ung dung dao ham tinh don dieu.pdf

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NHN:CHUYNLTH

BigingI.5ngdngcaohmTHHONGNGC SUNDAY,SEPTEMBER30,2012

BigingnythucKhanthiH2013.TholunvtcuhitiyXembigingI.1I.2I.3I.4

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Vicgiivbinlunphngtrnh,btphngtrnh,hphngtrnh,hbtphngtrnh,...bng ccphng phpnh:Bini tng ng,t nph, lnggic ha,hnhhc,...khquenthuciviccbnchunbthivoihc.Tuynhinimtvimtbitondngnyccbntnhiucnlngtng,chatmcligiihocxcnhcnglinhnglikhngaracktqucuicng!

Trongphmvibigingnytimunbn vmtphngphpgiiquytccbitonvphngtrnh,btphngtrnh,hphngtrnhvhbtphngtrnh,...khimccphngphpnutrngpkhkhnhocbtc!l"ngdngtnhniucahms".

Tnh cht 1:Gi s hm s lin tc v n iu trn tp th phng trnhcnhiunhtmtnghimthuc .

Tnhcht2:Nuphngtrnh cmtnghimtrntp thphngtrnhcnhiunhthainghimtrn .

Tnhcht3:Nu lintc,ngbintrn v lintc,nghchbin(hochmhng)trn thphngtrnh cnhiunhtmtnghimtrn .

Tnh cht 4:Nu hm s lin tc v n iu trn th vi ta c:.

Tnhcht5:Nu niutrn th lnghimcahphng

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Tnhcht6: ngbintrn thnghchbintrn th vimi .

Chcnnmcnhnghavhmsngbin,nghchbinccbnddngsuyratnhcht1,2,3,4v6.Ringtnhcht5SGKkhngcp,domikhisdngktqunyccbnphichngminhli.TisnichitithnvngthichngminhtnhchtnytrongVd2.2caVn2!

ReadMore

Vn1.ngdngtnhniucahmsgiiphngtrnhVd1.1.Giiccphngtrnhsau:

a)b)Nhnnh:ivicu1,cthbnnghnvicbinitngnghocsbnh

phng,tuynhinbnsgpkhtrongbini.Cu2,ccphngphp"truynthng"khngkhthi.

Nubnchukhquanstvchuynvnginthvtriulnhnghmsngbin(trnmttpno).Lcny,sdngtnhniugiiquytbitonnyratrongubn.Vncnlilonnghim!Cngvicnykhngkh,nhngnubncthtngsthsmtthigian.Hyutinnhnggitrca saochoccbiuthcdiducnnhngitrlschnhphng!Ligii.a)iukin:PhngtrnhchoXthms trn .Tathyrng:+Hms lintctrn+Cohm vi .

Dohm ngbintrn .+Ktlun: lnghimduynhtcaphngtrnhcho.(Xemtnhcht1!)b)iukin:PhngtrnhchoXthms trn .Tathyrng:+Hms lintctrn+Cohm vi .

Dohm ngbintrn .+Ktlun: lnghimduynhtcaphngtrnhcho.

Vd1.2.Giiphngtrnh:Nhnnh:CngvitduynhtrongVd1.1nhngskhkhngnhchmslin tc, niu trnTX ca n.Tuynhin, quan st k th thycc biu thc

diducn2vcchungmtmilinh:

Do,nut v thphngtrnhchotrthnh:

PLUS.MATHS.ORG

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ny,tachvicxtmthmscdng vvndngtnhcht4,bitoncgiiquyt!Ligii.Tpxcnh:t v thphngtrnhchotrthnh:

Xthms trn .Tathy:+ lhmlintctrn

+Cohm trn nn lhmngbintrntrn .

Do$$f(u)=f(v)&\Leftrightarrowu=v$$

Ktlun:Nghimcaphngtrnhl: v .

BitpBitp1.1.Giiccphngtrnhsau:

a)b)c)d)e)

Bitp1.2.Giiccphngtrnhsau:a)

b)

Bitp1.3.Giiccphngtrnhsau:a)b)

Vn2.ngdngtnhniucahmsgiibtphngtrnh,hphngtrnhVd2.1.Giiccbtphngtrnhsau:

a)b)

c)Nhnnh:Cu1,bnhontoncthsdngphngphpbnhphnghocbin

itngnggii.Tuynhin,timunhngbnnvicsdngtnhniucahmsgiiquyt,tuynhinoncmtnghimcaphngtrnhnymtkhnhiu thi gian(bn ch chnnhng s sao cho biuthc di du cnl s chnhphng).Cu2,cthtnph,nhngbinikhri.Bitonnginnusdngtnhniucahms.Cu3,khphctpvcngcthtnph.SongnuquanstkththycmiquanhtngtnhVd1.2.Ligii.a)iukin:BtphngtrnhchotngngviXthms trn .Ta c hm s l hm s lin tc v c o hm

nn ng bin trn

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Cngphugioha(TrchNhtBnDuyTnBaMiNm)oTrinhNht

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4/1/2015 HongNgcTh:BigingI.5ngdngcaohm

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Do,minnghimcabtphngtrnhchophithamn:

Ktlun:Tpnghimcabtphngtrnhl: .b)iukin:t .

Khi lin tc trn v c o hmnnnnghchbintrn .

Hnna .

Do,tpnghimcaphngtrnhphithamn:

Ktlun:Tpnghimcabtphngtrnhl:c)iukin:Btphngtrnhchotngng:

t v th(1)thnh hay.

Xthms trn .Khi lhmlin tctrn v co hm trn

nnhm ngbintrn .Vtnhngbinnnt suyra hay .Kthpviiukintac:Ktlun:Tphpnghimcabtphngtrnhl: .

Bitp2.1.Giiccbtphngtrnhsau:a)b)c)d)

Vd2.2.Giicchphngtrnhsau:

a)

b)

Nhnnh:Hthnhtbinivchnxtmthmsctrng.Hthhaicdnghonvvngquanh:

Gi s v cng ngbin (hoccng nghchbin) trnmin th khi ,nulnghimth .

Ligii.a)TacXthms vi .

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Khi l hm s cp nn lin tc trn TX ca n, hn na o hmnn tngtrn

DoHchotngngvih:

Ktlun:Hcnghimduynht:

b)Xthmsctrng .

Khihchotrthnh:

Vc: vimiVy lhmsngbintrn(Chngminhtnhcht5btutonny!)Giis lmtnghimcahvythchngphithamnh(*)!Khngmttnhtngquttacthgis .

Vy .

(Ktthcchngminhtnhcht5)Thayvohcho,tac:

Ktlun:Hcnghimduynhtl:Ch:Khithay trlihtathucmtphngtrnhbc3vin vvicgiiphngtrnhkh ddng.Tuynhin,khigiiphngtrnh nygpkhkhnthccbnnhthm,khost,onnghimvkhngnhphngtrnhcnghimduynhtnhtrnhbyVn1.CcbncthxemBitp2.2cud)!

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a)

b)

c)

d)

e)(D2006).Chngminhrngvimi hphngtrnhsaucnghimduynht

Vn3.ngdngtnhniucahmschngminhbtngthc

I.TMTTLTHUYT:Nh cc bn bit ,tnh n iu ca hm s ph thuc vo o hm ca hm s.Dngohm,ta cthxtc tnhngbinvnghch binca1hm strn1

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min no ,do chng c th ng dng chng minh kh nhiu Bt ngthc(BT).Taxtphngphpcthnhsau:Xthms trnon .* Nu hm s ng bin trn .Suy ra:

* Nu hm s nghch bin trn .Suy ra:

Lu:Khitasdngiukin hay thtaphimbophngtrnh chxyratihuhnim,tclphngtrnhnychchuhnnghimmthi.Lutrncbitquantrngkhitaxtncchmslng gic.iu ny cng d hiu bi khi ta cp n nghim ca 1 phng trnhlnggic,tachsdngkhinimtpnghimbiudinccgitrthamnphngtrnhlnggicchotrc.Nimtcchnmna,phngtrnhlnggicluncvhnnghim.DokhitamunchngminhccBTlinquanncchmlnggicphctp,ta phi s dng n phng php i s ha,vn s c trnh by trongchuynlnggic.

II.VDVBITP:Vd3.1:Cho$0

4/1/2015 HongNgcTh:BigingI.5ngdngcaohm

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nhhnggii:Taphingithuyt l3gctamgicnntacththay,nhvyc2vcaBTuxuthinccshngchaA,B,C,nnta

cthnhmBTvdngsau: vi.Taschngminh: .

Ligii:DotamgicABCnhnnn .Xthms$f(x)=2\sin{x}+\tan{x}3x\left(0

4/1/2015 HongNgcTh:BigingI.5ngdngcaohm

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Vd3.5.Chngminhrng: (1),vimiLigii.(1) .t ,taschngminhhms ,viTac: lhmlintctrn v$$f'(x)&=2x\frac{2}{x}=\frac{2(x^21)}{x}$$

trn .Lpbngbinthincahm trn nhsau:

Tbngbinthinsuyra

Vd3.6.Chngminhrngvimisthc ,tauc:

Ligii.Vi ,btngthc(1)tngngvi:

Xthms trnTathyhm lintctrn vc:$$f'(x)&=\frac{1}{x}x^{\frac{1}{2}}+\frac{1}{2}(x1){x^{\frac{3}{2}}}=\frac{2\sqrtxx1}{2x\sqrtx}:=\frac{g(x)}{2x\sqrtx}$$Duca lduca vi ,viHms lintctrn vc .Dohm nghchbintrn v suyraTtac nnhms nghchbintrn v

Vy:

Vd3.7.Chngminhrngvimi ,tauc:

Ligii.Xthms trnkhongTac vimi

Dohms ngbintrnkhongTsuyra vimi

tcl vimiMtkhctac\begin{align*}\frac{{\left|a\right|+\left|b\right|}}{{1+\left|a\right|+\left|b\right|}}&=\frac{{\left|a\right|}}{{1+\left|a\right|+\left|b\right|}}+\frac{{\left|b\right|}}{{1+\left|a\right|+\left|b\right|}}\\&\leqslant\frac{{\left|a\right|}}{{1+\left|a\right|}}+\frac{{\left|b\right|}}{{1+\left|b\right|}}\quad(2)\end{align*}T(1)v(2)tac

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4/1/2015 HongNgcTh:BigingI.5ngdngcaohm

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bivy

Vy,hms ngbintrnSuyravi th .

Bitptluyn:Bitp3.1.Chngminhrng:

a) vimib) vimi

Bitp3.2.Chngminh:

Bitp3.3.Chngminh:

Bitp3.4.Chngminh:

Bitp3.5.(D2007).Chngminh:

Vn4.ngdngtnhniubinlunphngtrnhvbtphngtrnhBitonvbinlunsnghim,cnghimtrnmttphpchotrclbitonm

hunh trongthi nmnocng c.Ngoivic yucuhc sinhnmvng cckinthccs,bitoncnihictduy,sngto.ividngtonnycngcnhiuphngngiiquyt,trongphngnsdngtnhniucahmsschotaligiingngn,covcnhgirtcao!

Vd4.1.Chohmsa)Tm phngtrnh: cnghimb)Tm btphngtrnh: nghimngc)Tm btphngtrnh: cnghimNhnnh:Mtcchrttnhin,khibinlunphngtrnhhocbtphngtrnhc

chathamsthvicutincnlmlcgngathamsvmtvclp.Sausdngcctnhchtniucahmsmbnbit.tinisnh,tixinnhclimts"hngngthc"sau:

a)Nghimca phngtrnh lhonh giaoimcath hmsvith

b)Nghimcabtphngtrnh lphnhonh tngngviphnth nmphatrnsoviphnth

c)Nghimcabt phngtrnh lphnhonh tngngviphnth nmphadisoviphnth

d) Nghim ca phng trnh l honh giao im ca ng thngvith

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e)Btphngtrnh ngvi

f)Btphngtrnh ngvi

g)Btphngtrnh cnghim

h)Btphngtrnh cnghim

Ligii.a)Biniphngtrnh saochothams nmvmtvclp:

t

cnghim th philnhnhocbngGTNNvnhhnhocbngGTLNcahm trnon (Xem"hngngthc"(d)vhnh1),tcl:

Bygixthm trn tngtnhvn1,cthnhsau:V lhmlintctrn ,cohm

nn lhmnghchbintrn .

Do,$$\mathop{min}\limits_{x\in[12]}g(x)=g(2)=\frac{3}{8}\\\mathop{Max}\limits_{x\in[12]}g(x)=g(1)=1$$Ktlun:Gitr thamnbitonl:b)Vi ,tac:$$f(x)\leqslant0\Leftrightarrowm{x^2}+2mx3\leqslant0\\\Leftrightarrowm(x^2+2x)\leqslant3\\\Leftrightarrowg(x):=\frac{3}{x^2+2x}\geqslantm,\forallx\in[14]$$ nghimngvimi th ,iunycckhi

(xem(e)vhnh1)

Vhm lintcvcohm trnon[14]nn l

hmnghchbintrn .Do

Ktlun:Gitr thamnbitonl:b)Vi ,tac:$$f(x)\geqslant0\Leftrightarrowm({x^2+2x})\geqslant3\Leftrightarrowm\geqslant\frac{3}{x^2+2x}\\(1)$$tvxthms (2)ytagpchtrcrilon chaim"cbit"mlmcho khngxcnhlim .Kthutltastchralmhaionvixtringim .Cth:Xtcckhnngsau:Nu ,thaytrctipvo(1),btphngtrnhtrthnh .Vnghim!Nu th

$$\text{btphngtrnhcho}\Leftrightarrowg(x)\leqslantm\text{cnghim}x\in(03]\Leftrightarrow\mathop{min}\limits_{x\in(03]}g(x)\leqslantm$$Do nghchbintrn nn

Nu th n=""span=""style="fontfamily:arial,helvetica,sansserif"datamcestyle="fontfamily:arial,helvetica,sansserif">$$\text{btphngtrnhcho}\Leftrightarrowg(x)\geqslantm\text{cnghim}x\in[10)\\\Leftrightarrow\mathop{Max}\limits_{x\in[10)}g(x)\geqslantm$$Do nghchbintrn nn .

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Kthpcctrnghptainktlun:

Vdsauytngtnhcu1caVd4.2,cungcpchoccbnmtkthutkhigpkhkhntrongvickhngnhtnhniucamthmsbngcngcohm.

Vd4.2.Tm pt: cnghim.

Ligii.iukin:Biniphngtrnhchovthm nhsau:

Vic tnho hm lkh phctp, tuynhinta dsuy ra v lhmtngtrnTX,cn vlhmgim(tcl vlhmtng)trnTX.

Suyra lmthmstngtrnon .Nhvy:

cnghim

Ktlun:

V d 4.3. Tm phng trnh cnghimthc.Ligii.iukin:Biniphngtrnh(1)vxthm nhsau:

Cthtnph th ,vi .

Khi(2)trthnh: .lmthmsbchai,vickhostkhngcgkhkhn!\parTac .Lpbngbinthinchohm nhsau:

Nhnvobngbinthinddngsuyragitrca cntml: .Ghich:

a)ivi nph trn, ccbn bucphitmmingi trcan, chc stbcny.

b)BitonnythikhiA2007vcnmtscchgiikhc,ccbnttmhiuthm!

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4/1/2015 HongNgcTh:BigingI.5ngdngcaohm

http://hoangngocthe.blogspot.com/2012/09/baigiangi5ungdungcuaaoham.html 13/15

Ligii.t th(1)trthnh:$$mt^2+4(m1)t+m1>0\Leftrightarrowm(t^2+4t+1)>4t+1\\\Leftrightarrow\frac{4t+1}{t^2+4t+1}

4/1/2015 HongNgcTh:BigingI.5ngdngcaohm

http://hoangngocthe.blogspot.com/2012/09/baigiangi5ungdungcuaaoham.html 14/15

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nghim.

Bi tp 4.3. Tm bt phng trnh: nghimng

Bi tp 4.4. Tm bt phng trnh:ng .

Bi tp4.5. Tm phng trnh: cnghim.

Bi tp 4.6.Chng minh rng: Vi mi phng trnhluncnghainghimphnbit.

Bi tp 4.7. Tm phng trnh: cnghainghimthcphnbit.

Bi tp 4.8. Tm bt phng trnh: ng vi

Bi tp 4.9.(A2002).Tm phng trnh:

ctnhtmtnghimthucon

Bivitlinquan:BigingI.2Cctrcahms(phn2)BigingI.2Cctrcahms(Phn1)BiI.1TnhniucahmsTmiukincathams hms niuBigingI.7BitonkhongcchBigingI.6BitontiptuynBigingI.4GTLNGTNN(TrnTrungKin)BiI.3Stnggiaocaccth

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4/1/2015 HongNgcTh:BigingI.5ngdngcaohm

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