17復丂戞2師曃摫娭悢偲2曄悢娭悢偺嬌抣丂


1.戞2師曃摫娭悢

2曄悢娭悢f(x,y)偺戞2師曃摫娭悢丄椺偊偽 fxy(x,y)傪媮傔傞偵偼


亜diff(f(x,y),x,y) 丂

偁傞偄偼

亜D[1,2](f) 丂


偲偟傑偡丅


幚廗 17.1 丂丂丂儮f亖0傪枮偨偡C2娭悢f(x,y)傪挷榓娭悢偲偄偄傑偡丄偙偙偱丄儮f亖fxx亄fyy偱偡丅乮儮偼儔僾儔僔傾儞偲傛偽傟傑偡丅乯 丂
師偺娭悢偑挷榓娭悢偱偁傞偐偳偆偐挷傋側偝偄丅
(1)f(x,y)亖x2亅y2
(2)f(x,y)亖log(x2亄y2)丂
拲堄 暋嶨側宍偺摎偊偑弌椡偝傟偨帪偵偼丄simplify僐儅儞僪傪梡偄偰丄傛傝娙扨側宍偱摎偊傪昞帵偱偒側偄偐帋偟偰傒傑偟傚偆丅

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2. 嬌戝丒嬌彫 丂 娭悢f(x,y)偵偍偄偰丄偁傞兟>0偵懳偟丂
側傜偽
f(x,y)亙f(a,b)丂(偁傞偄偼丄f(x,y)>f(a,b))
偑惉傝棫偮偲偒丄(a,b)傪嬌戝揰丂(偁傞偄偼丂嬌彫揰)偲偄偄丄f(a,b)傪f偺嬌戝抣(偁傞偄偼丂嬌彫抣)偲偄偄傑偡丅
娭悢f(x,y)亖x3亄y3亅3xy偺嬌抣傪媮傔偰傒傑偟傚偆丅


>f:亖(x,y)->x3+y3-3*x*y

>solve({diff(f(x,y),x)亖0,diff(f(x,y),y)亖0},{x,y})
(f偺椪奅揰傪媮傔傞)


{y亖0,x亖0},{y亖1,x亖1},{乧乧乧}


>evalf(%)

>det(H):=diff(f(x,y),x,x)*diff(f(x,y),y,y)-diff(f(x,y),x,y)2
(僿僢僙峴楍偺峴楍幃傪det(H)偲偍偔)




>subs(x=0,y=0,det(H))
((0,0)偵偍偗傞det(H)偺抣傪寁嶼)

亅9
(det(H)<0傛傝,(0,0)偱嬌抣傪偲傜側偄)



>subs(x=1,y=1,det(H))
((1,1)偵偍偗傞丂det(H)偺抣傪寁嶼)


27



>subs(x=1,y=1,diff(f(x,y),x,x))
(fxx(1,1)傪寁嶼)


6


(det(H)>0偐偮fxx(1,1)>0偱偁傞偐傜(1,1)偼嬌彫揰偱偁傞)


>f(1,1)
(嬌彫抣f(1,1)傪寁嶼)


>plot3d(fx,y),x=-0.5..1.5,y=-0.5..1.5,style=patchcontour,contours=30)

([-0.5,1.5]亊[-0.5,1.5]偵偍偗傞 z亖f(x,y)偺僌儔僼傪昞帵)丂




拲堄
(1)峴楍 偼僿僢僙峴楍偲傛偽傟傑偡丅
娭悢f(x,y)偺僿僢僙峴楍偼det(H)=fxxfyy-yxyfyx 偱偡丅摿偵丄f偑C2媺偺応崌偼 fxy=f偑惉傝棫偮偐傜丄det(H)=fxxfyy-fxy2偲側傝傑偡丅 丂

(2)娭悢f(x,y)偑揰(a,b)偱嬌抣傪偲傞側傜偽丄fx(a,b)=0偐偮,fy(a,b)偑惉傝棫偮丄偡側傢偪(a,b)偼椪奅揰偱偡丅偡側傢偪(a,b)偼椪奅揰偱偡丅
偙偙偱fx(a,b)=0偐偮fy(a,b)=0傪枮偨偡揰(a,b)傪椪奅揰偲偄偄傑偡丅
椪奅揰(a,b)偵偍偄偰丄
(i)det(H)>0偺偲偒丂
丂丂fxx>0側傜偽(a,b)偼嬌彫揰
丂丂fxx<0側傜偽(a,b)偼嬌戝揰
(ii)det(H)<0偺偲偒(a,b)偱f(x,y)偼嬌抣傪偲傜側偄
偙偲偑徹柧偱偒傑偡丅 (ii)偺応崌丄(a,b)偼埰揰偲傛偽傟傑偡丅丂

幚廗 17.2
娭悢f(x,y)=(x2+y2)2-2x2+2y2偺嬌抣傪媮傔側偝偄丅
傑偨丄娭悢偺僌儔僼傪摍崅慄傪偮偗偰昞帵偟丄寢壥傪妋偐傔側偝偄丅

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幚廗 17.3
娭悢f(x,y)=e-x2-y2(2x2+y2)偵偮偄偰埲壓偺栤偄偵摎偊傛丅
(1) 椪奅揰傪媮傔側偝偄丅
(2)嬌抣傪媮傔側偝偄丅
(3)z=f(x,y)偺僌儔僼傪嶌惉偟偰(1),(2)偺寢壥傪妋擣偟側偝偄丅

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