ËùÒÔº¯Êýf(x)ÔÚÇø¼ä,(0,+¡Þ)ÄÚµ¥µ÷µÝÔö,ÔÚÇø¼äÄÚµ¥µ÷µÝ¼õ;
µ±a<0ʱ,x¡Ê(-¡Þ,0)ʱ,f'(x)>0,xʱ,f'(x)<0,
ËùÒÔº¯Êýf(x)ÔÚÇø¼ä(-¡Þ,0),ÄÚµ¥µ÷µÝÔö,ÔÚÇø¼äÄÚµ¥µ÷µÝ¼õ.
(2)ÓÉ(1)Öª,º¯Êýf(x)µÄÁ½¸ö¼«ÖµÎªf(0)=b,fa3+b,
Ôòº¯Êýf(x)ÓÐÈý¸öÁãµãµÈ¼ÛÓÚf(0)¡¤f=b<0,´Ó¶ø
ÓÖb=c-a,ËùÒÔµ±a>0ʱ,
a3-a+c>0»òµ±a<0ʱ,a3-a+c<0.
Éèg(a)=a3-a+c,ÒòΪº¯Êýf(x)ÓÐÈý¸öÁãµãʱ,aµÄȡֵ·¶Î§Ç¡ºÃÊÇ
(-¡Þ,-3),
ÔòÔÚ(-¡Þ,-3)ÄÚg(a)<0,ÇÒÔÚÄÚg(a)>0¾ùºã³ÉÁ¢,´Ó¶øg(-3)=c-1¡Ü0,ÇÒ
g=c-1¡Ý0,Òò´Ëc=1.
3
2
2
´Ëʱ,f(x)=x+ax+1-a=(x+1)[x+(a-1)x+1-a],
2
Òòº¯ÊýÓÐÈý¸öÁãµã,Ôòx+(a-1)x+1-a=0ÓÐÁ½¸öÒìÓÚ-1µÄ²»µÈʵ¸ù,ËùÒÔ¦¤=(a-1)2-4(1-a)=a2+2a-3>0,ÇÒ(-1)2-(a-1)+1-a¡Ù0,
½âµÃa¡Ê(-¡Þ,-3)×ÛÉÏc=1.
4.Ö¤Ã÷(1)f'(x)=aesinx+ecosx=e(asinx+cosx)=axaxaxesin(x+¦Õ),ÆäÖÐ
axtan¦Õ=,0<¦Õ<
Áîf'(x)=0,ÓÉx¡Ý0µÃx+¦Õ=m¦Ð,
*¼´x=m¦Ð-¦Õ,m¡ÊN.
¶Ôk¡ÊN,Èô2k¦Ð Òò´Ë,ÔÚÇø¼ä((m-1)¦Ð,m¦Ð-¦Õ)Óë(m¦Ð-¦Õ,m¦Ð)ÉÏ,f'(x)µÄ·ûºÅ×ÜÏà·´.ÓÚÊǵ±x=m¦Ð-¦Õ(m¡ÊN*)ʱ,f(x)È¡µÃ¼«Öµ,ËùÒÔxn=n¦Ð-¦Õ(n¡ÊN*). a(n¦Ð-¦Õ)n+1a(n¦Ð-¦Õ) ´Ëʱ,f(xn)=esin(n¦Ð-¦Õ)=(-1)esin¦Õ. Ò×Öªf(xn)¡Ù0,¶øÊ×ÏîΪf(x1)=e a(¦Ð-¦Õ) =-ea¦ÐÊdz£Êý,¹ÊÊýÁÐ{f(xn)}ÊÇ a¦Ð sin¦Õ,¹«±ÈΪ-eµÄµÈ±ÈÊýÁÐ. (2)ÓÉ(1)Öª,sin¦Õ=,ÓÚÊǶÔÒ»ÇÐn¡ÊN,xn<|f(xn)|ºã³ÉÁ¢,¼´ *n¦Ð-¦Õ a(n¦Ð-¦Õ) ºã³ÉÁ¢,µÈ¼ÛÓÚ(*)ºã³ÉÁ¢(ÒòΪa>0). Éèg(t)=(t>0),Ôòg'(t)=Áîg'(t)=0µÃt=1. µ±0 Òò´Ë,Ҫʹ(*)ʽºã³ÉÁ¢,Ö»Ðè ¶øµ±a=ʱ,ÓÉtan¦Õ=ÇÒ0<¦Õ<Öª,<¦Õ< ÓÚÊǦÐ-¦Õ<,ÇÒµ±n¡Ý2ʱ,n¦Ð-¦Õ¡Ý2¦Ð-¦Õ> Òò´Ë¶ÔÒ»ÇÐn¡ÊN,axn=*1, ËùÒÔg(axn)>g(1)=e=¹Ê(*)ʽÒàºã³ÉÁ¢. ×ÛÉÏËùÊö,Èôa,Ôò¶ÔÒ»ÇÐn¡ÊN,xn<|f(xn)|ºã³ÉÁ¢. *5.(1)½âf'(x)=(x>0), µ±a>0ʱ,f(x)µÄµ¥µ÷µÝÔöÇø¼äΪ(0,1),µ¥µ÷µÝ¼õÇø¼äΪ(1,+¡Þ); µ±a<0ʱ,f(x)µÄµ¥µ÷µÝÔöÇø¼äΪ(1,+¡Þ),µ¥µ÷µÝ¼õÇø¼äΪ(0,1). (2)½âÁîF(x)=alnx-ax-3+ax+x+4-e=alnx+x+1-e, F'(x)=,ÁîF'(x)=0,µÃx=-a. Èô-a¡Üe,¼´a¡Ý-e, 22 ÔòF(x)ÔÚx¡Ê[e,e]ÉÏÊÇÔöº¯Êý,ҪʹF(x)¡Ü0¶ÔÈÎÒâx¡Ê[e,e]ºã³ÉÁ¢, ÔòÐèF(x)max=F(e)=2a+e-e+1¡Ü0,a2 2 22 ,ÎÞ½â; Èôe<-a¡Üe,¼´-e¡Üa<-e, ÔòF(x)ÔÚx¡Ê[e,-a]ÉÏÊǼõº¯Êý, 2 ÔÚx¡Ê[-a,e]ÉÏÊÇÔöº¯Êý,ÁîF(e)=a+1¡Ü0,µÃa¡Ü-1. ÁîF(e)=2a+e-e+1¡Ü0,µÃa22 , ¡à-e2¡Üa2 2 2 Èô-a>e,¼´a<-e,F(x)ÔÚx¡Ê[e,e]ÉÏÊǼõº¯Êý,ÁîF(x)max=F(e)=a+1¡Ü0,µÃa¡Ü-1,¡àa<-e2, ×ÛÉÏËùÊöa (3)Ö¤Ã÷Áîa=-1(»òa=1),´Ëʱf(x)=-lnx+x-3,µÃf(1)=-2, ÓÉ(1)Öªf(x)=-lnx+x-3ÔÚÇø¼ä(1,+¡Þ)ÄÚµ¥µ÷µÝÔö,ËùÒÔµ±x¡Ê(1,+¡Þ)ʱ,f(x)>f(1), ¼´-lnx+x-1>0, ËùÒÔlnx ÒòΪ 2 n¡Ý2,n¡ÊN*,ÔòÓÐ 2 2 2 ln *,ÒªÖ¤ ln(2+1)+ln(3+1)+ln(4+1)+¡+ln(n+1)<1+2lnn!(n¡Ý2,n¡ÊN), Ö»ÐèÖ¤lnÒò +ln+ln+¡+ln<1(n¡Ý2,n¡ÊN*), Ϊ ln+ln+ln+¡+ln+¡+=1-<1,¹ÊÔ²»µÈʽ³ÉÁ¢. 6.½â(1)ÓÉf(x)=,µÃf'(x)=, ÓÉÌâÒâµÃf'(1)=ab=ae.¡ßa¡Ù0,¡àb=e. (2)Áîh(x)=x[f(x)-g(x)]=x-(a+e)x+aelnx,ÔòÈÎÒâx2 ,f(x)Óëg(x)ÓÐÇÒÖ»ÓÐ Á½¸ö½»µã,µÈ¼ÛÓÚº¯Êýh(x)ÔÚÓÐÇÒÖ»ÓÐÁ½¸öÁãµã. ÓÉh(x)=x-(a+e)x+aelnx,µÃh'(x)=2 , ¢Ùµ±aʱ,ÓÉh'(x)>0µÃx>e; ÓÉh'(x)<0µÃ ´Ëʱh(x)ÔÚÇø¼äÄÚµ¥µ÷µÝ¼õ,ÔÚÇø¼ä(e,+¡Þ)ÄÚµ¥µ÷µÝÔö. ÒòΪh(e)=e-(a+e)e+aelne=-e<0, 22 h(e2)=e4-(a+e)e2+2ae=e(e-2)(e2-2a)e(e-2)>0(»òµ±x¡ú+¡Þʱ,h(x)>0Òà ¿É),ËùÒÔҪʹµÃh(x)ÔÚÇø¼äÄÚÓÐÇÒÖ»ÓÐÁ½¸öÁãµã, ÔòÖ»Ðèh+aeln0,¼´a ¢Úµ±0µÃ ´Ëʱh(x)ÔÚÇø¼ä(a,e)ÄÚµ¥µ÷µÝ¼õ,ÔÚÇø¼äºÍ(e,+¡Þ)ÄÚµ¥µ÷µÝÔö. ´Ëʱh(a)=-a-ae-aelna<-a-ae+aelne=-a<0, 222