n元函数极值的充分条件

n徐虎(泸州广播电视大学四川泸州646000):本文将一元函数极值的充分条件推广到n元函数上,得到了n元函数极值的两个判别定理并举例说明定理的应用:n元函数极值充分条件Abstract:Twotheoremstodiscriminateextremevalueofmultivariatefunctionsareob-tainedwhensufficientconditionsofextremevalueofone-variablefunctionaregeneralized.Andtheapplicationoftheoremsisillustratedwithexamples.KeyWord:Functionofn-variablesExtremevalueSufficientconditions:O1741n,,:f(x)x0f'(x)(x-x0)0(0),()n;:f(x),x0,f(x0)X0,f(x0)0(0),x0()nfij(i=1,2,,,n;j=1,2,,n),fij,,n:X=(x1,x2,,,xn),X0=(x10,x20,x10,,,xn0),hi=xi-xi0,(i=1,2,,n)H=X-X0:f'(X0)=0,X0f(X)n:f(X)=f(X0)+df(X0)+1/2d2f(X0)+,+1/(n+1)!dn+1f(X0+HH)(0H1)¹262008年第2期徐虎:n元函数极值的充分条件21f(X)X0,U0(X0)(),Eni=1fxi(x1,x2,,,xn)(xi-xi0(0),X0():(1)n=0,X0XP=(N1,N2,,,Nn)vf=f(X)-f(X0)=Eni=1fxi(N1,N2,,,Nn)hi,Ni=xi0+Hhi,(0H1)ºX0XnA1,A2,,,Anhi='X0X'cosAiNi-xi0='X0P'cosAi(2)vf=Eni=1fx1(N1,N2,,,Nn)(Ni-xi0)'X0X'/'X0P'PXIU0(X0),PIU0(X0),Xvf=Eni=1fx1(x1,x2,,,xn)(xi-xi0)'X0X'/'X0P',f(X)f(X0),X02f(X)X0U(X0),X0f(X)ai,j=fij(X0),(i,j=1,2,,,n),A=(Aij)n@nX0(1)A(),X0f(X)()(2)A,X0f(X):PX=(x10+h1+,+xn0+hn)=X0+HIU(X0),¹n=1,X0vf=f(X)-f(X0)=12Eni=1,j=1fij(X0+HH)hihj,(h1,h2,,,hn)limHy0fij(X0+HH)=fij(X0),fij(X0+HH)=fij(X0)+Eij,limQy0Eij=0,Q=h12+h22+,+hn2(Hy0,Qy0),vf=12Eni=1,j=1aijhihj+12Eni=1,j=1Eijhihj»(1)A,Eni=1,j=1aijhihj0,Eni=1,j=1aijhihjQ=h12+h22+,+hn2,m0,limQy0,Eni=1,j=1Eijhihj=0,U(X0)'Eni=1,j=1Eijhihj'm,vf»,f(X)f(X0),X0f(X)(2)A,Eni=1,j=1aijhihj,F(t)=f(x10+th1+,+xn0+thn)=f(X0+Ht),,F'(t)=Eni=1hifi(X0+Ht),F(t)=Eni,j=1fij(X0+Ht)hihj¼f(X)X0,X0x1=x10+th1,,,xn=xn0+thn,F(t)=f(x10+th1+,+xn0+thn)t=0,¼F(0)=Eni=1,j=1aijhihj,F'(0)X0,Eni=1,j=1aijhihj,X0f(X)2743成都电子机械高等专科学校学报CHENGDUELECTROMECHANICALCOLLEGE20082200861f(x,y)=1-X23-y35:f'x=-23x-13,f'y=-23y-35(0,0)(0,0),f'x#(x-0)+f'y#(y-0)0,f(0,0)=12f(x,y,z)=(1+ey)cosx-yey+2z2-4z(xI[-P/2,P/2])f'x=-sinx-eysinx,f'y=cosxey-ey-yey,f'z=4z-4,f'x=f'y=f'z=0(0,0,1),=A-2000-10004,A,3f(x1,x2,x3)=x12+2x1x2+6x22+3x32+10x2+1012(),n,,1,;2,21[1]()[M].,1983.4244[2].()(2)[M].,1983.11187-188[3](2)[M].,1988本文作者为大学讲师(52)[2].[M].:,1999.[3],,,.McMaster[M].:,2005.[4].$[M].:,2005.本文作者为副教授法学硕士282008年第2期徐虎:n元函数极值的充分条件