复旦大学高等数学教案08多元函数的无条件极值

112nfnR0x0)()(0xxff))()(0xxff),(0xxO0xf)(0xf222yxz)0,0(0)0,0()0,0(xyz)0,0()0,0(0Fermatf0x0xf0)(0xf),,()0()0(10nxxxnf2f0x0)(0xixfni,,2,1ix1n),,,,,,()()0()0(1)0(1)0(1niiiiixxxxxfxi)0(ix)0(ixiFermat,0)()0(iixixf0)(0xf07.7.17.7.2||),(xyxf),(yxfzOxyy),0(yffxnf0x0xffHessiannnnnnnxxxxxxxxxxxxxxxxxxfffffffff212221212111H0xH0xf0xH0xf0xf0)(0xixfni,,2,1f0xTaylorxxxHxxx)()(21)()(00Tff0xxx10)(0xHHessianjixxf0x0),(0xxO)(xH)(xH||||0x0)()(0xxxHxT0)()(0xxff0xfH0x)(0xH0xff3zxyzyxzyxfu26),,(223f,022,062,0632zuxyuyxuzyx)1,18,6(P)1,0,0(Qf),,(zyxHessian200026066xH2000260636)(PH200026060)(QH0|36|036266360722000260636)(PH7.7.2Pf)(QH7.7.2Q)(QH2137123713x)(QHixxxHxTiTQ)()()(HxT)(x)(QiHessian||||xxxxHxxx)()(21)()(00Tffi0,21030xx)()(0xxffx)()(0xxff0xf),(yxfz),(00yx),(00yxf2000000)],([),(),(yxfyxfyxfxyyyxx0),(00yxf0),(00yxfxx),(00yx0),(00yxfxx),(00yx0),(00yxf40),(00yx00),(00yxfxxf),(00yxHessian00),(00yxfxxf),(00yxHessian1f),(00yxHessian0),(00yxf0a)(),(yxaxyyxff0)(,0)(xyyxaxfxyyxayfyx)0,0()0,(a),0(a3,3aayyxfxx2),(22)22()2)(2(yxayxfffxyyyxxayaxyxa44442220|||2),0()0,()0,0(aaa)0,0()0,(a),0(af031|23,3aaaaaafxx323,30a273,33aaaf0a273,33aaaf44)()2(),(yxxyxf0)(4),(,0)(4)2(4),(333yxyxfyxxyxfyx)2,2(0xyyyxxfff0|)2,2(7.7.2f)2,2(),(yx)2,2()2,2(0)()2(),(44fyxxyxf)2,2(f54222),(yyxyxyxf5.0544,022432yyxyyfyxxf)0,0(322222220124,4,2yyxyfyyxfxf)0,0(02BAC0)0,0(f522)(),(yyxyxf0,2yyx0),(yxf0,2yyx0),(yxf0)0,0(f12cm7.7.1xFsin)]cos2212()212[(21),(xxxxxFcossinsin2sin1222xxx}0,60|),{()(xxFDF)(FD.0)sin(coscos2cos12),(,0cossin2sin4sin12),(2222xxxxFxxxFx0x0.0)1cos2(cos2cos12,0cos262xxxxy0,2yyxOx0,2yyx621cos4x34x8.203123,4F)cm(20x0),(xF018)2sin18max(),6(maxFF)(FD4x3),(iiyxni,,2,1xy),(iiyxiAni,,2,1yx,baxyiAbaxyixxiybaxi)(baxyiiiba,iiba,niiibaxyba12)]([),(),(ba),(baba,,0)1)((2,0))((211niiiniiiibaxybxbaxyanxxxniiniinii1112ba=niiniiiyyx11ba,baxy12-2x12xx7.7.173.343.43.363.323.343.363.383.43.43.4249.25049.3494949.549.849.950.250.2460xyyxy=ax+b7.7.2ixiy1072.3372.337136.113ba=1.496984.1672ba=04.381.13xyy=13.81x+3.043.343.403.363.323.343.363.383.403.403.4249.2050.0049.3049.0049.0049.5049.8049.9050.2050.2049.1749.9949.4448.8949.1749.4449.7249.9949.9950.27(%)x=4y58.2860xyaxbaxyeeebey~=yln8y~=ax+bxyyln~78495.326921.0~xyxy26921.0e02271.0bxay1y~=y1x~x1y~=ax~+babAx=b97AmnRxmRbnRAx=bx~niiiAb12])~([x=nRxminniiiAb12])([xx~AAAT0detAATnRx0x0AxAT0AxAxTT0AxAAAT7x~Ax=bx~AATx=TAbTmxx),,(~1xbAxx~m2111)(),,(mjjijniimxabxxf0kxfmk,,2,1))((211ikmjjijniikaxabxfniniiikmjjikijbaxaa111mk,,2,1bAAxATTbAxAATT~x~fbAxAATT~x~f)~))(~(~()~(21)~()(xxxxxHxxxxTfffH,212niiliklkaaxxfAAHT2)~()~()~()(xxAAxxxxTTffAxx~0)~(xxA0)~()(xxffx~f10xycbxaxy27.7.4112144139124111cba=982411991321278650786506084650608460710cba=316920312156430abcy=8.62x2+110.01x+15.751246178101113