пαê2018½ì¸ß¿¼Êýѧ¶þÂÖ¸´Ï°µÚÈý²¿·ÖÌâÐÍÖ¸µ¼¿¼Ç°Ìá·ÖÌâÐÍÁ·8´óÌâרÏîÁùº¯ÊýÓëµ¼Êý×ÛºÏÎÊÌâÀí ÏÂÔØ±¾ÎÄ

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¦Ð0;Èô(2k+1)¦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±01ʱ,g'(t)>0,ËùÒÔg(t)ÔÚÇø¼ä(1,+¡Þ)ÄÚµ¥µ÷µÝÔö. ´Ó¶øµ±t=1ʱ,º¯Êýg(t)È¡µÃ×îСֵg(1)=e.

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±0µÃe;ÓÉh'(x)<0µÃa

´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