Олон хувьсагчтай функцийн экстремум

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Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)

(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà

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Îëîí õóâüñàã÷òàé ôóíêö

Ä.Áàòò°ð

ÎËÎÍ ÓËÑÛÍ ÓËÀÀÍÁÀÀÒÀÐÛÍ ÈÕ ÑÓÐÃÓÓËÜ

2016.01.15

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1 Îëîí õóâüñàã÷òàé ôóíêö (ÎÕÔ)

(ÎÕÔ)-èéí Òåéëîðûí òîìú¼î

Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì

Ôóíêöèéí õàìãèéí èõ áà õàìãèéí áàãà óòãà

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Õî¼ð õóâüñàã÷òàé ôóíêöèéí Òåéëîðûí òîìú¼î

Íýã õóâüñàã÷òàé ôóíêöèéí àäèëààð z = f (x1; x2; ...; xn) ãýñýín õóâüñàã÷òàé ôóíêöèéã (x − a), (x − b)-èéí n-çýðãèéí îëîí

ãèø³³íò áà ÿìàð íýã ³ëäýãäýë ãèø³³íèé íèéëáýðò çàäëàæ

áîëíî.

Õýðýâ z = f (x ; y)-ôóíêöèéí n = 2 çýðãèéí Òåéëîðûí

òîìú¼î íü

f (x ; y) = A0 + B0(x − a) + C0(y − b)

2[A(x − a)2 + 2B(x − a)(y − b) + C (y − b)2 + R2] (1)

õýëáýðòýé áàéíà. �³íä A0,B0,C0,A,B,C -êîýôôèöèåíò íüx , y -ýýñ õàìààðàõã³é, ³ëäýãäýë ãèø³³í R2-íü íýã õóâüñàã÷òàé

ôóíêöèéí ³ëäýãäýë ãèø³³íèé á³òýöòýé àäèë áàéíà.

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Íýã õóâüñàã÷òàé ôóíêöèéí àäèëààð z = f (x1; x2; ...; xn) ãýñýín õóâüñàã÷òàé ôóíêöèéã (x − a), (x − b)-èéí n-çýðãèéí îëîí

ãèø³³íò áà ÿìàð íýã ³ëäýãäýë ãèø³³íèé íèéëáýðò çàäëàæ

áîëíî.

Õýðýâ z = f (x ; y)-ôóíêöèéí n = 2 çýðãèéí Òåéëîðûí

òîìú¼î íü

f (x ; y) = A0 + B0(x − a) + C0(y − b)

2[A(x − a)2 + 2B(x − a)(y − b) + C (y − b)2 + R2] (1)

õýëáýðòýé áàéíà. �³íä A0,B0,C0,A,B,C -êîýôôèöèåíò íüx , y -ýýñ õàìààðàõã³é, ³ëäýãäýë ãèø³³í R2-íü íýã õóâüñàã÷òàé

ôóíêöèéí ³ëäýãäýë ãèø³³íèé á³òýöòýé àäèë áàéíà.

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Òîäîðõîéëò

Áèä ÎÕÔ-èéí äýýä ýðýìáèéí óëàìæëàë àøèãëàí ì°í

R2 = α0∆ρ3 ãýæ òýìäýãëýâýë

f (x ; y) = f (a; b) + f ′x(a; b)∆x + f ′y (a; b)∆y

[f ′′xx(a; b)∆x2 + 2f ′′xy (a; b)∆x∆y + f ′′yy (a; b)∆y2

]+α0∆ρ3 (2)

áàéíà. (2) òîìú¼îã õî¼ð õóâüñàã÷òàé ôóíêöèéí Òåéëîðûí

òîìú¼î ãýíý.

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Æèøýý

f (x ; y) = 2x2 − xy − y2 − 6x − 3y + 5 ôóíêöèéã

A(1;−2) öýãèéí îð÷èíä Òåéëîðûí òîìú¼îãîîð çàäàë.

f (x ; y) = f (1;−2) + f ′x(1;−2)(x − 1) + f ′y (1;−2)(y + 2)+12

[f ′′xx(1;−2) · (x − 1)2 + 2f ′′xy (1;−2)(x − 1)(y + 2) +

f ′′yy (1;−2)(y + 2)2]

+ R2 õýëáýðòýé áàéíà.

f ′x(x ; y) = 4x − y − 6, f ′y (x ; y) = −x − 2y − 3,

f ′′xx(x ; y) = 4, f ′′xy (x ; y) = −1, f ′′yy (x ; y) = −2

áàéõ òóë 2-îîñ äýýø ýðýìáèéí á³õ òóõàéí óëàìæëàëóóä

òýãòýé òýíöýõ òóë ìàíàé òîõèîëäîëä R2 = 0 áàéíà.

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Æèøýý

f (x ; y) = f (1;−2) + f ′x(1;−2)(x − 1) + f ′y (1;−2)(y + 2)+12

[f ′′xx(1;−2) · (x − 1)2 + 2f ′′xy (1;−2)(x − 1)(y + 2) +

f ′′yy (1;−2)(y + 2)2]

f ′x(x ; y) = 4x − y − 6, f ′y (x ; y) = −x − 2y − 3,

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Æèøýý

f (x ; y) = f (1;−2) + f ′x(1;−2)(x − 1) + f ′y (1;−2)(y + 2)+12

[f ′′xx(1;−2) · (x − 1)2 + 2f ′′xy (1;−2)(x − 1)(y + 2) +

f ′′yy (1;−2)(y + 2)2]

f ′x(x ; y) = 4x − y − 6, f ′y (x ; y) = −x − 2y − 3,

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Æèøýý

f (1;−2) = 5, f ′x(1;−2) = 0, f ′y (1;−2) = 0,f ′′xx(1;−2) = 4, f ′′xy (1;−2) = −1, f ′′yy (1;−2) = −2 áàéõ òóë

ìàíàé ôóíêöèéí A(1;−2) öýãèéí îð÷èí äàõü Òåéëîðûí

òîìú¼î íü

f (x ; y) = 5 + 2(x − 1)2 − (x − 1)(y + 2)− (y + 2)2 .

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f (x ; y) = y x ôóíêöèéí n = 2 áàéõàä A(1; 2) öýãèéíîð÷èíä Òåéëîðûí òîìú¼îãîîð çàäàë.

I áà II-ýðýìáèéí òóõàéí óëàìæëàëóóäûã îëæ A(1; 2) öýãäýýð áîäîæ ãàðãàâàë

f (1; 2) = 2, f ′x(1; 2) = 2 ln 2, f ′y (1; 2) = 1, f ′′xx(1; 2) = 2 ln2 2,

f ′′xy (1; 2) = ln 2 + 1, f ′′yy (1; 2) = 0

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Æèøýý

f (x ; y) = y x ôóíêöèéí n = 2 áàéõàä A(1; 2) öýãèéíîð÷èíä Òåéëîðûí òîìú¼îãîîð çàäàë.

I áà II-ýðýìáèéí òóõàéí óëàìæëàëóóäûã îëæ A(1; 2) öýãäýýð áîäîæ ãàðãàâàë

f (1; 2) = 2, f ′x(1; 2) = 2 ln 2, f ′y (1; 2) = 1, f ′′xx(1; 2) = 2 ln2 2,

f ′′xy (1; 2) = ln 2 + 1, f ′′yy (1; 2) = 0

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Æèøýý

I áà II-ýðýìáèéí òóõàéí óëàìæëàëóóäûã îëæ A(1; 2) öýã äýýðáîäîæ ãàðãàâàë

f (1; 2) = 2, f ′x(1; 2) = 2 ln 2, f ′y (1; 2) = 1, f ′′xx(1; 2) = 2 ln2 2,

f ′′xy (1; 2) = ln 2 + 1, f ′′yy (1; 2) = 0

áîëîõ òóë

y x = 2 + 2(x − 1) ln 2 + (y − 2)

[2(x − 1)2 ln2 2 + (x − 1)(y − 2)(ln 2 + 1)

]+ R2 =

= (x − 1) ln 4 + y + (x − 1)2 ln2 2 + (x − 1)(y − 2)(ln 2 + 1) +R2

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Æèøýý

I áà II-ýðýìáèéí òóõàéí óëàìæëàëóóäûã îëæ A(1; 2) öýã äýýðáîäîæ ãàðãàâàë

f (1; 2) = 2, f ′x(1; 2) = 2 ln 2, f ′y (1; 2) = 1, f ′′xx(1; 2) = 2 ln2 2,

f ′′xy (1; 2) = ln 2 + 1, f ′′yy (1; 2) = 0

áîëîõ òóë

y x = 2 + 2(x − 1) ln 2 + (y − 2)

[2(x − 1)2 ln2 2 + (x − 1)(y − 2)(ln 2 + 1)

]+ R2 =

= (x − 1) ln 4 + y + (x − 1)2 ln2 2 + (x − 1)(y − 2)(ln 2 + 1) +R2

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Òîäîðõîéëò

Õýðýâ z = f (x ; y) ôóíêöèéí M0(x0; y0) öýã äýýðõ óòãà íü ýíýöýãèéí îð÷íû áóñàä á³õ M(x ; y) öýã³³ä äýýðõ óòãààñ èõ

(áàãà)

f (x0; y0) > f (x ; y) (f (x ; y) > f (x0; y0))

áàéâàë óóë ôóíêöèéã M0(x0; y0) öýã äýýð ìàêñèìóì

(ìèíèìóì)-òàé ãýíý. z = f (x ; y) ôóíêöèéí ìàêñèìóì,

ìèíèìóìûã õàìòàä íü ôóíêöèéí ýêñòðåìóì ãýíý.

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Æèøýý

z = 12 − sin(x2 + y2) ôóíêö O(0; 0) öýã äýýð

ìàêñèìóìòàé áà ìàêñèìóìûí óòãà íü

f (0; 0) =1

áàéíà.

x2 + y2 = π6 òîéðîã àâàõàä x 6= 0, y 6= 0 áàéõ ýíýõ³³

òîéðãèéí äîòîð îðøèõ 0 < x2 + y2 < π6 öýã á³õýí äýýð

sin(x2 + y2) > 0

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Æèøýý

z = 12 − sin(x2 + y2) ôóíêö O(0; 0) öýã äýýð

ìàêñèìóìòàé áà ìàêñèìóìûí óòãà íü

f (0; 0) =1

áàéíà.

x2 + y2 = π6 òîéðîã àâàõàä x 6= 0, y 6= 0 áàéõ ýíýõ³³

òîéðãèéí äîòîð îðøèõ 0 < x2 + y2 < π6 öýã á³õýí äýýð

sin(x2 + y2) > 0

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Æèøýý

f (x ; y) =1

2− sin(x2 + y2) <

áàéíà. �°ð°°ð õýëáýë 0 < x2 + y2 < π6 áàéõ á³õ M(x ; y) öýã

äýýð

f (x ; y) < f (0; 0) =1

áàéõ ó÷èð z = 12 − sin(x2 + y2) ôóíêö O(0; 0) öýã äýýð

ìàêñèìóìòàé.

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Æèøýý

yr√π6

z = 12 − sin(x2 + y2)

7-ð çóðàã

��

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Àãóóëãà

Õî¼ð õóâüñàã÷òàé ôóíêö ýêñòðåìóìòàé áàéõ çàéëøã³é

í°õöëèéã òîãòîî¼.

Òåîðåì (Ôóíêö ýêñòðåìóìòàé áàéõ çàéëøã³é í°õö°ë)

Õýðýâ z = f (x ; y) ôóíêö M0(x0; y0) (x = x0; y = y0) öýã äýýðýêñòðåìóìòàé áàéâàë ýíý ôóíêöèéí íýãä³ãýýð ýðýìáèéí

òóõàéí óëàìæëàëóóä íü òýãòýé òýíö³³ ýñâýë îðøèõã³é áàéíà.

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Àãóóëãà

Õî¼ð õóâüñàã÷òàé ôóíêö ýêñòðåìóìòàé áàéõ çàéëøã³é

í°õöëèéã òîãòîî¼.

Òåîðåì (Ôóíêö ýêñòðåìóìòàé áàéõ çàéëøã³é í°õö°ë)

Õýðýâ z = f (x ; y) ôóíêö M0(x0; y0) (x = x0; y = y0) öýã äýýðýêñòðåìóìòàé áàéâàë ýíý ôóíêöèéí íýãä³ãýýð ýðýìáèéí

òóõàéí óëàìæëàëóóä íü òýãòýé òýíö³³ ýñâýë îðøèõã³é áàéíà.

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Æèøýý

z = x2− y2 ôóíêöèéí õóâüä ∂z∂x = 2x ; ∂z

∂y = −2y áàéíà.

∂z∂x = 2x = 0 ãýäãýýñ x = 0, ∂z∂y = −2y = 0 ãýäãýýñ y = 0

áàéíà.Ãýòýë O(0; 0) öýã íü z = x2 − y2-ôóíêöèéíýêñòðåìóìèéí öýã áàéæ ÷àäàõã³é.

Ó÷èð íü zx=0 = 0 áàéõ á°ã°°ä O(0; 0) öýãèéí ÿìàð ÷

æèæèã îð÷èíä ∆z-íü ýåpýã áàéæ áîëíî, ñ°ð°ã ÷ áàéæ

áîëíî.

Öààøäàà áèä z = f (x ; y) ôóíêöèéí I-ýðýìáèéí á³õ

óëàìæëàëóóä òýãòýé òýíöýõ þìóó ýñâýë îðøèõã³é áàéõ

öýã³³äèéã ôóíêö ýêñòðåìóìòàé áàéõ ñýæèãòýé öýã³³ä

ãýæ íýðëýå.

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Æèøýý

áàéíà.

Ãýòýë O(0; 0) öýã íü z = x2 − y2-ôóíêöèéíýêñòðåìóìèéí öýã áàéæ ÷àäàõã³é.

áîëíî.

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Àãóóëãà

Æèøýý

áîëíî.

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Àãóóëãà

Æèøýý

áîëíî.

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Ä.Áàòò°ð

Àãóóëãà

Æèøýý

áîëíî.

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Àãóóëãà

Òåîðåì (Ôóíêö ýêñòðåìóìòàé áàéõ õ³ðýëöýýòýé í°õö°ë)

z = f (x ; y) ôóíêö M0(x0; y0)-ñýæèãòýé öýãèéã àãóóëñàí ìóæ

äýýð II-ýðýìáèéã äóóñòàë òàñðàëòã³é òóõàéí óëàìæëàëóóäòàé

áàéã.

�°ð°°ð õýëáýë f ′x(x0; y0) = 0, f ′y (x0; y0) = 0.

1)∂2f (x0; y0)

∂x2· ∂

2f (x0; y0)

∂y2−[∂2f (x0; y0)

∂x∂y

]2> 0;

∂2f (x0; y0)

∂x2< 0

áîë z = f (x ; y) ôóíêö M0(x0; y0) öýã äýýð ìàêñèìóìòàé.

2)∂2f (x0; y0)

∂x2· ∂

2f (x0; y0)

∂y2−[∂2f (x0; y0)

∂x∂y

]2> 0;

∂2f (x0; y0)

∂x2> 0

áîë z = f (x ; y) ôóíêö M0(x0; y0) öýã äýýð ìèíèìóìòàé.

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áàéã.

1)∂2f (x0; y0)

∂x2· ∂

2f (x0; y0)

∂y2−[∂2f (x0; y0)

∂x∂y

]2> 0;

∂2f (x0; y0)

∂x2< 0

2)∂2f (x0; y0)

∂x2· ∂

2f (x0; y0)

∂y2−[∂2f (x0; y0)

∂x∂y

]2> 0;

∂2f (x0; y0)

∂x2> 0

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áàéã.

1)∂2f (x0; y0)

∂x2· ∂

2f (x0; y0)

∂y2−[∂2f (x0; y0)

∂x∂y

]2> 0;

∂2f (x0; y0)

∂x2< 0

2)∂2f (x0; y0)

∂x2· ∂

2f (x0; y0)

∂y2−[∂2f (x0; y0)

∂x∂y

]2> 0;

∂2f (x0; y0)

∂x2> 0

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Àãóóëãà

áàéã.

3)∂2f (x0; y0)

∂x2· ∂

2f (x0; y0)

∂y2−[∂2f (x0; y0)

∂x∂y

áàéâàë z = f (x ; y) ôóíêö M0(x0; y0) öýã äýýð ýêñòðåìóì

áàéõã³é.

4)∂2f (x0; y0)

∂x2· ∂

2f (x0; y0)

∂y2−[∂2f (x0; y0)

∂x∂y

]2= 0 áîë z = f (x ; y)

ôóíêö M0(x0; y0) öýã äýýð ýêñòðåìóìòàé ýñýõ íü òîäîðõîéã³é.

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Àãóóëãà

áàéã.

3)∂2f (x0; y0)

∂x2· ∂

2f (x0; y0)

∂y2−[∂2f (x0; y0)

∂x∂y

áàéâàë z = f (x ; y) ôóíêö M0(x0; y0) öýã äýýð ýêñòðåìóì

áàéõã³é.

4)∂2f (x0; y0)

∂x2· ∂

2f (x0; y0)

∂y2−[∂2f (x0; y0)

∂x∂y

]2= 0 áîë z = f (x ; y)

ôóíêö M0(x0; y0) öýã äýýð ýêñòðåìóìòàé ýñýõ íü òîäîðõîéã³é.

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Æèøýý

z = x2 − xy + y2 + 3x − 2y + 1 ôóíêöèéí ýêñòðåìóèéã

Ýõëýýä ñýæèãòýé öýã³³äèéã îëú¼.

∂x= 2x − y + 3,

∂y= −x + 2y − 2

{2x − y + 3 = 02y − x − 2 = 0

ñèñòåìèéí øèéäèéã

îëáîëx = −43 ; y = 1

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Àãóóëãà

Æèøýý

∂x= 2x − y + 3,

∂y= −x + 2y − 2

{2x − y + 3 = 02y − x − 2 = 0

îëáîëx = −43 ; y = 1

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Àãóóëãà

Æèøýý

∂x= 2x − y + 3,

∂y= −x + 2y − 2

{2x − y + 3 = 02y − x − 2 = 0

îëáîë

x = −43 ; y = 1

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Ä.Áàòò°ð

Àãóóëãà

Æèøýý

∂x= 2x − y + 3,

∂y= −x + 2y − 2

{2x − y + 3 = 02y − x − 2 = 0

îëáîëx = −43 ; y = 1

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Àãóóëãà

Æèøýý

Õî¼ðäóãààð ýðýìáèéí òóõàéí óëàìæëàëóóäûã îëæ M(−43 ; 1

3)öýã äýýð áîäâîë

A =∂2z

∂x2= 2, B =

∂x∂y= −1, C =

∂y2= 2

áàéõ áà

AC − B2 = 2 · 2− (−1)2 = 3 > 0 A = 2 > 0 òóë

°ã°ãäñ°í ôóíêö M(−43 ; 1

3) öýã äýýð ìèíèìóìòàé,ìèíèìóìûí

óòãà

zmin = −4

áàéíà.

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Ä.Áàòò°ð

Àãóóëãà

Æèøýý

A =∂2z

∂x2= 2, B =

∂x∂y= −1, C =

∂y2= 2

áàéõ áàAC − B2 = 2 · 2− (−1)2 = 3 > 0 A = 2 > 0 òóë

3) öýã äýýð ìèíèìóìòàé,

ìèíèìóìûí

óòãà

zmin = −4

áàéíà.

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Ä.Áàòò°ð

Àãóóëãà

Æèøýý

A =∂2z

∂x2= 2, B =

∂x∂y= −1, C =

∂y2= 2

áàéõ áàAC − B2 = 2 · 2− (−1)2 = 3 > 0 A = 2 > 0 òóë

3) öýã äýýð ìèíèìóìòàé,ìèíèìóìûí

óòãà

zmin = −4

áàéíà.

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Àãóóëãà

Æèøýý

z = x3 + y3 − 3xy ôóíêöèéí ýêñòðåìóìûã îë.

∂x= 3x2 − 3y ,

∂y= 3y2 − 3x

{3x2 − 3y = 03y2 − 3x = 0

ñèñòåìèéí øèéäèéã îëáîë

O(0; 0), M1(1; 1) ãýñýí õî¼ð ñýæèãòýé öýã ãàðíà.

II-ýðýìáèéí òóõàéí óëàìæëàëóóäûã îëáîë

∂x2= 6x ,

∂x∂y= −3,

∂y2= 6y

áàéíà.

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Ä.Áàòò°ð

Àãóóëãà

Æèøýý

∂x= 3x2 − 3y ,

∂y= 3y2 − 3x

{3x2 − 3y = 03y2 − 3x = 0

∂x2= 6x ,

∂x∂y= −3,

∂y2= 6y

áàéíà.

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Ä.Áàòò°ð

Àãóóëãà

Æèøýý

∂x= 3x2 − 3y ,

∂y= 3y2 − 3x

{3x2 − 3y = 03y2 − 3x = 0

∂x2= 6x ,

∂x∂y= −3,

∂y2= 6y

áàéíà.

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Ä.Áàòò°ð

Àãóóëãà

Æèøýý

Îäîî ñýæèãòýé öýã³³äèéã øèíæèëüå.à) M1(1; 1) öýãèéí õóâüä

A =(∂2z∂x2

)x = 1y = 1

= 6, B =( ∂2z

∂x∂y

)x = 1y = 1

= −3

C =(∂2z∂y2

)x = 1y = 1

AC − B2 = 36− 9 = 27 > 0, A > 0

òóë M(1; 1) öýã äýýð °ã°ãäñ°í ôóíêö ìèíèìóìòàé áàéõ á°ã°°ä

ìèíèìóìûí óòãà íü

Zmin = −1

áàéíà.

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Ä.Áàòò°ð

Àãóóëãà

Æèøýý

A =(∂2z∂x2

)x = 1y = 1

= 6, B =( ∂2z

∂x∂y

)x = 1y = 1

= −3

C =(∂2z∂y2

)x = 1y = 1

AC − B2 = 36− 9 = 27 > 0, A > 0

Zmin = −1

áàéíà.

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Ä.Áàòò°ð

Àãóóëãà

Æèøýý

A =(∂2z∂x2

)x = 1y = 1

= 6, B =( ∂2z

∂x∂y

)x = 1y = 1

= −3

C =(∂2z∂y2

)x = 1y = 1

AC − B2 = 36− 9 = 27 > 0, A > 0

Zmin = −1

áàéíà.

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Àãóóëãà

Æèøýý

á) O(0; 0) öýãèéã øèíæèëüå.

A =(∂2z∂x2

)x = 0y = 0

= 0, B =( ∂2z

∂x∂y

)x = 0y = 0

= −3

C =(∂2z∂y2

)x = 0y = 0

AC − B2 = 0− (−3)2 = −9 < 0 òóë °ã°ãäñ°í ôóíêö O(0; 0)öýã äýýð ýêñòðåìóì áàéõã³é.

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Àãóóëãà

Æèøýý

á) O(0; 0) öýãèéã øèíæèëüå.

A =(∂2z∂x2

)x = 0y = 0

= 0, B =( ∂2z

∂x∂y

)x = 0y = 0

= −3

C =(∂2z∂y2

)x = 0y = 0

AC − B2 = 0− (−3)2 = −9 < 0 òóë °ã°ãäñ°í ôóíêö O(0; 0)öýã äýýð ýêñòðåìóì áàéõã³é.

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Àãóóëãà

Ôóíêöèéí õàìãèéí èõ áà õàìãèéí áàãà óòãà

Çààãëàãäñàí áèò³³ ìóæ äýýð òîäîðõîéëîãäñîí z = f (x ; y) òàñðàëòã³éôóíêö óóë ìóæ äýýðýý õàìãèéí èõ áà õàìãèéí áàãà óòãàà íààä çàõ íü

íýã öýã äýýð àâíà ãýñýí òàñðàëòã³é ôóíêöèéí ÷àíàð áàéäàã.

Çààãëàãäñàí áèò³³ ìóæ äýýð òîäîðõîéëîãäñîí òàñðàëòã³é ôóíêöèéí

ýíý ìóæ äýýðõ õàìãèéí èõ áà õàìãèéí áàãà óòãûã îëîõ áîäëîãî

ïðàêòèêò èõýýõýí òîõèîëääîã.

Çààãëàãäñàí áèò³³ ìóæ äýýð òîäîðõîéëîãäñîí òàñðàëòã³é ôóíêöèéí

õàìãèéí èõ áà õàìãèéí áàãà óòãûã îëîõäîî ýíý ìóæèéí äîòîîä öýã³³ä

äýýðõ ýêñòðåìóì óòãóóäûã îëæ òýäãýýðèéã ìóæèéí õèë äýýðõ óòãóóäòàé

æèøèæ òýäãýýðèéí äîòðîîñ õàìãèéí èõ íü ìóæ äýýðõ õàìãèéí èõ,

õàìãèéí áàãà íü ìóæ äýýðõ õàìãèéí áàãà óòãà íü áàéäàã.

Олон хувьсагчтай функцийн экстремум

Education

Transcript of Олон хувьсагчтай функцийн экстремум