(2)¢ÙÈôa=0,Ôòf(x)=e2x,ËùÒÔf(x)¡Ý0.
¢ÚÈôa>0,ÔòÓÉ(1)µÃ,µ±x=ln aʱ,f(x)È¡µÃ×îСֵ,×îСֵΪf(ln a)=-a2ln a.´Ó¶øµ±ÇÒ½öµ±-a2ln
a¡Ý0,¼´a¡Ü1ʱ,f(x)¡Ý0.
¢ÛÈôa<0,ÔòÓÉ(1)µÃ,µ±x=ln(-)ʱ,f(x)È¡µÃ×îСֵ,×îСֵΪf(ln(-))=a2[-ln(-)].´Ó¶øµ±ÇÒ
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a2[34-ln(-??
2)]¡Ý0,¼´
a¡Ý-2e3
4ʱ
f(x)¡Ý0.
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µÄÈ¡Öµ·¶Î§ÊÇ[-2e3
4,1].
µäÌâÑÝÁ·ÌáÄÜ¡¤Ë¢¸ß·Ö
1.(2019ºþ±±°ËУÁª¿¼Ò»)ÒÑÖªº¯Êýf(x)=x3+3
2x2-4ax+1(a¡ÊR). (1)Èôº¯Êýf(x)ÓÐÁ½¸ö¼«Öµµã,ÇÒ¶¼Ð¡ÓÚ0,ÇóaµÄÈ¡Öµ·¶Î§; (2)Èôº¯Êýh(x)=a(a-1)ln x-x3+3x+f(x),Çóº¯Êýh(x)µÄµ¥µ÷Çø¼ä.
½â (1)ÓÉf(x)ÓÐÁ½¸ö¼«ÖµµãÇÒ¶¼Ð¡ÓÚ0,µÃf'(x)=3x2+3x-4a=0ÓÐÁ½¸ö²»ÏàµÈµÄ¸ºÊµ¸ù,
??=9+48??>0,¡à{??1+??2=-1<0,4
??1??2=-3
??>0,½âµÃ-3
16 ¹ÊaµÄÈ¡Öµ·¶Î§Îª-3 16,0. (2)ÓÉh(x)=a(a-1)ln x+32x2-(4a-3)x+1,x>0, Ôòh'(x)= ??(??-1)??+3x-(4a-3)=1 ?? (3x-a)[x-(a-1)]. Áî(3x-a)[x-(a-1)]=0,µÃx=?? »òx=a-1,Áî??=a-1,µÃa=33 3 2 . ?? ¢Ùµ±{3¡Ü0, ??-1¡Ü0, ¼´a¡Ü0ʱ,ÔÚ(0,+¡Þ)ÉÏh'(x)>0ºã³ÉÁ¢; ¢Úµ±{0? ??-1¡Ü3,0, ¼´0???? 3ʱ,h'(x)>0,µ±0 ¢Ûµ±??3???? 3>a-1>0,¼´13ʱ,h'(x)>0,µ±a-1 3 3=a-1>0,¼´a=2ʱ,h'(x)>0ºã³ÉÁ¢; ¢Ýµ±a-1>?? 3 ?? 3>0,¼´a>2,µ±0 3 9 ×ÛÉÏËùÊö:µ±a¡Ü0»òa=ʱ,h(x)ÔÚ(0,+¡Þ)Éϵ¥µ÷µÝÔö; ?? ?? 32µ±02ʱ,h(x)ÔÚ0,3,(a-1,+¡Þ)Éϵ¥µ÷µÝÔö,ÔÚ3,a-1Éϵ¥µ÷µÝ¼õ. 2.ÒÑÖªº¯Êýf(x)= -??2+????-?? (x>0,a¡ÊR). e??3 ?? ?? 3 ?? ?? (1)µ±a=1ʱ,Çóº¯Êýf(x)µÄ¼«Öµ; (2)Éèg(x)=(1)½â µ±??(??)+??'(??)4,Èôº¯Êýg(x)ÔÚ(0,1)¡È(1,+¡Þ)ÄÚÓÐÁ½¸ö¼«Öµµãx1,x2,ÇóÖ¤:g(x1)g(x2)<2. ??-1e -??2+??-1 a=1ʱ,f(x)=e??(x>0). (-2??+1)e??-(-??2+??-1)e?? e2??¡àf'(x)== (??-1)(??-2) (x>0). e??µ±x¡Ê(0,1),(2,+¡Þ)ʱ,f'(x)>0,f(x)µ¥µ÷µÝÔö; µ±x¡Ê(1,2)ʱ,f'(x)<0,f(x)µ¥µ÷µÝ¼õ. ËùÒÔf(x)ÔÚ(0,+¡Þ)ÉÏÓм«´óÖµf(1)=-e,¼«Ð¡Öµf(2)=-e2. (2)Ö¤Ã÷ ÓÉÌâÒâµÃg(x)= ??(??)+??'(??) ??-11 3 =(??-1)e??, -2??+?? ¡àg'(x)= 2??2-(2+??)??+2 2(??-1)e?? , Éèh(x)=2x2-(2+a)x+2, ¡ßº¯Êýg(x)ÔÚ(0,1)¡È(1,+¡Þ)ÄÚÓÐÁ½¸ö¼«Öµµãx1,x2, ¡à·½³Ìh(x)=2x2-(2+a)x+2=0ÔÚ(0,1)¡È(1,+¡Þ)ÉÏÓÐÁ½¸ö²»ÏàµÈµÄʵ¸ùx1,x2,ÇÒ1²»ÄÜÊÇ·½³ÌµÄ ¸ù,¦¤=(2+a)2-16>0. ??+??2=2>0,¡à{1½âµÃa>2. ??1??2=1>0, ??+2 ¡àg(x1)g(x2)=(?? == (-2??1+??)(-2??2+??) ???? 1-1)e1(??2-1)e2 4??1??2-2??(??1+??2)+??2 [??1??2-(??1+??2)+1]e??1+??2 2(2-??)??+22(2-2)e ??+2= ??+2, e24 ¡ßa>2,¡à4??+2e2 4 10 ¡àg(x1)g(x2)= e2??+2< 4 4. e23.(2019°²»Õ½»´Ê®Ð£Áª¿¼Ò»)ÒÑÖªº¯Êýf(x)=ax2+xln x(aΪ³£Êý,a¡ÊR,eΪ×ÔÈ»¶ÔÊýµÄµ×Êý,e=2.718 28¡). (1)Èôº¯Êýf(x)¡Ü0ºã³ÉÁ¢,ÇóʵÊýaµÄÈ¡Öµ·¶Î§; (2)ÈôÇúÏßy=f(x)ÔÚµã(e,f(e))´¦µÄÇÐÏß·½³ÌΪy=(2e+2)x-e2-e,k¡ÊZÇÒk?(??) ¶ÔÈÎÒâx>1¶¼³ÉÁ¢,ÇókµÄ ??-1 ×î´óÖµ. ½â (1)º¯Êýf(x)¡Ü0ºã³ÉÁ¢,¼´ax2+xln x¡Ü0ºã³ÉÁ¢,¿ÉµÃa¡Ü-ln?? ?? ºã³ÉÁ¢. Éèg(x)=-ln????,g'(x)=ln??-1?? 2. µ±0 ¿ÉµÃµ±x=eʱg(x)È¡µÃ×îСֵ,ÇÒg(e)=-ln e 1 e=-e, ËùÒÔa¡Ü-1 e. (2)f(x)µÄµ¼ÊýΪf'(x)=2ax+1+ln x, ÇúÏßy=f(x)ÔÚµã(e,f(e))´¦µÄÇÐÏßбÂÊΪ2ae+2=2e+2, ¿ÉµÃa=1,¼´f(x)=x2+xln x. ÓÖÓÉk?(??) ??-1¶ÔÈÎÒâx>1¶¼³ÉÁ¢,¿ÉµÃ k?2+??ln????-1¶Ôx>1ºã³ÉÁ¢. Éè h(x)=??2+??ln????2-??-ln??-1??-1,x>1,h'(x)=(??-1) 2. Éèk(x)=x2-x-ln x-1,x>1, k'(x)=2x-1-1 = (??-1)(2??+1) ?? ?? >0, ¿ÉµÃk(x)ÔÚ(1,+¡Þ)ÄÚµÝÔö,ÓÉk(1.8)=0.44-ln 1.8. ÓÉ1.8>¡Ìe¿ÉµÃln 1.8>1 2, ¼´ÓÐk(1.8)<0,k(2)=1-ln 2>0,Ôò´æÔÚm¡Ê(1.8,2),ʹµÃk(m)=0, Ôò1 h(x)min=h(m)=??2+??ln?? ??-1. ÓÖk(m)=m2-m-ln m-1=0, ¼´ÓÐm2-1=m+ln m, 11 ¿ÉµÃh(x)min=m2+mÔÚ(1.8.2)µÝÔö, ¿ÉµÃh(x)min¡Ê(5.04,6), ÓÉk 4.(2019ɽ¶«Î«·»¶þÄ£)ÒÑÖªº¯Êýf(x)=xex-aln x(ÎÞÀíÊýe=2.718¡). (1)Èôf(x)ÔÚ(0,1)µ¥µ÷µÝ¼õ,ÇóʵÊýaµÄÈ¡Öµ·¶Î§; (2)µ±a=-1ʱ,Éèg(x)=x(f(x)-xex)-x3+x2-b,Èôº¯Êýg(x)´æÔÚÁãµã,ÇóʵÊýbµÄ×î´óÖµ. ½â (1)f'(x)=(x+1)ex-?? +??)e??-?? ??= (??2?? . ÓÉÌâÒâ¿ÉµÃf'(x)¡Ü0,x¡Ê(0,1)ºã³ÉÁ¢. ¼´(x2+x)ex-a¡Ü0,Ò²¾ÍÊÇa¡Ý(x2+x)exÔÚx¡Ê(0,1)ºã³ÉÁ¢. Éèh(x)=(x2+x)ex,Ôòh'(x)=(x2+3x+1)ex. µ±x¡Ê(0,1)ʱ,x2+3x+1>0,h'(x)>0ÔÚx¡Ê(0,1)µ¥µ÷µÝÔö. ¼´h(x) (2)µ±a=-1ʱ,f(x)=xex+ln x.g(x)=xln x-x3+x2-b, ÓÉÌâÒâµÃÎÊÌâµÈ¼ÛÓÚ·½³Ìb=xln x-x3+x2,ÔÚ(0,+¡Þ)ÉÏÓнâ. ÏÈÖ¤Ã÷ln x¡Üx-1.Éèu(x)=ln x-x+1,x¡Ê(0,+¡Þ),Ôòu'(x)=1 1-?? ??-1=??. ¿ÉµÃµ±x=1ʱ,º¯Êýu(x)È¡µÃ¼«´óÖµ, ¡àu(x)¡Üu(1)=0.Òò´Ëln x¡Üx-1, ËùÒÔb=xln x-x3+x2¡Üx(x-1)-x3+x2=-x(x2-2x+1)¡Ü0.µ±x=1ʱ,È¡µÈºÅ. ¹ÊʵÊýbµÄ×î´óֵΪ0. 5.(2019Ïæ¸ÓÊ®ËÄУÁª¿¼Ò»)ÒÑÖªº¯Êýf(x)=ln x-mx-n(m,n¡ÊR). (1)Èôn=1ʱ,º¯Êýf(x)Óм«´óֵΪ-2,Çóm; (2)Èô¶ÔÈÎÒâʵÊýx>0,¶¼ÓÐf(x)¡Ü0,Çóm+nµÄ×îСֵ. ½â (1)µ±n=1ʱ,f(x)=ln x-mx-1. ¡ßº¯Êýf(x)Óм«´óֵΪ-2, ÓÉf'(x)=1-m=0Öªx=1?? ?? >0, ¡àf1 ?? =-ln m-1-1=-2, ¡àm=1.¾¼ìÑé,m=1Âú×ãÌâÒâ. (2)º¯Êýf(x)µÄ¶¨ÒåÓòΪ(0,+¡Þ), f'(x)=1 ??-m. ¢Ùµ±m<0ʱ,µ±x¡Ê(0,+¡Þ)ʱ,f'(x)>0, ¡àf(x)ÔÚ(0,+¡Þ)Éϵ¥µ÷µÝÔö,Áîx=en, Ôòf(en)=ln en-men-n=-men>0,ÉáÈ¥; 12