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LUẬN VĂN THẠC SĨ TOÁN HỌC

Người hướng dẫn khoa học

❚❙✳ ▲➊ ❱❿◆ ❉Ô◆●

✣➔ ◆➤♥❣ ✲ ✷✵✷✵

◆❤➙♥ ❞à♣ ♥➔②✱ t→❝ ❣✐↔ ①✐♥ ❜➔② tä sü ❦➼♥❤ trå♥❣ ✈➔ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥

❚❤➛②✳ ❚→❝ ❣✐↔ ❝ô♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ q✉þ ❇❛♥ ❧➣♥❤ ✤↕♦ ❚r÷í♥❣ ✣↕✐

❤å❝ ❙÷ ♣❤↕♠ ✣➔ ◆➤♥❣✱ P❤á♥❣ ✣➔♦ t↕♦ ❙❛✉ ✣↕✐ ❤å❝✱ ❑❤♦❛ ❚♦→♥✱ ❝ò♥❣ q✉þt❤➛② ❝æ ❣✐→♦ ❣✐↔♥❣ ❞↕② ❧î♣ ❝❛♦ ❤å❝ ❑✸✻ t↕✐ ◗✉↔♥❣ ◆❣➣✐ ✤➣ ❞➔② ❝æ♥❣ ❣✐↔♥❣

❞↕②✱ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ❝❤♦ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝

❤✐➺♥ ✤➲ t➔✐✳

◆❤➙♥ ✤➙②✱ t→❝ ❣✐↔ ❝ô♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ sü ❤é trñ ✈➲ ♠➦t t✐♥❤t❤➛♥ ❝õ❛ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ ❧✉æ♥ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï ✤➸ t→❝ ❣✐↔

❤♦➔♥ t❤➔♥❤ tèt ❦❤â❛ ❤å❝ ✈➔ ❧✉➟♥ ✈➠♥ ♥➔②✳

▼➦❝ ❞ò ❧✉➟♥ ✈➠♥ ✤÷ñ❝ t❤ü❝ ❤✐➺♥ ✈î✐ sü ♥é ❧ü❝ ❝è ❣➢♥❣ ❤➳t sù❝ ❝õ❛ ❜↔♥t❤➙♥✱ ♥❤÷♥❣ ❞♦ ✤✐➲✉ ❦✐➺♥ t❤í✐ ❣✐❛♥ ❝â ❤↕♥✱ tr➻♥❤ ✤ë ❦✐➳♥ t❤ù❝ ✈➔ ❦✐♥❤

♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤â tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉sât✳ ❚→❝ ❣✐↔ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷ñ❝ ♥❤ú♥❣ ❣â♣ þ ❝õ❛ q✉þ t❤➛② ❝æ ❣✐→♦ ✤➸

❧✉➟♥ ✈➠♥ ✤÷ñ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳

✭✐✐✮ ◆➳✉ tç♥ t↕✐ sè h > 0 s❛♦ ❝❤♦ f (x) ≥ f(x0) ✈î✐ ♠å✐ x t❤✉ë❝ ❦❤♦↔♥❣(x0 − h; x0 + h) t❤➻ t❛ ♥â✐ ❤➔♠ sè f (x) ✤↕t ❝ü❝ t✐➸✉ t↕✐ ✤✐➸♠ x0✳

✭✐✮ ◆➳✉ f′(x) > 0 ✈î✐ ♠å✐ x ∈ (a; x0) ✈➔ f′(x) < 0 ✈î✐ ♠å✐ x ∈ (x0; b)t❤➻ ❤➔♠ sè y = f (x) ✤↕t ❝ü❝ ✤↕✐ t↕✐ x0✳

✭✐✐✮ ◆➳✉ f′(x) < 0 ✈î✐ ♠å✐ x ∈ (a; x0) ✈➔ f′(x) > 0 ✈î✐ ♠å✐ x ∈ (x0; b)t❤➻ ❤➔♠ sè y = f (x) ✤↕t ❝ü❝ t✐➸✉ t↕✐ x0✳

✣à♥❤ ❧þ ✶✳✶✳✸ ✭❬✽❪✱ ✣à♥❤ ❧➼ ✸✮✳ ●✐↔ sû ❤➔♠ sè f ❝â ✤↕♦ ❤➔♠ tr➯♥ ❦❤♦↔♥❣(a; b) ❝❤ù❛ ✤✐➸♠ x0✱ f′(x0) = 0 ✈➔ f ❝â ✤↕♦ ❤➔♠ ❝➜♣ ❤❛✐ ❦❤→❝ ✵ t↕✐ ✤✐➸♠

❚ø ✤ç t❤à✱ t❛ t❤➜② ❤➔♠ sè f ✤↕t ❝ü❝ ✤↕✐ t↕✐ x = 0 ♥❤÷♥❣ ❦❤æ♥❣ ✤↕t ❣✐→trà ❧î♥ ♥❤➜t t↕✐ ✤✐➸♠ ✤â✳

✭✐✐✮ ◆❣✉②➯♥ ❧➼ ❲❡✐❡rstr❛ss✿ ◆➳✉ ❤➔♠ sè f ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a; b] t❤➻ ♥â

❧✉æ♥ ✤↕t ❣✐→ trà ❧î♥ ♥❤➜t ✈➔ ❣✐→ trà ♥❤ä ♥❤➜t tr➯♥ ✤♦↕♥ ✤â✳

• ❇÷î❝ ✶✿ ❚➼♥❤ f′(x)✳ ❚➻♠ ♥❤ú♥❣ ✤✐➸♠ xi ∈ (a; b), i = 1, 2, 3 ♠➔t↕✐ ✤â ✤↕♦ ❤➔♠ ❜➡♥❣ 0 ❤♦➦❝ ✤↕♦ ❤➔♠ ❦❤æ♥❣ ①→❝ ✤à♥❤✳

5 .

❇↔♥❣ ❜✐➳♥ t❤✐➯♥

❱➟② ❤➔♠ sè ✤↕t ❝ü❝ t✐➸✉ t↕✐ x = 2

√5

5 ✈➔ yCT = √

5✳

❱➼ ❞ö ✶✳✶✳✻✳ ❚➻♠ ♥❤ú♥❣ ✤✐➸♠ ❝ü❝ trà ❝õ❛ ❤➔♠ sè

f (x) = 3 − 2 cos x − cos 2x

a ∈ (−∞; −1), b ∈ (−1; 0), c ∈ (0; 1) ✈➔ d ∈ (1; +∞).

❚❛ ❝â ❜↔♥❣ ❜✐➳♥ t❤✐➯♥ ❝õ❛ ❤➔♠ sè g(x) = 4x2 + 4x ♥❤÷ s❛✉

❉ü❛ ✈➔♦ ❜↔♥❣ ❜✐➳♥ t❤✐➯♥ ❝õ❛ ❤➔♠ sè g(x)✱ t❛ t❤➜② ♣❤÷ì♥❣ tr➻♥❤ ✭✶✮ ✈æ

♥❣❤✐➺♠✱ ❝á♥ ♠é✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✮✱ ✭✸✮✱ ✭✹✮ ✤➲✉ ❝â ✷ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t✱ ❝→❝

♥❣❤✐➺♠ ♥➔② ✤➲✉ ❦❤→❝ ♥❤❛✉ ✈➔ ❦❤→❝ −1

2✳ ❉♦ ✤â ❤➔♠ sè y′ s➩ ✤ê✐ ❞➜✉ ❦❤✐ xq✉❛ 7 ♥❣❤✐➺♠ ✤â✳ ❱➟② ❤➔♠ sè ✤➣ ❝❤♦ ❝â 7 ✤✐➸♠ ❝ü❝ trà✳ ❚❛ ❝❤å♥ ♣❤÷ì♥❣

5 ✭✈➻ x ∈ [0; 1] ✮✳

❱➻ ❤➔♠ sè f (x) ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ ❬✵❀✶❪ ♥➯♥ ❧✉æ♥ tç♥ t↕✐ ❣✐→ trà ❧î♥ ♥❤➜ttr➯♥ ✤♦↕♥ ✤â✳ ❚❛ ❝â

f (0) = 0; f 2

√55

√

1 − t2 −

√10p

3 − 2√2t;

g′(t) = 0 ⇔ −

√2t

√

1 − t2 =

√10p

3 − 2√2t ✭✣✐➲✉ ❦✐➺♥ t ≤ 0✮✳

⇔ 2√2t3 − 8t2 + 5 = 0

⇔ t = −

√2

2 ✭❱➻ t ≤ 0✮✳

6 − x −

1

24

q(6 − x)3, x ∈ (0; 6).

2(x + y)

2

− 1

(x + y) − 3

1

❚❛ ❝â

f′(t) = 32t − 2

f′(t) = 0 ⇔ t = 1

16.

• ❇÷î❝ ✹✿ ❑➳t ❧✉➟♥ ❣✐→ trà ❧î♥ ♥❤➜t ✈➔ ❣✐→ trà ♥❤ä ♥❤➜t tr♦♥❣ ❝→❝ ❣✐→trà ð ❜÷î❝ ✸ ❝❤➼♥❤ ❧➔ ❣✐→ trà ❧î♥ ♥❤➜t ✈➔ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❤➔♠ sè ftr➯♥ t➟♣ D✳

2; 1

, M3

2; 1



✱ t❛ ❝â ∆

1

❱➟② ❤➔♠ sè ✤↕t ❝ü❝ t✐➸✉ t↕✐ ✤✐➸♠

1

▲➜② ✭✶✮ trø ✭✷✮ ✈➳ t❤❡♦ ✈➳✱ t❛ ✤÷ñ❝ x = y✳ ❙✉② r❛ ❤➺ ❝â ❜❛ ♥❣❤✐➺♠ ♥➯♥t❛ ✤÷ñ❝ ❜❛ ✤✐➸♠ ❞ø♥❣ M1(−1; −1), M2(1; 1), M3(0; 0)✳ ❚❛ ❝â

• ❚↕✐ M3(0; 0)✱ t❛ ❝â ∆(0; 0) = 0 ♥➯♥ ❦❤æ♥❣ t❤➸ ❦➳t ❧✉➟♥ ✤÷ñ❝✳ ❉♦ ✤ât❛ ♣❤↔✐ sû ❞ö♥❣ ✤à♥❤ ♥❣❤➽❛ ✤➸ ①→❝ ✤à♥❤ ✤✐➸♠ M3 ❝â ♣❤↔✐ ✤✐➸♠ ❝ü❝ trà ❤❛②

❦❤æ♥❣✳ ▲➜② ✤✐➸♠ M (x, y) t❤✉ë❝ ❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ (0; 0)✱ ❣✐↔ sû

x = y = 1

n, n ∈ N∗t❤➻

f (x, y) = 2

1n

4

− 4

1n

2

= 2 1

n2

1

●✐↔✐✳ ✣➦t f (x, y) = (4x2 + 3y)(4y2 + 3x) + 25xy✳

❚❛ ❝â D = {x ≥ 0, y ≥ 0, x + y = 1} ❧➔ ♠ët ♠✐➲♥ ✤â♥❣ ✈➔ ❜à ❝❤➦♥ ♥➯♥ s➩tç♥ t↕✐ ❣✐→ trà ❧î♥ ♥❤➜t ✈➔ ❣✐→ trà ♥❤ä ♥❤➜t tr➯♥ ♠✐➲♥ ✤â✳ ❑❤✐ ✤â f ✤↕t

1

2;

12

4 .

❑❤✐ ✤â f 2 +

√3

4 ;

2 −√34

!

= 191

16 .

✯ ❚↕✐ ❝→❝ ✤✐➸♠ ❜✐➯♥ x = 0 ⇒ y = 1✱ t❛ ❝â f (0; 1) = 12✳

✳

• ❚÷ì♥❣ tü✱ t↕✐ y = 0 t❤➻ 1 ≤ x ≤ 2 t❛ ❝ô♥❣ ✤÷ñ❝ ❦➳t q✉↔ ♥❤÷ tr➯♥✿max f (x, y) = −2 ✈➔ min f (x, y) = −9

4✳

12



✳

• ❚r➯♥ ❜✐➯♥ x + y = 3 t❤➻ 1 ≤ x ≤ 2✳ ❑❤✐ ✤â f (x, y) = x2 − 3x ❝â ❣✐→trà ❧î♥ ♥❤➜t ❜➡♥❣ −2 t↕✐ (x, y) = (1; 2) ❤♦➦❝ (x, y) = (2; 1) ✈➔ ❣✐→ trà ♥❤ä

♥❤➜t ❜➡♥❣ −9

4 t↕✐ (x, y) =

3

2;

32

❚÷ì♥❣ tü ♥❤÷ ❝ü❝ trà tü ❞♦✱ t❛ ❝â ✤à♥❤ ♥❣❤➽❛ ❝ü❝ trà ❝â ✤✐➲✉ ❦✐➺♥ ♥❤÷s❛✉✿

✈î✐ λ ✤÷ñ❝ ❣å✐ ❧➔ ♥❤➙♥ tû ▲❛❣r❛♥❣❡✳

✣à♥❤ ❧þ s❛✉ ✤➙② ❝❤♦ ❜✐➳t ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❤➔♠ sè ▲❛❣r❛♥❣❡ ✭✶✳✸✮ ✈➔ ❜➔✐t♦→♥ ❝â ❝ü❝ trà ✤❛♥❣ ①➨t✳

❑❤✐ ✤â t❛ t➼♥❤ ✤à♥❤ t❤ù❝✿

D =

0 ϕ1 ϕ2

ϕ1 L11 L12

ϕ2 L21 L22

... ●å✐ A ❧➔ ✈à tr➼ ❝õ❛ ✈✐➯♥ ❣↕❝❤ ❧ó❝ ❜➢t ✤➛✉ ❦❤↔♦ s→t✳ ❈❤å♥ ❤➺ trư❝ tå❛

✤ëOxy s❛♦ ❝❤♦ ❣è❝ O trị? ??❣ ✈ỵ✐ A tOx trũ ợ ữợ

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