3✳ ❈➙✉ ✻✵✳ ❱ỵ✐ ❣✐→ trà t ỗ t số ỗ tr [2, +) ỗ t số ỗ ❜✐➳♥ tr➯♥ [2, +∞) ⇔ y,(x) ≥ 0, ∀x > ⇔ f (x) = x2 − 2x + m − ≥ 0, ∀x > ✰❚❍✶✿ ,... tr➻♥❤ 3m −2 X2 + = ❝â ✷ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t ⇔ < ⇔ m > 0✳ 3m 3m ❱ỵ✐ tr số ỗ tr [2, +∞)✳ ❈➙✉ ✷✽✳ ❈➙✉ ✷✾✳ ❱ỵ✐ ❣✐→ trà ♥➔♦ số ỗ tr [, 0) ✸✵✳ ❱ỵ✐ ❣✐→ trà ♥➔♦ ❝õ❛ ♠ ❤➔♠ sè ♥❣❤à❝❤ ❜✐➳♥ tr➯♥ [0,