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x^2-mlnx-x^2+x=x-mlnxâ¥0(x>1),
xâ¥mlnx,mâ¤x/lnx,令g(x)=x/lnx,g'(x)=
(lnx-x*1/x)/(lnx)^2=(lnx-1)/(lnx)^2,åg'(x)=0ï¼è§£å¾lnx=1,x=e,
å 为g(x)å¨xâ(1ï¼e)ä¸åè°éåï¼å¨xâï¼eï¼+âï¼ä¸åè°éå¢ï¼
æ以å¨x=eå¤åå¾æå°å¼ï¼gmin(x)=g(e)=e,
æ以æmâ¤eï¼
ï¼2ï¼
kï¼xï¼=-2lnx+x-a=0,设两é¶ç¹ä¸ºx1â¥1,x2â¤3,a=-2lnx1+x1=-2lnx2+x2;
设g(x1)=-2lnx1+x1,y(x2)=-2lnx2+x2,
g'(x1)=-2/x1+1,(x1â¥1),å¾g(x1)â¥g(2)=-2ln2+2;
y'(x2)=-2/x2+1,(x2â¤3)ï¼å¾y(x2)â¤y(3)=-2ln3+3;
æ以æ-2ln2+2â¤aâ¤-2ln3+3
(3)
f'(x)=2x-m/x,
h'(x)=2x-1,
åf'(x)=0,å¾m=2x^2;x=âm/2,
åh'(x)=0,å¾x=1/2,
è¦æ»¡è¶³f(x)åh(x)å¨å
Œ
±å®ä¹åä¸å
·æç¸åçåè°æ§ï¼
âm/2=1/2,æ以m=1/2
f(x)çä¸é¶å¯¼æ°f'(x)=-2*(2x^2 - tx -2)/(x^2 + 1)^2
f'(x)çåæ¯æ大äº0ï¼åå为æ£çé¨åæ£å¥½æ¯ãαãβãã
æ以f'(x)å¨åºé´ãαãβãä¸æ大äº0
æ以f(x)å¨åºé´ãαãβãä¸åè°éå¢
æ以A=f(β)=(4β-t)/(β^2 +1)ï¼B=f(α)=(4α-t)/(α^2 +1)
g(t)=A-B=[4αβ(α-β)-4(α-β)-t(α-β)(α+β)]/(α^2β^2+α^2+β^2+1)
å 为αãβæ¯æ¹ç¨çä¸¤ä¸ªæ ¹ï¼æ以α+β=t/2ï¼Î±*β=-1
α-β=-sqrt(α^2 + β^2 -2αβ)=-sqrt[(α+β)^2-4αβ]=-[sqrt(t^2+16)]/2
带å
¥g(t)=sqrt(t^2 +16)
åå 为æ¹ç¨æ两个å®æ ¹ï¼æ以delt=t^2 +16 æ大äº0
æ以g(t)æå°å¼ä¸ºt=0æ¶g(0)=4
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