导数题型2隐零点代换及解析

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ᦪ⚪2◚ᣚ1.ᙠឋᳮজ᝞ᦪy=/(x)ᙠ!,#$Ḅ&'()*+,Ḅ-ᩩ/0123ᨵ5)<0,89ᦪ:=/(x)ᙠ(஺<)ᑁᨵ1ᓽᙠce(a,b),CD/(c)=0,EFcGH(IJ/(x)=0Ḅ᪷.ঝ᝞-F$ᦪᓫN3Oᔠᙠឋᳮ189QᙠEF$ᨵ3Rᨵ-F஺2.ST/(x)UVᦪ(EW-Xឤᡂ[S⚪ᨵ\]^ᜐ1`(ᨵabcᱯe^ᜐজ/(X)g(ᢣᦪᦪijᦪᦪ2ঝS⚪ᨵkWl᦮ᦪᨵᐵ)3.pTজq1ᑨ,ᦪᓫNឋ(ss◤⌕qv▤ஹy▤ᦪ)ঝz{ᦪḄᙠឋ12q|QḄᙠ(}~})ঞᦪḄᐭᦪ1Dᑮ-FឤটঞḄᜐᳮᐭᦪ1CDᦪᡂ-F+ᢣᦪᦪijᦪᦪḄᦪঠঝḄᩭ1qটḄᦪᙠEF$Ḅ1-aᡂz{஺◚Ạ᱐1.ᦪ/(x)(x+2),qzf3>1.2.ᦪ/'(x)-Inx.(/஻20.6931,yfe«1.649),z{£x>0k1+/(x)>1.35ឤᡂ[.3.ᦪ/(x)=1+ᔙ+1)(x>0),x)>¦ឤᡂ[1q᦮ᦪ$Ḅᨬᜧ©4.ªᦪ/(x)—ex-ax-2.(I)qf(x)ḄᓫN©(II)°=1,¯°᦮ᦪ13£x>0k1(x-k)/(x)+x+l>0,q¯Ḅᨬᜧ.5.ᦪ/(x)=1-/”(X+/M).£1"W2k1z{/(X)>0.◚ᓣ¸᱐1.ªp:/(x)=e*+/nx+2x?+"?x+lᙠ(0,+oo)ᑁᓫN⌴À1q:-5,ᑣp(1Ḅ()A.ᐙᑖ+g⌕ᩩÅB.g⌕+ᐙᑖᩩÅC.ᐙᑖg⌕ᩩÅD.É+ᐙᑖG+g⌕ᩩÅ2.(I)ÊËᦪ஻x)=3e'ḄᓫNឋ12z{£x>0k1(x-2)e'+x+2>0©x+2P'—HY—fl(II)z{£஺w[0,1)kஹᦪg(x)=--------(x>0)ᨵᨬÏ.ªg(x)ḄᨬÏ°஻(a),qᦪ஻(a)xḄ.

13.ᦪ-ax-x/nx,3/(x)U0.(1)qa©(2)z{/(x)ᙠÐ-Ḅ᩽ᜧ3e<0k1f(x)^2a+aln—.a6.ªᦪ/(x)=(x-a)2/nx,awR(I)x=e°y=/(x)Ḅ᩽1qÖᦪa©(II)qÖᦪaḄ×Ù1CDjÚÛḄxe(0,3e],ឤᨵ/(x)W4e2ᡂ[.Ýe°ÞᯠjᦪḄàᦪ.7.ᦪ/(x)=ae'T-M+/஻a.(1)£“=ek1q/0y=/(x)ᙠ(1,f(1))ᜐḄᑗ0Wcᙶ᪗æÙᡂḄyçèḄ☢ê;(2)qaḄ×Ù.8.ᦪf{x}=aex-lnx-\.(1)ªx=2(஻x)Ḅ᩽1qa,2q஻x)ḄᓫN©(2)z{£í1

2◚v-.⌱ï⚪(ᐳ1Ï⚪)1.ªp:/(x)=e'+/஻x+2U2+s+lᙠ(0,”)ᑁᓫN⌴À1q:&-5,ᑣp(qḄ()A.ᐙᑖ+g⌕ᩩÅB.g⌕+ᐙᑖᩩÅC.ᐙᑖg⌕ᩩÅD.É+ᐙᑖG+g⌕ᩩÅv,pó⚪(ᐳ7Ï⚪)2.(I)ÊËᦪ/(x)=ve'ḄᓫNឋ12z{£x>0k1(x-2)eஹ+x+2>0©x+2p'—fly—77(H)z{£ae[O,1)k1ᦪg(x)=——=—(x>0)ᨵᨬÏ.ªg(x)ḄᨬÏ°÷x(a),qᦪ¯(a)Ḅ.3.ᦪ/(x)=ax?-ox-x/஻x,3/(x)U0.(1)qa;(2)z{f(x)ᙠÐ-Ḅ᩽ᜧ%,3e-20k,f(x)^2a+aln—.a6.ªᦪ/(x)=(x-a)2/"x,a&R(I)x=e°y=/(x)Ḅ᩽1qÖᦪa;(II)qÖᦪaḄ×Ù1CDjÚÛḄxe(0,3e],ឤᨵ/(x)W4e2ᡂ[.Ýe°ÞᯠjᦪḄàᦪ.7.ᦪJ\x)=-Inx+Ina.(1)£a=ek1q/0y=/(x)ᙠ(1,f(I))ᜐḄᑗ0Wcᙶ᪗æÙᡂḄyçèḄ☢ê;û1⚓(ᐳ13⚓)

3(2)qaḄ×Ù.8.ᦪ/(x)=ae*.(1)ªx=2(/(x)Ḅ᩽1*a,2q/(x)ḄᓫN;(2)z{£aU1k1/(x)>0.eû2⚓(ᐳ13⚓)

4◚vὃᫀ⚪᪆-.⌱ï⚪(ᐳ1Ï⚪)1.ªp:/a)=e"+ᔊx+Z/+?Xᵫ@A+4824,BCDBE=4x,ᓽx=1FGHᡂ>?ᦑJK"ḄX£(0,+oo),ᨵxx2x1/5x/.---4xQ#RS8T5xUB72—5F?ᨵ?+1+4x+780ᡂ>?ᓽ/'পᓾᙠxw(0,+8)Aᡂ>x/.p᜛Ḅᐙᑖᩩ?p0Ḅ⌕ᩩ?ᓽp4Ḅ⌕ᐙᑖᩩᦑ⌱B.\,⚪(ᐳ7^⚪)2.(I)`abᦪ/(%)=\Ḅᓫ2ឋ?fghBx>0F?a-2)e'+x+2>0jx+2px—/7V—a(H)ghBae[O,1)F?bᦪg(x)=?(x>0)ᨵᨬ^s.tg(x)Ḅᨬ^su஻x(a),wbᦪ஻(a)Ḅs;.ூ௃(1)gh/(x)=—e'x+2ஹY/X—24x2exJM=e(--+7—ry)=;—rrx+2(x+2)(x+2)••Bx£(-oo,-2)D(-2,+oo)F?/'(x)80.\/(x)ᙠ(-8,-2)~(-2,+00)Aᓫ2⌴4.•.x>0F?^^ex>/(0)=-lx+2ᓽ(x—2)e'+x+2>03⚓(ᐳ13⚓)

5(ex-a)x2-2x(e'-ax-a)(2)g'(x)=x{xex-2ex+ax+2a)_+x+2,£+")_(8+2)(/(x)+a)a&[0,1)ᵫ(1)?/(x)+aᓫ2⌴4,JK"Ḅ“e[0,1),/(0)+a=a-l<0,f(2)+a=,ᙠTḄfw(O,2],#/(/)+a=0,BX€(O,/)F?g'(x)<0,g(x)ᓫ2jBxe(f,+oo),g'(x)>0,g(x)ᓫ24jt-2/+(z+l)&d£-^£+l)A(?)=e(?+஺)=/T?ᙠfe(O,2]F?k\t)=V>0?t+2(f+2)2ᦑk(f)ᓫ2⌴4?ᡠ஻(a)=%(/)€(j,(].3.bᦪ/(>¥)=♦-or-x/nx,C/(x)20.(1)wa;(2)gh/(X)ᙠTḄ᩽ᜧs,C"20),ᑣ/(x)80G@h(x)=ax-a-Inx^O,w஻(x)=a-E.XᑣB஻W0Fh\x)<0,ᓽy=஻(x)ᙠ(0,+8)Aᓫ2⌴,ᡠB%>1F?h(x)0.QuB0LF"(x)>0,aaᡠᐭ஺)“=//(-),a;<=6(1)=a-a-ln\=0fᡠL=l,#a=ljau/(1)=0,ᡠ/(x)80G@/(x)ᙠx>0FḄᨬ^su/1),4⚓(ᐳ13⚓)

6ᡠG@f(x)ᙠX=1ᜐ᩽^s,ᡠ#4=1j(2)ᵫ(1)=f-x-r/nx,frM=2x-2-lnx,¤/'(x)=஺?#2%T2—/஻x=0?iB/(x)=2x-2-Inx,WOf(x)=2-—,x¤r(x)=o,#x=j,ᡠr(x)ᙠ§¨(O,j)Aᓫ2⌴?ᙠ(g,+00)Aᓫ2⌴4?ᡠ"x)”“"=?d)=/஻2-l<0,«“¬=¬>0,ᡠ)ᙠ(0,3AᙠT¯?2ee2ᡠf(x)=Oᨵ?ᓽ/'(x)=0ᙠ°᪷²,“C³t´'(X)ᙠ(0,%)Auµஹᙠ(஽,X?)Au·ஹᙠ(E2?+8)Auµ?ᡠ/(X)ᙠT᩽ᜧsX஺?C2X()-2-lnx=0,0ᡠ/&>)=*TTxlnx=x2-x+2x-2k=x-x2,oo00000ᵫ//(x)<(x-V)=--^-+^=^-j00mavᵫf'(~)<0/<—<—>ee2ᡠ/G)ᙠ(0,x0)Aᓫ2⌴4?ᙠ(x஺?1)Aᓫ2⌴?eᡠ/®)>/d)=ljee½Aᡠ¾?/(x)ᙠTḄ᩽ᜧs/,Ce“

7#f'{x)=(2ax+l)e*-(ax?+x-l)e*=(ar+l)(x-2)¤/''(x)=0,#᳝=2,X2=-<(),Bxe(-8,-L)F?/((%)<0,*€(-,,2)F?f\x)>0,xe(2,+a))F?f\x)<0.aa.•./(X)ᙠ(-8,-3?(2,+oo)⌴?ᙠ(-1,2)⌴4?aaÐ"ᑮF,bᦪg(x)=ax2+x-lᙠ(2,+8)ᓫ2⌴4,Cg(2)=4a+1>0>ᦑg(x)ᙠ(2,+00)Aឤᜧ@¯?ᓽ/(x)=ax2+x?lexᙠ(2,+00)Aឤᜧ@¯.bᦪ/(X)ḄÒÓ᝞Õf(x)=-e"8-e,ml.,.BaÖF?f(x)+e8O.5.tbᦪf(x)=e2x-alnx.(I)`aḄbᦪ×(x)¯ḄØᦪ;(II)ghBaு0F?/(%)82஺+஺ᔊ—.aூ௃(I)/(%)=/T஻7Ḅ9:;u(0,+8),:.f\x)=2^xXBF?/'(x)>0ឤᡂ>?ᦑ/'(X)Ûᨵ¯?Ba>0F?•.J=e2ஹuᓫ2⌴4?᜛=-@ᓫ2⌴4?X.•/(X)ᙠ(0,+8)ᓫ2⌴4?«×(a)>0,Ꮇtᙠbßà0<6<âF?Hb<-,f(b)<0,6⚓(ᐳ13⚓)

8ᦑBQ>0F?bᦪ/(X)ᙠTḄ¯,(II)ᵫ(I)?tbᦪ(ಘᙠ(0,+00)AḄT¯uM,Bᑍ£(O,x)F?/'(X)<0,oBE£(%,+00)F?f\x)>0,ᦑ/(x)ᙠ(0,/)ᓫ2⌴?ᙠ(%,”)ᓫ2⌴4,ᡠBX=X0F?/(X)ç#ᨬ^s?ᨬ^su/(%)?ᵫ@2/3--=0,a22ᡠ/(x)=+2aX஺+aln—22a+aln—.02xaa02ᦑBa>0F?/(X)22Q+aln—.a6.tbᦪ/(x)=(x-a)2/஻x,aeR(I)èx=eu8=/(ஹ)Ḅ᩽s?wéᦪaj(II)wéᦪ஺Ḅçsë?#JK"Ḅxe(0,3e],ឤᨵᡂ>.ÐeuíᯠJᦪḄïᦪ.ூ௃(I)w#/'(x)=2(x-஻)/஻x+S—ߟ=(x-a)(2lnx+1--),xxux=e/(x)Ḅ᩽s?ᡠ/'(e)=0,#஺=eᡈ஺=3e.óôõ?a=eᡈa=3eöᔠ⚪"?ᡠa=e,ᡈa=3e.(II)জB0ঝBl0f3e3e3j/஻3e7⚓(ᐳ13⚓)

9஻(X)ᙠ(0,+00)ᑁᓫ⌴ᡠᦪ஻(X)ᙠᙠ(0,+8)ᑁᨵ!"#ᑣl0,0-/£(%,a).,f\x)<0,-2w(a,+8).,/Xx)>0,ᓽ/(x)ᙠ(0,%)ᑁ6ᦪ,ᙠ(7஻)ᑁ68ᦪᙠ(a,+Q0)ᑁ6ᦪ.ᡠ⌕<=>?@Ḅxe(0,3e],ឤᨵ/(x)^4e2ᡂHI⌕ᨵj/Uo)=(/[/(3e)=(3e-a)2ln3e^4e2ᨵh(x)=2lnx+1-—=0=a=2xlnx+x,PQRᐭ/(x)=(x-a)2lnx^4e2=Q0QQ0000%4xj/zz3x^4e2(0%>1,U@ᑮᦪ4//஻3%ᙠ(I,+QO)W6ᦪᦑiீ/We,\ᵫ஻=2x°JX஺+x,^ᦪ2x/஻x+xᙠ(l,+oo)W6ᦪ_=1/ln3ecWaḄdeg"3e——.4lnie7.ijᦪ/(x)=aex~]-Inx-^-Ina.(1)-஺=e.lmny=/(x)ᙠ(1,f(1))ᜐḄᑗn7rᙶ᪗ugᡂḄvwxḄ☢z{(2)|l஺Ḅdeg.ூa~௃M:(1)-a=e.f(x)=ex-lnx+\,:.f\x)=exX:.f(1)=e-l,•/f(1)=e+1,mny=/(x)ᙠ(1,f(1))ᜐḄᑗn"y-(e+l)=(e-l)a-l),-x=0.y=2,-y=0.x=——,8⚓(ᐳ13⚓)

10mny=/(x)ᙠ(1,f(1))ᜐḄᑗn7rᙶ᪗ugᡂḄvwxḄ☢z1225=±x2x—=—2e—\e-\(2)ᵫ/(ᑍ)21,_=ae'T-/஻x+ᔊ஺21,ᓽ"".ᔊx+,ᓽeN+Ina+x-X^lnx+x=elnx+Inx,g(t)=d+ᑣg=d+l>0,g(஺ᙠRWᓫ⌴g(lna+x_:.lna+x-Y^bvc,KPIna^lnx-x+l,஻(x)=/஻x-x+l,.,v11-xh(zx)=—t1=----,XX-0cx<1.h\x)>0,ᦪ஻(x)ᓫ⌴-/>1.h\x)<0,ᦪ஻(x)ᓫ⌴8h(x)4h(1)=0,:./“a20,ᦑQḄg"1,+00).ᵫ_=-/஻/+/^,x>0,a>0fᓽaex~'-X^lnx-Inag(x)=ex-x-lf.•.᝞)=/1>0ឤᡂH.,.g(x)ᙠ(0,+oo)ᓫ⌴,.*.g(x)>g(0)=l-0-l=0e'—x—1>0,ᓽ>x+1,9⚓(ᐳ13⚓)

11\h(x)=x-\-Inx,1Y-1XX-00,ᦪ/?()ᓫ⌴h(x)^h(1)=0,Inx^O,ᓽx-Y^lnxex~l^x,ᑣaex~x^ax,!.I◤⌕ax^x-Ina,ᓽx(a-,-./.x[a-1)>0>-InaឤᡂH-00,a¨ᙠ/w(J)<=/(X஺)=o,a-xw(l,Xo).f(x)<0,ᦪ/(x)ᓫ⌴8/.f(x)0,Ina>0,f(x)^ex~1-Inx,g(x)=e"T_/nx,N.g'(x)=ex-'--,X᧕jg'(x)ᙠ(0,+oo)W"ᦪ,10⚓(ᐳ13⚓)

12•g'(1)=0,.•.-R£(02).g'(x)0,ᦪg(x)ᓫ⌴•••g(x)gপ=1ᓽ/(X)>1,cWᡠ¡஺Ḅdeg"1+O0).°/(x)=aex~l-Inx+Ina,x>0,a>0,N.f\x)=ae-'--,᧕j±(x)ᙠ(0,+8)W"ᦪX••,y=ae*Tᙠ(0,+oo)W"ᦪyᙠ0,+8)W"8ᦪ,X.•)=²17᜛=1ᙠ0,+8)Wᨵ´XN.¨ᙠX஺e(0,+oo),<=ff(x)==0,0%µlj,µ+/1=/஻¶0,BPIna=1-x—lnx,0Q%-xw(O,Xo).f\x)<0,ᦪ/(x)ᓫ⌴8,-xwQo,+<»).,f(x)>0,ᦪ/(x)ᓫ⌴•/(%)2/1᳝))=-ᔊ¹)+=-----lnx04-1—XQ—IHXQ=-------2,1HXQ+1—XQ^\///------2//7XQ—ᑍ஺஺Xog(x)=~——21nx-x,x᧕jᦪg(x)ᙠ(0,+oo)Wᓫ⌴8ºg(1)=1-0-1=0,.-/£(0,1
.g(x)OxG(0,1
.---21PQ»)20,0%h(x)=1—x-Inx,xG(0,1
,.•.஻(x)=T—LீOឤᡂHx.•.¼(%)ᙠ(0,1
Wᓫ⌴8h(x)^h(1)=1-1-M1=O,11⚓½ᐳ13⚓¾

13-x->0.஻(x)->+oo,¿À0=ln\,Á/(X)2lÂÃÄ-¿+/஻Å1,ƶឤᡂH.-X=1.ᨵQ+ᔊQ2l,ᐸÈQ>0.g(a)=஻+/஻஺1,ᑣg'(a)=1+->0,ᑣg()ᓫ⌴ºg(1)=0.aᡠ|a+Ina^lᡂHᑣÉᨵÊË☢Ì-஺Í.f(x)^lᡂH.h{x}=ex-x-\,h\x)=ex-\9ᙠ(YO,O)ᓫ⌴8,ᙠ(o,+8)ᓫ⌴.\/7(x)^/?(0)=l-0-l=0,e*~x—1^0,ᓽ+1,Ïxᣚᡂ/1=Ñvx-l^lnx,/.x-Inx^X.-•/(x)-aeX]~lnx+lna^ex~x-Inx^x-lnx^\,-x=1.ÂÒᡂH.cW8.ijᦪf(x)=aex-lnx-\.(1)x=26/(x)Ḅ᩽elÔÕlḄᓫÖ×{(2)Ì-a[./(x)^0.eூa~௃a(1),ᦪ/஺)=᝞ஹ¿x—l.x>0,fr(x)=aex-11x:x=26/(x)Ḅ᩽e,12⚓(ᐳ13⚓)

14ff(2)=ae2--=0,a=a=—22e2-f(x)=~2eX-/""-I=Jy"~~2e2ex-02.f(x)>09.•.ᓫ⌴8Ö×6(0,2),ᓫ⌴Ö×6(2,+oo).(2)-a!.f(x)^--lnx-i,eexx1eeg(x)=-----lnx-1,ᐺJg'(x)=--------,eexᵫg'(x)=2■•-1=0=x=l,ex-01.gr(x)>0,.♦.x=l6g(x)ḄᨬÜeᦑ-x>0.g(x)g(1)=0.,.-Å/.f(x)=aex-Inx-1^0.e•.•ᦪ/(x)=oe'-/஻Ý1,Þ./'(x)0,ᓽÅ"+1,%>0,ex1,τ——iHX-1g(x)=᝞B,µ|Jg\x)=-——-----x>0,Þ.g'(1)=0,ee-00,-Inx>0,g'(x)>0,x-x>l.--1<0,-Inx<0,g'(x)<0,xg(x)ᙠ(0,1)ᓫ⌴ᙠ(l,+oo)ᓫ⌴8g(x)(g(1)=1ea2,Þ.a^g(x).e...-a21./(x)0.e13⚓½ᐳ13⚓¾