Ôn tập giải tích thi Olympic sinh viên

ữỡ ợ số Pữỡ t ỡ ỵ tt é ú t sỷ ỵ s t ợ ởt số õ ỵ sè {a } ✤ì♥ ✤✐➺✉ t➠♥❣ ✈➔ ❜à ❝❤➦♥ tr➯♥ t tỗ t ợ lim n an n+ ỳ ❤ì♥ lim an = supan n→+∞ n∈N ✭❜✮ ❉➣② số {an } ỡ ữợ t tỗ t ợ n+ lim an ỡ ỳ lim an = inf an n→+∞ n∈N ⑩♣ ❞ö♥❣✳ ❱➼ ❞ư ✶✳✷✳ ❈❤♦ ❞➣② sè {u } ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿ n   u1 =  un+1 = un + ❚➻♠ ❣✐ỵ✐ ❤↕♥ lim n→+∞ u2n 1999 u2 un u1 + + ··· + u2 u3 un+1 ❚ø ❣✐↔ t❤✐➳t t❛ ❝â ❚ø ✤â s✉② r❛ un u2n 1999(un+1 − un ) = = = 1999 un+1 un+1 un un+1 un 1 − un un+1 u1 u2 uk + + ··· + = 1999 u2 u3 uk+1 ✶ 1 − u1 uk+1 ✭✶✳✶✮ ✷ ❚❛ t❤➜② {un } ❧➔ ❞➣② ✤ì♥ ✤✐➺✉ t➠♥❣✱ ❤ì♥ ♥ú❛ un ≥ 1, ∀n✳ ◆➳✉ {un } ❧➔ ❞➣② ❜à ❝❤➦♥ tr➯♥✱ ❦❤✐ ✤â t❤❡♦ ỵ tử ỡ tỗ t ❣✐ỵ✐ ❤↕♥ n→+∞ lim un = L ❚ø ❝ỉ♥❣ t❤ù❝ tr ỗ L q ợ n → ∞ t❛ ♥❤➟♥ ✤÷đ❝ L = L + 1999 L = ổ ỵ un ≥ ❱➟② {un } ❧➔ ❞➣② ❦❤æ♥❣ ❜à ❝❤➦♥ tr➯♥✳ ❉♦ ✤â n→+∞ lim un = +∞ ❈❤♦ k tr ú ỵ tr t ♥❤➟♥ ✤÷đ❝ lim k→+∞ u1 u2 uk + + ··· + u2 u3 uk+1 = lim 1999 k→+∞ 1 − u1 uk+1 = 1999 − lim ▼ët sè ❜➔✐ t➟♣ →♣ ❞ö♥❣✳ ❇➔✐ ✶✳ ❈❤♦ ❞➣② sè {u } ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿ n   u1 =  un+1 = u2n +1999un , 2000 ❚❤➔♥❤ ❧➟♣ ❞➣② {Sn } ✈ỵ✐ n Sn = k=1 ❚➼♥❤ ❣✐ỵ✐ ❤↕♥ n ∈ N uk uk+1 − lim Sn n→+∞ ❇➔✐ ✷✳ ❈❤♦ ❞➣② sè {u } ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿ n   u1 ∈ N  un+1 = ln + u2n , n ∈ N ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❞➣② sè {xn } ❤ë✐ tö✳ ❇➔✐ ✸✳ ❈❤♦ ❞➣② sè {xn} ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿   0 < xn <  xn+1 (1 − xn ) ≥ , ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ lim xn = n→+∞ n ∈ N ❇➔✐ ✹✳ ❈❤♦ f : [0, ∞) → [0, ∞) ❣✐↔♠ ✈➔ ❧✐➯♥ tö❝✳ ●✐↔ sû r➡♥❣ ❤➺    f (α) = β,    f (β) = α,     α, β ≥ 0, ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t α = β = a ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❞➣② {xn+1 = f (xn )} ✈ỵ✐ x0 > ❤ë✐ tư ✈➲ a k→+∞ uk+1 = 1999 ✸ ❇➔✐ ✺✳ ❈❤♦ ❞➣② sè {a } ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿ n   an = an−1 + 2015 an−1 , n ≥ 2,  a1 = 2016 √ lim an = 2015 ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ n→+∞ ❇➔✐ ✻✳ ●✐↔ sû r➡♥❣ ❞➣② {an} ❜à ❝❤➦♥ ✈➔ t❤ä❛ ♠➣♥ an+2 ≤ an+1 + an 3 ✈ỵ✐ n ≥ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❞➣② tr➯♥ ❤ë✐ tö✳ ✶✳✷ ỵ ỵ tt ố ợ t t sỷ ỵ ữợ t ỵ số {a , b , c } t❤ä❛ ♠➣♥ a ≤ b ≤ c ●✐↔ sû r➡♥❣ n lim cn = A n→+∞ n n n n n ❑❤✐ ✤â t❛ ❝â n→+∞ lim bn = A lim an = n→+∞ ✣è✐ ✈ỵ✐ ❜➔✐ t➟♣ ②➯✉ ❝➛✉ t➼♥❤ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ♠ët ❞➣② n→+∞ lim bn ♥➔♦ ✤â✱ t❛ ❝â t❤➸ t➻♠ ❝→❝ ❞➣② an , cn s❛♦ ❝❤♦ an ≤ bn ≤ cn ✈ỵ✐ n ✤õ ❧ỵ♥✱ ✈➔ ❤ì♥ ♥ú❛ ❝→❝ ❣✐ỵ✐ ❤↕♥ n→+∞ lim an , lim cn ♣❤↔✐ ❞➵ t➼♥❤✱ ✈➔ n→+∞ ♣❤↔✐ ❜➡♥❣ ♥❤❛✉✳ ❑❤✐ ✤â →♣ ỵ tr t s r ữủ ợ ❝õ❛ ❞➣② n→+∞ lim bn ❇➯♥ ❝↕♥❤ ✤â✱ ❝❤ó♥❣ t❛ ❝ơ♥❣ ❝➛♥ tỵ✐ ♠ët sè ❜➜t ✤➥♥❣ t❤ù❝ s❛✉✳ ❱ỵ✐ x > t❛ ❝â ✭❛✮ 1− x2 x4 x2 < cos x < − + 2 4! x− x3 x3 x5 < sin x < x − + 3! 3! 5! ✭❜✮ ✭❝✮ ✭❞✮ √ 1 1 1 + x − x2 < + x < + x − x2 + x3 8 16 x− ⑩♣ ❞ö♥❣✳ x2 x3 x4 x2 x3 + − < ln(1 + x) < x − + ❱➼ ❞ö ✶✳✹✳ ❈❤♦ ❜✐➳t r➡♥❣ n Sn = ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ( 1+ k=1 k − 1) n2 lim Sn = n→+∞ ✹ ự rữợ t t ú ỵ r ợ x > −1, t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ √ x x < 1+x−1< 2+x ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥✱ t❤❛② x ❧➛♥ ❧÷đt ❜ð✐ nk , ❝❤ó♥❣ t❛ ♥❤➟♥ ✤÷đ❝ k < 2n2 + k 1+ k k − < n2 2n ▲➜② tê♥❣ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ t❤❡♦ k tø ✤➳♥ n t❛ ♥❤➟♥ ✤÷đ❝ n k=1 ❚❛ ❝â n k=1 k = 2n2 2n k < Sn < 2n + k n n k=1 ❦❤✐ n → ∞ n(n + 1) → 4n2 k= k=1 ✭✶✳✷✮ k 2n2 ✭✶✳✸✮ ▼➦t ❦❤→❝✱ t❛ ❧↕✐ ❝â lim { n→+∞ ▲↕✐ ❝â n k=1 2n2 n n k− k=1 k2 < 2n (2n2 + k) k=1 n k=1 ❉♦ ✤â lim { n→+∞ k } = lim n→+∞ 2n2 + k n k=1 k2 2n2 (2n2 k2 n(n + 1)(2n + 1) = →0 4n 24n4 2n2 n n k k=1 ú ỵ ✭✶✳✸✮ t❛ ♥❤➟♥ ✤÷đ❝ k=1 n lim n→+∞ k=1 + k) ❦❤✐ n → ∞ k } = 0, +k 2n2 k = 2n ✭✶✳✹✮ ▲➜② ❣✐ỵ✐ ❤↕♥ tr♦♥❣ ✭✶✳✷✮ ❦❤✐ n → ∞ ✈➔ ú ỵ t ỵ t ♥❤➟♥ ✤÷đ❝ lim Sn = n→+∞ ▼ët sè ❜➔✐ t➟♣ →♣ ❞ö♥❣✳ ❇➔✐ ✶✳ ❈❤♦ ❞➣② sè {x } ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ n xn = (1 + ❚➻♠ n→+∞ lim lnxn n )(1 + ) · · · (1 + ) n n n ✺ ❇➔✐ ✷✳ ❳➨t sü ❤ë✐ tö✱ ♣❤➙♥ ❦➻ ❝õ❛ ❞➣② sè s❛✉ xn = √ n2 n2 n2 +√ + ··· + √ 6 n +1 n +2 n6 + n ❇➔✐ ✸✳ ❳➨t sü ❤ë✐ tö✱ ♣❤➙♥ ❦➻ ❝õ❛ ❞➣② sè s❛✉ xn = [α] + [2α] + · · · + [nα] , n2 tr♦♥❣ ✤â [x] ❧➔ ❝❤➾ sè ♥❣✉②➯♥ ợ t ổ ữủt q x, ởt sè t❤ü❝ ❜➜t ❦➻✳ ❇➔✐ ✹✳ ❚➼♥❤ ❣✐ỵ✐ ❤↕♥ s❛✉ ✐✳ n lim n→+∞ ✐✐✳ lim ( lim n→+∞ ❇➔✐ ✺✳ ❈❤♦ {a } ✈➔ {b } ✤à♥❤ ♥❣❤➽❛ ❜ð✐ n 1+ k=1 n k2 − 1) n3 12 + 22 + · · · + n2 n→+∞ ✐✐✐✳ n + + ··· + n2 + n2 + n +n n a1 = 3, b1 = 2, an+1 = an + 2bn ❍ì♥ ♥ú❛✱ ❝❤♦ ✈➔ bn+1 = an + bn an , n ∈ N bn √ √ |cn+1 − 2| < 21 |cn − 2|, n ∈ N cn = ✐✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✐✐✳ ❚➼♥❤ n→+∞ lim cn ❇➔✐ ✻✳ ❚➼♥❤ ❣✐ỵ✐ ❤↕♥ ln2 n n→+∞ n n−2 lim k=2 ln k ln (N − k) số tờ qt ỵ tt ố ợ t tổ tữớ s ❧➔ t➻♠ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ♠ët ❞➣② sè lim an n tr õ {an } ữủ ữợ tr ỗ t t sỷ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ s❛✐ ♣❤➙♥ ✤➸ t➻♠ sè ❤↕♥❣ tê♥❣ q✉→t ❝õ❛ ❞➣② sè✱ tù❝ ✤÷❛ ❞➣② sè {an } ✈➲ ❞↕♥❣ af (n) s❛✉ ✤â sû ❞ö♥❣ ❝→❝ ❦✐➳♥ t❤ù❝ ✤➣ ❤å❝ ✤➸ t➼♥❤ ❣✐ỵ✐ ❤↕♥ lim f (n) n→∞ ✻ ⑩♣ ❞ö♥❣✳ ❱➼ ❞ö ✶✳✺✳ ❈❤♦ α ∈ (0, 2) ❚➻♠ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❞➣② sè {xn } ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ lim xn n→+∞ xn+1 = αxn + (1 − α)xn−1 t❤❡♦ α, x0 , x1 ❚❤❡♦ ❣✐↔ t❤✐➳t✱ t❛ ❝â xn − xn−1 n−1 xn − xn−1 = (α − 1) n = 1, 2, · · · = (α − 1)(xn−1 − xn−2 ) (x1 − x0 ) ❉♦ ✤â n xn − x0 = ❇➡♥❣ q✉② ♥↕♣ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ n (α − 1)k−1 (xk − xk−1 ) = (x1 − x0 ) k=1 ✭✶✳✺✮ k=1 ❉♦ α ∈ (0, 2) ♥➯♥ |α − 1| < ▼➦t ❦❤→❝ t❛ ❧↕✐ ❝â n (α − 1)k−1 = k=1 − (α − 1)n − (α − 1)n = − (α − 1) 2−α ▲➜② ❣✐ỵ✐ ❤↕♥ ❦❤✐ n → tr ú ỵ tự tr t❛ ♥❤➟♥ ✤÷đ❝ lim xn = x0 + lim (x1 − x0 ) n→+∞ n→+∞ − (α − 1)n (1 − α)x0 + x1 = 2−α 2−α ▼ët sè ❜➔✐ t➟♣ →♣ ❞ö♥❣✳ ❇➔✐ ✶✳ ❈❤♦ ❞➣② ❋✐❜♦♥❛❝✐ ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿ f ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❣✐ỵ✐ ❤↕♥ lim n→+∞ = 1, f2 = 2, fn+1 = fn + fn−1 , fn+1 fn , ∀n ≥ tỗ t ợ tr x0 = a, x1 = b ✈➔ xn+2 = 13 (xn + 2xn+1), n ∈ N ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❣✐ỵ✐ n+ lim xn tỗ t ợ õ ❇➔✐ ✸✳ ❈➙✉ ❤ä✐ t÷ì♥❣ tü ❜➔✐ tr➯♥ ❝❤♦ ❞➣②✿ x0 = a, x1 = b, xn = (1 − ❇➔✐ ✹✳ ●✐↔ sû r➡♥❣ b ∈ R, a n →a∈R ✈➔ xn+1 = an + bxn ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✿ ✐✳ xn → ✐✐✳ xn → 1 )xn−1 + xn−2 , n ∈ N n n a 1−b a 1−b ♥➳✉ |b| < 1; ♥➳✉ |b| > ✈➔ x1 + ak = bk ✼ ✶✳✹ P❤÷ì♥❣ t ỵ tt ❚➼♥❤ ❣✐ỵ✐ ❤↕♥ 1 + + ··· + n+1 n+2 3n lim n→+∞ ❚❛ ❝â Sn := ❳➨t ❤➔♠ sè f (x) = ✤♦↕♥ ♥➔②✳ ❉♦ ✤â 1 +x 1 1 + + ··· + = n+1 n+2 3n 2n k=1 k + 2n tr➯♥ ✤♦↕♥ [0, 1] ❍➔♠ f ❧➔ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ [0, 1] ♥➯♥ ❦❤↔ t➼❝❤ tr➯♥ 1 n→+∞ 2n 2n f( f (x)dx = lim ▲↕✐ ❝â 2n k=1 k ) = lim n→+∞ 2n 2n 2n k=1 k + 2n f (x)dx = ❉♦ ✤â✱ t❛ ❝â 1 dx = ln ( + x)|10 = ln − ln = ln 2 +x 1 + + ··· + n+1 n+2 3n lim n→+∞ = ln ▼ët sè ❜➔✐ t➟♣ →♣ ❞ö♥❣✳ ❇➔✐ ✶✳ ❚➼♥❤ ❝→❝ ❣✐ỵ✐ ❤↕♥ s❛✉✳ ✐✳ lim n2 n→+∞ n3 ✐✐✳ lim n→+∞ ✐✐✐✳ 1 + + ··· + 3 +1 n +2 n + n3 k + k + · · · + nk , nk+1 n→+∞ n (n + 1)(n + 2) · · · (n + n) lim ✐✈✳ lim n→+∞ sin k ≥ n n n + sin + · · · + sin n2 + n + 22 n + n2 ✈✳ lim n→+∞ n 2n 2n 2n + + ··· + n+1 n+ n + n1 ✽ ❇➔✐ ✷✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❣✐ỵ✐ ❤↕♥ π sin n+1 lim n→+∞ + 2π sin n+1 + ··· + nπ sin n+1 n ❧➔ ♠ët sè ❞÷ì♥❣✳ ❇➔✐ ✸✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ f ❧➔ ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ tr➯♥ [0, 1] t❤➻  lim n n→+∞  n i f( ) − n i=1 f (x)dx = f (1) − f (0) ❙û ❞ö♥❣ ❦➳t q✉↔ tr➯♥✱ ❤➣② t➼♥❤✿ lim n n→+∞ 1k + 2k + · · · + nk − , nk+1 k+1 ❇➔✐ ✹✳ ❱ỵ✐ k ≥ 0, ❤➣② t➼♥❤ lim n→+∞ k ≥ 1k + 3k + · · · + (2n 1)k nk+1 Pữỡ ợ tr ữợ ỵ tt r ố ự ợ ởt số tỗ t ổ t ❝❤ù♥❣ ♠✐♥❤ ❣✐ỵ✐ ❤↕♥ tr➯♥ ❝õ❛ ❞➣② sè ✤â ❜➡♥❣ ợ ữợ số õ ổ tự ❧➔ t❛ ❦✐➸♠ tr❛ ❝â ✤➥♥❣ t❤ù❝ s❛✉ ❤❛② ❦❤æ♥❣ lim an n→+∞ lim inf an = lim sup an n→∞ n→∞ ⑩♣ ❞ö♥❣✳ ❱➼ ❞ö ✶✳✼✳ ●✐↔ sû r➡♥❣ ❞➣② {a } ❧➔ ♠ët ❞➣② sè t❤ü❝ s❛♦ ❝❤♦ n t❤ü❝ ❜à ❝❤➦♥✳ ◆➳✉ k ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ s❛♦ ❝❤♦ lim an = n→+∞ ✈➔ {bn } ❧➔ ❞➣② sè lim (bn − an bn+k ) = l, n→+∞ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ l = ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t b = lim inf bn , n→∞ B = lim sup bn n→∞ ❉♦ {bn } ❧➔ ❞➣② sè ❜à ❝❤➦♥ ♥➯♥ b, B ❧➔ ❝→❝ sè ❤ú✉ ❤↕♥✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ lim inf n→∞ an , lim supn an tỗ t {bp }, {bq } ❝õ❛ ❞➣② {bn } s❛♦ ❝❤♦ bp → b, bq → B ❦❤✐ r → ∞ ❉♦ an → ✈➔ bn − an bn+k → l ❦❤✐ n → ∞ ♥➯♥ ❞➣② ❝♦♥ {bp +k }, {bq +k } {bn } tữỡ ự t tợ b − l ✈➔ B − l ❦❤✐ r → ∞ ❍➺ q✉↔ ❧➔ b − l ≥ l ✈➔ B − l ≤ l ❉♦ ✤â l = r r r r r r ✾ ▼ët sè ❜➔✐ t➟♣ →♣ ❞ö♥❣✳ ❇➔✐ ✶✳ ❈❤♦ ❞➣② sè t❤ü❝ {a } s❛♦ ❝❤♦ a n n ≥ 1, ∀n ≥ ✈➔ ❞➣② {an + a−1 n } ❤ë✐ tö✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❞➣② {an } ❤ë✐ tö✳ ❇➔✐ ✷✳ ❈❤♦ ❞➣② sè t❤ü❝ {an} s❛♦ ❝❤♦ n→+∞ lim (2an+1 − an ) = l ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ lim an = l n→+∞ ❇➔✐ ✸✳ ❈❤♦ ❞➣② sè ❞÷ì♥❣ {an} s❛♦ ❝❤♦ n→+∞ lim an = L ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ lim (a1 a2 · · · an ) n = L n→+∞ ❈❤÷ì♥❣ ✷ ❚➼❝❤ ♣❤➙♥ ✷✳✶ ❚➼♥❤ t➼❝❤ ♣❤➙♥ ✣è✐ ✈ỵ✐ ❞↕♥❣ ❜➔✐ t➟♣ ♥➔② t❤➻ ❝❤õ ②➳✉ ❧➔ ❞ò ♣❤➨♣ ✤ê✐ ❜✐➳♥ ✈➔ t➼❝❤ ♣❤➙♥ ✤➸ ✤÷❛ t➼❝❤ ♣❤➙♥ ❜❛♥ ✤➛✉ ✈➲ t➼❝❤ ♣❤➙♥ ❞➵ t➼♥❤ ❤ì♥✳ ❇➯♥ ❝↕♥❤ ✤â ♠ët sè ❜➔✐ t➟♣ s➩ →♣ ❞ö♥❣ ❝→❝ ❦➳t q✉↔ s❛✉ ✤➸ t➼♥❤ t➼❝❤ ♣❤➙♥✳ ▼➺♥❤ ✤➲ ✷✳✶✳ ❈❤♦ f : [−a, a] → R ❧➔ ❤➔♠ ❧✐➯♥ tö❝ ✈➔ ❝❤➤♥ ✭a > 0.✮ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ a a f (x) dx = + ex −a ✭✷✳✶✮ f (x)dx ▼➺♥❤ ✤➲ ✷✳✷✳ ❈❤♦ f : [−a, a] → R ❧➔ ❤➔♠ ❧✐➯♥ tö❝ ✭a > 0.✮ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✶✳ a a f (x)dx = −a ✷✳ ♥➳✉ f ❧➔ ❤➔♠ ❝❤➤♥ f (x)dx ✭✷✳✷✮ a f (x)dx = ♥➳✉ f ❧➔ ❤➔♠ ❧➫ ✭✷✳✸✮ −a ▼➺♥❤ ✤➲ ✷✳✸✳ ❈❤♦ ❤➔♠ f : R → R ❧➔ ❤➔♠ ❧✐➯♥ tö❝ ✈➔ t✉➛♥ ❤♦➔♥ ❝❤✉ ❦➻ T > ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✶✳ ✈ỵ✐ ♠é✐ sè t❤ü❝ a t❛ ❝â a+T T f (x)dx = a f (x)dx, ✭✷✳✹✮ ✷✳ ✈ỵ✐ ♠é✐ sè t❤ü❝ a < b t❛ ❝â b lim n→+∞ b−a f (nx)dx = T a T f (x)dx ✶✵ ✭✷✳✺✮ ✶✶ ▼➺♥❤ ✤➲ ✷✳✹✳ ❈❤♦ f : [0, 1] → R ❧➔ ❤➔♠ ❧✐➯♥ tö❝✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ π π π xf (sin x)dx = ✭✷✳✻✮ f (sin x)dx ❱➼ ❞ö ✷✳✺✳ ❚➼♥❤ t➼❝❤ ♣❤➙♥ ✶✳ π I1 = −π cos2014 (x) √ dx 1−x+ x2 +1 ❈❤ù♥❣ ♠✐♥❤✳ ✣è✐ ✈ỵ✐ ❜➔✐ t➟♣ ♥➔②✱ ❝❤ó♥❣ t❛ s➩ ❞ò♥❣ ♣❤➨♣ ✤ê✐ ❜✐➳♥ t t st ữợ t ♣❤➙♥✱ t❛ t❤➜② ð ♠➝✉ sè ❝â ❞↕♥❣ ❧✐➯♥ ❤ñ♣✳ õ trữợ t t tỷ ủ ❝❤ó♥❣ t❛ ♥❤➟♥ ✤÷đ❝ ❝→✐ ❣➻❄ ❚❛ ❝â 1−x+ √ √ √ −(1 − x) + x2 + −(1 − x) + x2 + √ √ = = 2x x2 + (−(1 − x) + x2 + 1)(1 − x + x2 + 1) ◆➳✉ t❛ t❤❛② ✤➥♥❣ t❤ù❝ ♥➔② ✈➔♦ t➼❝❤ ♣❤➙♥ ❜❛♥ ✤➛✉✱ s❛✉ ✤â ❞ò♥❣ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ ✤➸ t➼♥❤✱ ✤✐➲✉ ♥➔② ♥â✐ ❝❤✉♥❣ ❧➔ ❦❤â✱ ✈➻ ❜✐➸✉ t❤ù❝ tr➯♥ ❝á♥ ❝❤ù❛ ❝➠♥✱ ❧↕✐ ❝â ❞↕♥❣ ❝❤✐❛ ❝❤♦ ✤❛ t❤ù❝ ð ♠➝✉✳ ✣➸ t➼♥❤ ✤÷đ❝ t➼❝❤ ♣❤➙♥ tr➯♥✱ ✤✐➲✉ t❛ ❝➛♥ ❧➔ ♣❤↔✐ ❦❤û ✤✐ ✤÷đ❝ ❝➠♥ ✈➔ ❝❤✐❛ ❝❤♦ ♠➝✉ số t ữ ữợ t ✈➲ ❞↕♥❣ xn cos2014 (x) tr♦♥❣ ✤â n ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣ ♥➔♦ ✤â✳ ❚❛ t❤û ♥❤➻♥ t❤❡♦ ❝→❝ ❦❤→❝ ①❡♠ s❛♦✳ ❚❤❛② ✈➻ ①➨t ❤➔♠ f (x) = 1−x+√1 x +1 t❛ t❤û ①❡♠ ❤➔♠ √ √ = f (−x) f (x) = −x + x2 + ❑❤✐ ✤â t❛ q✉❛♥ s→t t❤➜② f (x) = x+√1x +1 = −(−x)+ (−x) +1 ✣✐➲✉ ♥➔② ❝â t❤➸ ủ ỵ ú t õ tr ✤÷❛ t➼❝❤ ♣❤➙♥ ✤➣ ❝❤♦ ✈➲ t➼❝❤ ♣❤➙♥ ❞➵ t➼♥❤ ❤ì♥✳ Ð ✤➙② t❛ q✉❛♥ s→t t❤➜② ❤➔♠ f (x) õ ủ ỵ t ✈✐➺❝ ✤ê✐ ❜✐➳♥ t = −x ❱ỵ✐ ♣❤➨♣ ✤ê✐ ❜✐➳♥ ♥➔②✱ t❛ s➩ ♥❤➟♥ ✤÷đ❝ ❦➳t q✉↔ 2 π π 2014 cos (x) dx = − + f (x) I1 = −π −π π =− −π π =− cos2014 (−t) √ dt + t + t2 + cos2014 (t) √ dt + t + t2 + f (x) cos2014 (x) dx + f (x) −π ❚ø ✤â✱ s✉② r❛ π 2I1 = −π cos2014 (x) dx + + f (x) π −π f (x) cos2014 (x) dx = + f (x) π cos2014 (x)dx −π ✳ ợ sỷ ổ tự tr ỗ ✈➔ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥✱ t❛ ❤♦➔♥ t♦➔♥ ❝â t❤➸ t➼♥❤ ✤÷đ❝ t➼❝❤ ♣❤➙♥ I1 ❱✐➺❝ ❝á♥ ❧↕✐ ①✐♥ ❞➔♥❤ ❝❤♦ ❜↕♥ ✤å❝✳ ✶✷ ✷✳ π 2014 I2 = π ✸✳ I3 = ✹✳ I4 = 1 1+ecos 2014x dx ex sin x (cos x+sin x)2 dx x2014 2014 1+x+ x2 +···+ x2014! dx t tự t ỵ ❈❤♦ f, g ❧➔ ❝→❝ ❤➔♠ ①→❝ ✤à♥❤ ✈➔ ❦❤↔ t➼❝❤ tr➯♥ ✤♦↕♥ [a, b] s❛♦ ❝❤♦ f ≥ g ❑❤✐ ✤â t❛ ❝â b b f (x)dx ≥ f (x)dx a a ỵ t tự ✲ ❙❝❤✇❛r③✮✳ ❈❤♦ f, g ❧➔ ❝→❝ ❤➔♠ ①→❝ ✤à♥❤ ✈➔ ❦❤↔ t➼❝❤ tr➯♥ ✤♦↕♥ [a, b] ❑❤✐ ✤â t❛ ❝â  2 b b a b f (x)g(x)dx ≤  g (x)dx f (x)dx a a ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â✱ ✈ỵ✐ ♠å✐ x ∈ R t❤➻ b b 0≤ (xf (t) + g(t)) dt =x a b b (f (t)) dt + 2x a g (t)dt f (t)g(t)dt + a a =Ax2 + Bx + C, tr♦♥❣ ✤â b b A= (f (t)) dt, B= a ❚❛♠ t❤ù❝ ❜➟❝ ❤❛✐✱ Ax b f (t)g(t)dt, a + Bx + C  a ❦❤ỉ♥❣ ➙♠ ✈ỵ✐ ♠å✐ x ∈ R ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ B − AC ≤ 0, tù❝ ❧➔ 2 b b a b f (t)g(t)dt ≤  g (t)dt (f (t)) dt a ✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ a b (xf (t) + g(t)) dt = ❤❛② g (t)dt C= a xf (t) = g(t), t ∈ [a, b] ✶✸ ❱➼ ❞ö ✷✳✽✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ f ❧➔ ❝→❝ ❤➔♠ ①→❝ ✤à♥❤ ✈➔ ❦❤↔ t➼❝❤ tr➯♥ ✤♦↕♥ [a, b], t❤➻  2 b  a b f (x)dx f (x) cos xdx ≤ (b − a) f (x) sin xdx +   2 b a a ❈❤ù♥❣ ♠✐♥❤✳ ỵ ợ g(x) = sin x ✈➔ g(x) = cos x t❛ ♥❤➟♥ ✤÷đ❝  2 b b  a 2 b a b a b f (x) cos xdx ≤  sin2 (x)dx f (x)dx a ✈➔ b f (x) sin xdx ≤  cos2 (x)dx f (x)dx a a ❉♦ ✤â  b 2  a b a b f (x)dx a b 2 f (x) cos xdx ≤ f (x) sin xdx +   2 b sin (x)dx + a a sin2 (x)dx + a a (sin2 x + cos2 x)dx a b b = cos2 (x)dx b f (x)dx =  b a b a b f (x)dx  = cos2 (x)dx f (x)dx a  b b f (x)dx a dx a b f (x)dx =(b − a) a ❱➼ ❞ö ✷✳✾✳ ●✐↔ sû r➡♥❣ f : [a, b] → [m, M ] ✈➔ b f (x)dx = a ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ b f (x)dx ≤ −mM (b − a) ✭✷✳✼✮ a P❤➙♥ t➼❝❤✿ Ð ✤➙② t❛ ❝è ❣➢♥❣ →♣ ởt tr ỵ ỵ ỵ ự t tr t ỵ ự ✤â t❛ ❝❤÷❛ t❤➜② ①✉➜t ❤✐➺♥ ❤➔♠ g, tr♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔② ♠✉è♥ ①✉➜t ❤✐➺♥ ❤➔♠ g t❛ ❝â t❤➸ ♣❤➙♥ t➼❝❤ f t❤➔♥❤ t➼❝❤ ❝õ❛ ❤❛✐ ❤➔♠ ♥➔♦ ✤â✳ ✣➸ ❝â ♣❤➙♥ t➼❝❤ f t❤➔♥❤ t➼❝❤ ❝õ❛ ❤❛✐ ❤➔♠ t❤➻ f ❝❤➾ ❝â t❤➸ ❝â ♣❤➙♥ t➼❝❤ ❞↕♥❣ f = αf α, α = ✭t↕✐ s❛♦ ❄❄❄❄❄❄❄❄❄✮✳ ❑❤✐ ✤â t❛ ❝â ✶✹  2 b 0=  b f (x)dx =  a 2 f (x) αdx α a b b f (x) ( ) dx α ≤ a α2 dx a b =α2 (b − a) ( f (x) ) dx α a b f (x)dx =(b − a) a ✣➳♥ ✤➙②✱ ❝❤ó♥❣ t❛ ❝❤➥♥❣ ❣✐↔✐ q✉②➳t ✤÷đ❝ ✈➜♥ ✤➲ ❣➻✳ ◆➳✉ ♥❤÷ t❛ ❦❤ỉ♥❣ ♣❤➙♥ t➼❝❤ f = αf × α ♠➔ t➻♠ ởt t ỵ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ✭✷✳✼✮ ♥â✐ ❝❤✉♥❣ ❧➔ r➜t ❦❤â✳ ✣➳♥ t tỷ tợ ỵ s ỵ t ❝➛♥ t➻♠ ❤➔♠ g(x) ≥ 0, ∀x ∈ [a, b] ú ỵ r f : [a, b] [m, M ] ❉♦ ✤â✱ t❛ ❝â f (x) − m ≥ 0, M − f (x) ≥ 0, ∀x ∈ [a, b] ◆❤÷ ✈➟② ❤➔♠ g tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② ❝â t❤➸ ❧➔ f (x) − m ❤♦➦❝ M − f (x) ❤♦➦❝ ♠ët ❜✐➸✉ t❤ù❝ ♥➔♦ ✤â ❝õ❛ ♠ët ❤♦➦❝ ❝↔ ❤❛✐ t❤ø❛ sè tr➯♥✳ ▼➦t ❦❤→❝✱ t❛ ❧↕✐ ú ỵ tr õ t m M, ❝❤♦ ♥➯♥ t❛ ❞ü ✤♦→♥ g ❝➛♥ ①✉➜t ❤✐➺♥ ❝↔ ❤❛✐ t❤ø❛ sè f (x) − m ✈➔ M − f (x) ◆❤÷ ✈➟② t❛ ❞ü ✤♦→♥ ❤➔♠ g ❧➔ (f (x) − m)(M − f (x)) ❙❛✉ ✤â t tỷ ỵ t õ ❝❤ù♥❣ ♠✐♥❤ ♥❤÷ s❛✉✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t g(x) = (f (x) m)(M f (x)), ỵ ✷✳✻ t❛ ❝â b 0≤ ∀x ∈ [a, b] ❑❤✐ ✤â g(x) ≥ 0, ∀x ∈ [a, b] b (f (x) − m)(M − f (x))dx g(x)dx = a a b −f (x) + (n + M )f (x) − mM dx = a b b f (x)dx + =− a b a b f (x)dx − mM (b − a) =− a ❉♦ ✤â✱ t❛ ❝â ✭✷✳✼✮✳ −mM dx (n + M )f (x)dx + a ❑❤✐ ✤â ✶✺ ❱➼ ❞ö ✷✳✶✵✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✱ ♥➳✉ f ❧➔ ❝→❝ ❤➔♠ ①→❝ ✤à♥❤✱ ❞÷ì♥❣ ✈➔ ❦❤↔ t➼❝❤ tr➯♥ ✤♦↕♥ [a, b], t❤➻ b b (b − a) ≤ dx f (x) f (x)dx a a ❍ì♥ ♥ú❛✱ ♥➳✉ < m ≤ f (x) ≤ M, t❤➻ b b f (x)dx (m + M )2 dx ≤ (b − a)2 f (x) 4mM a a ❱➼ ❞ö ✷✳✶✶✳ ❈❤♦ f ∈ C ([a, b]), f (a) = f (b) = ✈➔ b f (x)dx = a ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ b xf (x)f (x)dx = − ✭✷✳✽✮ a ✈➔ b ≤ b x2 f (x)dx (f (x)) dx a a ữợ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ ❝❤♦ ✈➳ tr→✐ ❝õ❛ ✭✷✳✽✮ ✈➔ sû ❞ö♥❣ ❣✐↔ t❤✐➳t✳ P❤➛♥ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝✱ t ỵ f (x) ❧➔ ❤➔♠ f (x) ✈➔ ❤➔♠ g(x) ❧➔ ❤➔♠ xf (x) ❱➼ ❞ö ✷✳✶✷✳ ❚➻♠ (1 + x2 )f (x)dx f ∈A tr♦♥❣ ✤â A = {f ∈ C[0, 1] : f (x)dx = 1} ✈➔ t➻♠ ❤➔♠ f s❛♦ ❝❤♦ t❛ ♥❤➟♥ ✤÷đ❝ tr ọ t ữợ ỵ ✷✳✼ ✈ỵ✐ ❤➔♠ f (x) ❝❤➼♥❤ ❧➔ ❤➔♠ √1 + x2f (x) ✈➔ ❤➔♠ g(x) ❝❤➼♥❤ ❧➔ ❤➔♠ √1+x ❱➼ ❞ö ✷✳✶✸✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ f ❧➔ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ [0, 1], ❦❤↔ ✈✐ tr➯♥ (0, 1), f (0) = ✈➔ < f (x) ≤ tr➯♥ (0, 1) t❤➻  2 f (x)dx f (x)dx ≥  0 ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f (x) = x ❱➼ ❞ö ✷✳✶✹✳ ❚➻♠ f (x)dx A := f ∈A ✶✻ tr♦♥❣ ✤â A = {f ∈ R[0, 1] : f (x)dx = 3, xf (x)dx = 2} ✈➔ t➻♠ ❤➔♠ f s❛♦ ❝❤♦ t❛ ♥❤➟♥ ✤÷đ❝ ❣✐→ trà ♥❤ä ♥❤➜t✳ P❤➙♥ t➼❝❤✿ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ❞♦ ✈➲ ❝ì ❜↔♥ t❛ ❦❤ỉ♥❣ ❝â t❤ỉ♥❣ t✐♥ ❣➻ ✈➲ ❤➔♠ f ❧ỵ♥ ❤ì♥ ❤❛② ❜➨ ❤ì♥ ❤➔♠ ❤♦➦❝ số ỵ ♥â✐ ❝❤✉♥❣ s➩ r➜t ❦❤â✳ ❚r÷í♥❣ ❤đ♣ ♥➔② t❛ t❤û ỵ ổ tữớ õ t❤➸ ❝â ♠ët sè ❜↕♥ s✐♥❤ ✈✐➯♥ s➩ ❧➔♠ ♥❤÷ s❛✉✳ ❚❛ ❝â  2 9=  f (x)dx =  2 f (x) 1dx 1 ≤ 12 dx f (x)dx 0 f (x)dx = ❉♦ ✤â A = ú ỵ r s ổ ú ỵ ữ s ổ t❤➸ t➻♠ ✤÷đ❝ ❤➔♠ f ♥➔♦ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✤➛✉ ❜➔✐ ✤➸ ❝❤♦ A = ❱➻ ♥➳✉ ❦❤æ♥❣ t❤➻ ❦❤✐ ✤â t❛ ♣❤↔✐ ❝â    f (x) = α,     1 f (x)dx = 3,        xf (x)dx = 2, ✤✐➲✉ ♥➔② ❧➔ ✈æ ỵ ố ợ t t tr ợ t ❤❛② ♥❤ä ♥❤➜t✱ ❝❤ó♥❣ t❛ ❝➛♥ ♣❤↔✐ t➻♠ ✤÷đ❝ ❤➔♠ s❛♦ ❝❤♦ t❛ t❤➟t sü ♥❤➟♥ ✤÷đ❝ ❣✐→ trà ♥❤ä ♥❤➜t ❤♦➦❝ ❧ỵ♥ ♥❤➜t✳ ✣è✐ ✈ỵ✐ ❜➔✐ t♦→♥ tr➯♥✱ t❛ ữ s ự ỵ t❛ ❝â ✈ỵ✐ ♠å✐ b ∈ R t❤➻  2 f (x)dx ≥ (x + b)f (x)dx ≤ (2 + 3b) ≤  ❉♦ ✤â 3(2 + 3b)2 , 3b2 + 3b + ∀b ∈ R ❱➻ ✈➟② f (x)dx ≥ max b∈R f (x)dx (x + b) dx 3(2 + 3b)2 = 12 3b2 + 3b + ✶✼ ✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐   3(2+3b)2  max 3b  +3b+1 = 12,   b∈R      f (x) = α(x + b),   f (x)dx = 3,          xf (x)dx = 2, ●✐↔✐ ❤➺ tr➯♥ t❛ t➻♠ ✤÷đ❝ f (x) = 6x ❱➼ ❞ö ✷✳✶✺✳ ❚➻♠ (f (x))2 dx f ∈A tr♦♥❣ ✤â A = {f ∈ C [0, 1] : f (0) = f (1) = 0, f (0) = a} ✈➔ t➻♠ ❤➔♠ f s t ữủ tr ọ t ữợ ỵ ợ f (x) = x ỏ g(x) = f ỵ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✣è✐ ✈ỵ✐ ❞↕♥❣ ❜➔✐ t➟♣ ♥➔②✱ ú t ú ỵ tợ ỵ s ỵ ỵ sỷ r f (x) ❧✐➯♥ tö❝ tr♦♥❣ [a, b], ✭✐✐✮ f (a)f (b) < õ tỗ t c (a, b) s f (c) = ỵ ỵ sỷ r f (x) ❧✐➯♥ tö❝ tr♦♥❣ [a, b], ✭✐✐✮ ❤➔♠ f (x) ❦❤↔ ✈✐ tr♦♥❣ (a, b) ✭✐✐✐✮ f (a) = f (b) õ tỗ t c (a, b) s f (c) = ỵ ỵ r ●✐↔ sû r➡♥❣ (x) ✶✽ ✭✐✮ ❤➔♠ f (x) ❧✐➯♥ tö❝ tr♦♥❣ [a, b], ✭✐✐✮ ❍➔♠ f (x) ❦❤↔ ✈✐ tr (a, b) õ tỗ t c (a, b) s❛♦ ❝❤♦ f (b) − f (a) = f (c)(b a) ỵ ỵ ❤❛✐ ❤➔♠ sè f, g : [a, b] → R ●✐↔ sû r➡♥❣ ✭✐✮ ❍➔♠ f (x), g(x) ❧✐➯♥ tö❝ tr♦♥❣ [a, b] ✭✐✐✮ ❍➔♠ f (x), g(x) ❦❤↔ ✈✐ tr♦♥❣ (a, b) ✭✐✐✐✮ g (x) = ∀x ∈ (a, b) õ tỗ t c (a, b) s❛♦ ❝❤♦ f (b)−f (a) g(b)−g(a) = f (c) g (c) ◆❤➟♥ ①➨t ✷✳✷✵✳ ❈❤♦ f ❧➔ ❤➔♠ sè ❧✐➯♥ tö❝ tr➯♥ [a, b] ❑❤✐ ✤â ❤➔♠ sè F (x) = x ❧➔ ❤➔♠ sè ❦❤↔ ✈✐ tr➯♥ [a, b] ❚r♦♥❣ tr÷í♥❣ ❤đ♣ f ❝❤➾ ❣✐→♥ ✤♦↕♥ t↕✐ ♠ët sè ❤ú✉ ❤↕♥ ✤✐➸♠ ✈➔ ❜à ❝❤➦♥ t❤➻ ❤➔♠ F (x) ❧➔ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ [a, b] f (t)dt a ❱➼ ❞ö ✷✳✷✶✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ f : [a, b] → R ❧➔ ❤➔♠ sè ❦❤↔ t➼❝❤✱ ❦❤✐ õ tỗ t [a, b] s b f (t)dt f (t)dt = a θ P❤➙♥ t➼❝❤ ổ tữớ ố ợ t ú t sỷ ỵ r ú t tỷ ỵ t ỵ t t❛ ❝➛♥ ①➙② ❞ü♥❣ ❤➔♠ F ❦❤↔ ✈✐✳ ▼ët tr♦♥❣ sè ❝→❝ ❝→❝❤ ✤➸ t↕♦ r❛ ❤➔♠ ❦❤↔ ✈✐ tø ❤➔♠ f ❜❛♥ x ✤➛✉ t❛ ❝â t❤➸ ♥❣❤➽ tỵ✐ ❧➔ ①➨t ❤➔♠ F (x) = f (t)dt, t✉② ♥❤✐➯♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② ❤➔♠ f ❝❤➾ ❦❤↔ t➼❝❤✱ a ♥➯♥ ❤➔♠ F ❝❤➾ ❧✐➯♥ tö❝✳ ❉♦ ✤â ✤➸ →♣ ỵ trỹ t F ✤✐➲✉ ❦❤ỉ♥❣ t❤➸✳ ◆➳✉ ✈➝♥ t✐➳♣ tư❝ t❤❡♦ ❝→❝❤ ♥➔②✱ t❛ ❝â t❤➸ ①➙② ❞ü♥❣ ❤➔♠ ❦❤→❝✱ ❤♦➦❝ ❝â t❤➸ tø ❤➔♠ F t❛ ♥❤➙♥ ✈ỵ✐ ♠ët ❤➔♠ sè ❦❤→❝ s❛♦ ❝❤♦ ❤➔♠ sè ✈ø❛ t↕♦ t❤➔♥❤ ❧➔ ❤➔♠ ❦❤↔ ✈✐✳ ❚✉② ♥❤✐➯♥✱ t❤❡♦ ❝→❝❤ ♥➔② ❝â t❤➸ ❦❤â ❦❤➠♥ tr♦♥❣ ✈✐➺❝ t➻♠ ♠ët ❤➔♠ ♥❤÷ ✈➟②✳ ❚❛ t❤û t➻♠ s tỷ sỷ ỵ ✷✳✶✻ ✳ ❚❛ ❜✐➳♥ ✤ê✐ b b θ f (t)dt − f (t)dt = a θ ❚ø ❣✐↔ t❤✐➳t✱ t❛ s✉② r❛ f (t)dt a b θ f (t)dt = a f (t)dt a ❉♦ ✤â✱ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ tr t ỗ t (a, b) s ❝❤♦ b θ f (t)dt = a f (t)dt a ✶✾ ❚❤❛② θ ❜ð✐ x, ✈➔ t❛ ①➨t ❤➔♠ b h(x) = 2F (x) − f (t)dt, a tr♦♥❣ ✤â x F (x) = ✭✷✳✾✮ f (t)dt a ❚❛ ú ỵ r b h(a)h(b) = 2F (a) − f (t)dt 2F (b) − a  = 0 − b a a f (t)dt a 2 b =−  b f (t)dt − f (t)dt 2  f (t)dt a  b  b f (t)dt ≤ a ữ ỵ õ t❤➸ →♣ ❞ư♥❣ ✤÷đ❝✳ ❚ø ✤â✱ t❛ ❝â ❝→❝❤ ❣✐↔✐ ♥❤÷ s❛✉✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t b h(x) = 2F (x) − f (t)dt a tr♦♥❣ ✤â x F (x) = x ∈ [a, b] f (t)dt, a ❑✐➸♠ tr❛ ✤÷đ❝ h ❧➔ ❤➔♠ sè ❧✐➯♥ tö❝ tr➯♥ [a, b] ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ b −h(a) = h(b) = f (t)dt a ❉♦ ✤â✱  2 b h(a)h(b) = − f (t)dt a ỵ tỗ t (a, b) s h() = 0, tù❝ ❧➔ θ b f (t)dt = a f (t)dt θ ✷✵ ❱➼ ❞ö ✷✳✷✷✳ ❈❤♦ f : [a, b] → R ❧➔ ❤➔♠ sè ❧✐➯♥ tö❝✱ ✈➔ sỷ r tỗ t (a, b) s ❝❤♦ b f (t)dt = a ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✱ θ f (t)dt = f (θ) a P❤➙♥ t➼❝❤✳ ❚÷ì♥❣ tỹ tr t tỷ ỵ ❤❛② ▲❛❣r❛♥❣❡ ✤➸ ❝❤ù♥❣ ♠✐♥❤✳ ❈ơ♥❣ ✈ỵ✐ ❤➔♠ F ①→❝ ✤à♥❤ ♥❤÷ tr♦♥❣ ✭✷✳✾✮ t❤➻ t❛ ❝â F (x) = f (x) ✈➔ F (a) = F (b) ❞♦ ✤â ỵ ữủ tọ trỹ t ỵ t t ữủ tỗ t (a, b) s❛♦ ❝❤♦ F (θ) = 0, tr♦♥❣ ❦❤✐ ❝→✐ t tỗ t (a, b) s ❝❤♦ F (θ) = F (θ) ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② t❛ ❝➛♥ t↕♦ r❛ ♠ët ❤➔♠ ❦❤→❝ ❤➔♠ F (x) ✈➔ t➜t ♥❤✐➯♥ ❝â t❤➸ ❞ü❛ tr➯♥ ❤➔♠ F (x) õ t ữủ ỵ t t❤û ♥❤➙♥ ❤➔♠ F (x) − F (x) ✈ỵ✐ ♠ët h(x) ữỡ s tỗ t G(x) s ❝❤♦ G (x) = (F (x) − F (x)) h(x) ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ t❛ ❝â t❤➸ t➻♠ ✤÷đ❝ ❤➔♠ h(x) = e−x , x ∈ [a, b] ❚❛ ❝â ❝→❝❤ ❣✐↔✐ ♥❤÷ s❛✉✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t G(x) = F (x)e−x , tr♦♥❣ ✤â F (x) ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ tr♦♥❣ ✭✷✳✾✮✳ ❑❤✐ ✤â tø ❣✐↔ t❤✐➳t t❛ ❝â ✶✳ G(x) ❧➔ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ [a, b], ✷✳ G(x) ❧➔ ❤➔♠ ❦❤↔ ✈✐ tr➯♥ (a, b) ✸✳ G(a) = G(b) = ỵ tỗ t θ ∈ (a, b) s❛♦ ❝❤♦ G (θ) = ▼➦t ❦❤→❝ G (x) = (F (x) − F (x)) e−x ❉♦ ✤â F (θ) − F (θ) ❤❛② θ f (t)dt = f (θ) a ❱➼ ❞ö ✷✳✷✸✳ ❈❤♦ f : [a, b] → R ❧➔ ❤➔♠ sè ❧✐➯♥ tö❝✱ a > ✈➔ ❣✐↔ sû r➡♥❣ r➡♥❣✱ tỗ t (a, b) s f (t)dt = θf (θ) a ❈❤ù♥❣ ♠✐♥❤✳ ❈❤å♥ ❤➔♠ F (x) = x1 x f (t)dt a b f (t)dt = a ❈❤ù♥❣ ♠✐♥❤ ✷✶ ❱➼ ❞ö ✷✳✷✹✳ ❈❤♦ f, g : [a, b] → R ❧➔ ❤❛✐ ❤➔♠ số tửự r tỗ t (a, b) s❛♦ ❝❤♦ b b a a ❈❤ù♥❣ ♠✐♥❤✳ ❈❤å♥ ❤➔♠ F (x) = g(t)dt f (t)dt = f (θ) g(θ) x f (t)dt a ✈➔ G(x) = x g(t)dt a ỵ f, g : [a, b] → R ❧➔ ❤❛✐ ❤➔♠ số tửự r tỗ t (a, b) s❛♦ ❝❤♦ b θ f (t)dt = f (θ) g(θ) a ❈❤ù♥❣ ♠✐♥❤✳ ❈❤å♥ ❤➔♠ F (x) = θ b x g(t)dt f (t)dt x a g(t)dt ⑩♣ ❞ö♥❣ ỵ f, g : [a, b] → R ❧➔ ❤❛✐ ❤➔♠ sè ❞÷ì♥❣✱ ❧✐➯♥ tửự r tỗ t (a, b) s ❝❤♦ f (θ) − θ g(θ) g(t)dt f (t)dt a ❈❤ù♥❣ ♠✐♥❤✳ ❈❤å♥ ❤➔♠ F (x) = e−x = b θ x b f (t)dt g(t)dt a x ⑩♣ ỵ ởt số t ✶✳ ❈❤♦ f : [0, 1] → [−1, 1] ❦❤↔ ✈✐ ❧✐➯♥ tö❝ tr➯♥ [0, 1] ✈➔ f (0) = f (1) = ự r tỗ t c ∈ (0, 1) s❛♦ ❝❤♦ f (x) = 2c tan f (c) ✷✳ ❈❤♦ ❤➔♠ sè ❧✐➯♥ tö❝ tr➯♥ (2013, 2015) s❛♦ ❝❤♦ f [2013, 2015] ✈➔ 2015 f (t)dt = 2013 ự r tỗ t c c 2014 f (t)dt = cf (c) 2013 1 ✸✳ ❈❤♦ f : [0, 1] → R ❧✐➯♥ tö❝ s❛♦ ❝❤♦ f (t)dt = tf (t)dt ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ tỗ t c (0, 1) s 0 c f (c) = 2014 f (t)dt b ✹✳ ❈❤♦ ❤➔♠ f ❧✐➯♥ tö❝ tr➯♥ [a, b] ✈➔ ❦❤↔ ✈✐ tr➯♥ (a, b) ✈ỵ✐ a > ✈➔ f (t)dt = 0.ự r a tỗ t c (a, b) s❛♦ ❝❤♦ c c f (t)dt − 2013cf (c) + 2012 2014c a f (t)dt = a ✷✷ ✺✳ ❈❤♦ ❤➔♠ sè f : [−1, 1] → R ❧✐➯♥ tư❝✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ xf 2014 (x) − 2014f (x) = −2013x ❝â ♥❣❤✐➺♠ tr➯♥ ✤♦↕♥ [1, 1] t st ỵ I ❧➔ ❦❤♦↔♥❣ ♠ð ❦❤→❝ tr♦♥❣ R, ❣✐↔ sû x ✈➔ f, g : I → R ❧➔ ❤❛✐ ❤➔♠ ∀x ∈ I ●✐↔ sû r➡♥❣ ♠ët tr♦♥❣ ❤❛✐ ✤✐➲✉ s ữủ tọ ợ g ỡ t➠♥❣ ✈➔ g (x) = ♠➣♥ ✶✳ f (x), g(x) → ✷✳ g(x) → ±∞ ∈ R ∪ {±∞} ❦❤✐ x → x0 , ❦❤✐ x → x0 ❑❤✐ ✤â✱ ♥➳✉ ❣✐ỵ✐ ❤↕♥ lim x→x0 t❤➻ t❛ ❝â f (x) = l ∈ R ∪ {±∞} g (x) f (x) = l x→x0 g(x) lim ❱➼ ❞ö ✷✳✷✽✳ ❈❤♦ ❤➔♠ f ❧✐➯♥ tö❝ tr➯♥ (0, +∞) ✈➔ lim f (x) = 2016 x→+∞ ❚➼♥❤ x lim x→+∞ x f (t)dt ❈❤ù♥❣ ♠✐♥❤✳ ✣➸ →♣ ❞ö♥❣ q✉② t➢❝ ▲✬❍♦s♣✐t❛❧✱ t❛ ❝➛♥ ❝â ♣❤➨♣ ❝❤✐❛ ♠ët ❤➔♠ ❝❤♦ ♠ët ❤➔♠✳ ❚r♦♥❣ x t❤÷í♥❣ ❤đ♣ ♥➔②✱ t❛ ❝â t❤➸ t❤➜② ✤÷đ❝ ♠ët ❤➔♠ ❧➔ f (t)dt ✈➔ ❤➔♠ ỏ x ỵ ❤➔♠ f (x) ð tr➯♥ ❝❤➼♥❤ ❧➔ ❤➔♠ x f (t)dt, ỏ g(x) ỵ tr ❝❤➼♥❤ ❧➔ ❤➔♠ x ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ỵ ữủ tọ ỡ ỳ ợ G(x) = x, F (x) = F (x) f (x) = = f (x) G (x) ❉♦ ✤â✱ t❤❡♦ tt ỵ t õ x lim x→+∞ x f (t)dt = lim f (x) = 2016 x→+∞ ❱➼ ❞ö ✷✳✷✾✳ ❈❤♦ f : (0, +∞) → R ❧➔ ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ❝➜♣ ✷ s❛♦ ❝❤♦ |f (x) + 2xf (x) + (x2 + 1)f (x)| ≤ 1, ∀x ∈ R x f (t)dt t❤➻ ✷✸ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ lim f (x) = x→+∞ ❱➼ ❞ö ✷✳✸✵✳ ❱➼ ❞ö ✷✳✸✶✳