Олон хувьсагчтай функцийн экстремум
TRANSCRIPT
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Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
ÌÀÒÅÌÀÒÈÊ-2
Îëîí õóâüñàã÷òàé ôóíêö
Ä.Áàòò°ð
ÎËÎÍ ÓËÑÛÍ ÓËÀÀÍÁÀÀÒÀÐÛÍ ÈÕ ÑÓÐÃÓÓËÜ
2016.01.15
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Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
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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Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
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)
+1
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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Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
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)
+1
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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Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Õî¼ð õóâüñàã÷òàé ôóíêöèéí Òåéëîðûí òîìú¼î
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
+1
2
[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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Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Õî¼ð õóâüñàã÷òàé ôóíêöèéí Òåéëîðûí òîìú¼î
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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Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Õî¼ð õóâüñàã÷òàé ôóíêöèéí Òåéëîðûí òîìú¼î
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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Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Õî¼ð õóâüñàã÷òàé ôóíêöèéí Òåéëîðûí òîìú¼î
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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Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Õî¼ð õóâüñàã÷òàé ôóíêöèéí Òåéëîðûí òîìú¼î
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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Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Õî¼ð õóâüñàã÷òàé ôóíêöèéí Òåéëîðûí òîìú¼î
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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Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Õî¼ð õóâüñàã÷òàé ôóíêöèéí Òåéëîðûí òîìú¼î
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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Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Õî¼ð õóâüñàã÷òàé ôóíêöèéí Òåéëîðûí òîìú¼î
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)
+1
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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Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Õî¼ð õóâüñàã÷òàé ôóíêöèéí Òåéëîðûí òîìú¼î
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)
+1
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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Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
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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Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
Æèøýý
z = 12 − sin(x2 + y2) ôóíêö O(0; 0) öýã äýýð
ìàêñèìóìòàé áà ìàêñèìóìûí óòãà íü
f (0; 0) =1
2
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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Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
Æèøýý
z = 12 − sin(x2 + y2) ôóíêö O(0; 0) öýã äýýð
ìàêñèìóìòàé áà ìàêñèìóìûí óòãà íü
f (0; 0) =1
2
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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Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
Æèøýý
f (x ; y) =1
2− sin(x2 + y2) <
1
2
áàéíà. �°ð°°ð õýëáýë 0 < x2 + y2 < π6 áàéõ á³õ M(x ; y) öýã
äýýð
f (x ; y) < f (0; 0) =1
2
áàéõ ó÷èð z = 12 − sin(x2 + y2) ôóíêö O(0; 0) öýã äýýð
ìàêñèìóìòàé.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
Æèøýý
0
z
x
yr√π6
z = 12 − sin(x2 + y2)
7-ð çóðàã
-
6
��
���
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
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.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
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.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
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.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
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.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
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.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
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.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
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.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
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.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
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.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
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.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
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.
3)∂2f (x0; y0)
∂x2· ∂
2f (x0; y0)
∂y2−[∂2f (x0; y0)
∂x∂y
]2< 0
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.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
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.
3)∂2f (x0; y0)
∂x2· ∂
2f (x0; y0)
∂y2−[∂2f (x0; y0)
∂x∂y
]2< 0
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.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
Æèøýý
z = x2 − xy + y2 + 3x − 2y + 1 ôóíêöèéí ýêñòðåìóèéã
îë.
Ýõëýýä ñýæèãòýé öýã³³äèéã îëú¼.
∂z
∂x= 2x − y + 3,
∂z
∂y= −x + 2y − 2
{2x − y + 3 = 02y − x − 2 = 0
ñèñòåìèéí øèéäèéã
îëáîëx = −43 ; y = 1
3 .
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
Æèøýý
z = x2 − xy + y2 + 3x − 2y + 1 ôóíêöèéí ýêñòðåìóèéã
îë.
Ýõëýýä ñýæèãòýé öýã³³äèéã îëú¼.
∂z
∂x= 2x − y + 3,
∂z
∂y= −x + 2y − 2
{2x − y + 3 = 02y − x − 2 = 0
ñèñòåìèéí øèéäèéã
îëáîëx = −43 ; y = 1
3 .
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
Æèøýý
z = x2 − xy + y2 + 3x − 2y + 1 ôóíêöèéí ýêñòðåìóèéã
îë.
Ýõëýýä ñýæèãòýé öýã³³äèéã îëú¼.
∂z
∂x= 2x − y + 3,
∂z
∂y= −x + 2y − 2
{2x − y + 3 = 02y − x − 2 = 0
ñèñòåìèéí øèéäèéã
îëáîë
x = −43 ; y = 1
3 .
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
Æèøýý
z = x2 − xy + y2 + 3x − 2y + 1 ôóíêöèéí ýêñòðåìóèéã
îë.
Ýõëýýä ñýæèãòýé öýã³³äèéã îëú¼.
∂z
∂x= 2x − y + 3,
∂z
∂y= −x + 2y − 2
{2x − y + 3 = 02y − x − 2 = 0
ñèñòåìèéí øèéäèéã
îëáîëx = −43 ; y = 1
3 .
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
Æèøýý
Õî¼ðäóãààð ýðýìáèéí òóõàéí óëàìæëàëóóäûã îëæ M(−43 ; 1
3)öýã äýýð áîäâîë
A =∂2z
∂x2= 2, B =
∂2z
∂x∂y= −1, C =
∂2z
∂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
3
áàéíà.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
Æèøýý
Õî¼ðäóãààð ýðýìáèéí òóõàéí óëàìæëàëóóäûã îëæ M(−43 ; 1
3)öýã äýýð áîäâîë
A =∂2z
∂x2= 2, B =
∂2z
∂x∂y= −1, C =
∂2z
∂y2= 2
áàéõ áàAC − 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
3
áàéíà.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
Æèøýý
Õî¼ðäóãààð ýðýìáèéí òóõàéí óëàìæëàëóóäûã îëæ M(−43 ; 1
3)öýã äýýð áîäâîë
A =∂2z
∂x2= 2, B =
∂2z
∂x∂y= −1, C =
∂2z
∂y2= 2
áàéõ áàAC − 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
3
áàéíà.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
Æèøýý
z = x3 + y3 − 3xy ôóíêöèéí ýêñòðåìóìûã îë.
Ýõëýýä ñýæèãòýé öýã³³äèéã îëú¼.
∂z
∂x= 3x2 − 3y ,
∂z
∂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
∂2z
∂x2= 6x ,
∂2z
∂x∂y= −3,
∂2z
∂y2= 6y
áàéíà.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
Æèøýý
z = x3 + y3 − 3xy ôóíêöèéí ýêñòðåìóìûã îë.
Ýõëýýä ñýæèãòýé öýã³³äèéã îëú¼.
∂z
∂x= 3x2 − 3y ,
∂z
∂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
∂2z
∂x2= 6x ,
∂2z
∂x∂y= −3,
∂2z
∂y2= 6y
áàéíà.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
Æèøýý
z = x3 + y3 − 3xy ôóíêöèéí ýêñòðåìóìûã îë.
Ýõëýýä ñýæèãòýé öýã³³äèéã îëú¼.
∂z
∂x= 3x2 − 3y ,
∂z
∂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
∂2z
∂x2= 6x ,
∂2z
∂x∂y= −3,
∂2z
∂y2= 6y
áàéíà.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
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
= 6
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.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
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
= 6
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.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
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
= 6
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.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
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
= 0
AC − B2 = 0− (−3)2 = −9 < 0 òóë °ã°ãäñ°í ôóíêö O(0; 0)öýã äýýð ýêñòðåìóì áàéõã³é.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
Îëîí õóâüñàã÷òàé ôóíêöèéí ýêñòðåìóì
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
= 0
AC − B2 = 0− (−3)2 = −9 < 0 òóë °ã°ãäñ°í ôóíêö O(0; 0)öýã äýýð ýêñòðåìóì áàéõã³é.
ÌÀÒÅÌÀÒÈÊ-
2
Ä.Áàòò°ð
Àãóóëãà
Îëîíõóâüñàã÷òàéôóíêö(ÎÕÔ)
(ÎÕÔ)-èéíÒåéëîðûíòîìú¼îÎëîíõóâüñàã÷òàéôóíêöèéíýêñòðåìóìÔóíêöèéíõàìãèéí èõáà õàìãèéíáàãà óòãà
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.