TAP CHi KHOA HQC - BAI HQC BONG NAI, SO 03 - 2016 ISSN 2354-1482 VAN DVNG TRI THtTC HAM DE GIAI M Q T SO BAI TOAN
Of P H 6 THONG
TS. Bink Quang Mink^
TdM TAT
Cku de hdm duge xudt hipn xuyin sudt tiong chuang trinh Todn hge phd thdng, vi vgy viic van d^tng tri thirc hdm di gidi lodn vd thdng qua dd ren luyen ky nang gidi todn Id rdt cdn thiit. Bdi viit tap trung vdo viic chi ra nhieng dgng todn & phd thong CO thi gidi dugc nha van dung tri thue hdm vd niu ra cdc dinh hu&ng giup gido viin cd thi hu&ng ddn hgc sinh tap luyin nhdm hinh thdnh mpt sd ky ndng gidi todn nh& vdo viic van dting tri thuc hdm. Cdc dgng todn ndy cd thi dua ra tic l&p 10 vd no xudt hiin khd nhieu trong cdc di thi vdo dgi hgc, cao ddng vd trung hgc phd thdng qudc gia.
Tir khda: Tri thtrc hdm, ky ndng gidi todn. npi dung, y tu&ng, hoqt dpng Trong nhidu nara ttd lgi ddy vide todn d phd thdng.
xudt hien nhieu bai toan (BT) kho ttong cac ky thi vao Dgi hgc & Cao ddng hay ky thi tmng hgc phd thdng qudc gia md vide gidi nd nhieu luc phai v$n dgng cdc kidn thftc ve hara dang khd phd bidn.
Cac BT ndy lien quan den cdc vdn de nhu; gidi hay bien luan phuong trinh (PT), he PT, chimg minh bdt ddng thuc..., day Id nhung BT Dgi so tuy nhien khi gidi nd thudng vdn dung kidn ttiuc cua Gidi tich. Vi thd khi day hgc, gido vidn (OV) cdn cd nhiing BT ma viec gidi nd phdi van dgng kien thftc lidn phdn mdn. Cdc dgng toan nay cd thd xudt hidn tii ldp 10, khi rad mgt sd tii thuc ham (TTH) dugc ttang bi kha ddy dft. Ndu dugc tap luyen sdm va ed chu dinh thi se hinh thanh cho hgc sinh (HS) mdt sd ky ndng (KN) giai toan nhd vao vdo vi?c vgn dung TTH. Van dd la GV cln chii y nhung dgng toan nao cd thd gidi dugc nhd van dgng vao TTH? Va cd nhiing hudng ddn ndo de giup HS bidt van dung TTH vdo gidi
'Tmong eai hgc Dong Nai
1. Mdt so dang toan d pho thdng cd the giai dirge nhd vdn dung tri thurc ham
Q phd thdng cdc BT gidi dugc theo hudng van dung TTH cd the chia so bd thanh hai loai: Logi 1: Nhiing BT cd ndi dung de cap trgc tiep den ehu de ham, chdng hgn nhu: BT khao sat su bien thien cua hdm sd, chftng rainh radt diera ndo dd thudc hay khdng thudc do thi cua radt hara sd da cho,... Xoa/2: Nhung BT nhin be ngodi khd cd thd nhan ra cac radi lien he ddn hara. Nhiing BT thudng cd chfta radt frong cdc dgc trung cfta hdra:
tuong ling, bien thidn, phu thudc (cd the tiidng rainh, hay khdng tudng minh), Chang ban nhu: BT chung minh bdt dang thftc thdng qua vide van dgng tinh don dieu cua hdm sd, hay khai thde dac tnmg tuong ftng, bidn thidn phg tiiudc dd gidi todn.
Vdi loai BT thu nhdt HS de dang hon frong vide van dung TTH dd gidi 103
TAP CHJ KHOA HQC - BAI HQC BONG NAI, SO 03 - 2016 ISSN 2354-1482 chftng, bdi le khi gidi loai toan nay HS
da dugc ddt frong "tmh hudng ham".
Ddi vdi loai thft 2 thi khdng dd dang nhu vdy, ldm the nao dd HS cd the gidi duge ragt BT nhin be ngoai khd nhdn ra mdi lien he vdi hdm bang each van dung TTH? Dd gidi quydt van de nay thi frong qua tiinh dgy hgc thdng qua cdc chu de (true tidp hay gian tiep lidn quan ddn TTH) GV cd thd dua vao mdt sd dinh hudng giup HS van dung TTH frong gidi todn, ddng thdi xera ddy la co sd de GV hudng dan HS gidi cdc bdi todn lidn quan den TTH.
2. Mgt so dinh hudng giiip HS vdn dung TTH vao giai toan nham phat trien ky nang giai toan
Dinh hirdng 1: Tap trung vdo hu&ng ddn HS van dung dugc TTH vdo viic gidi nhicng BT da dgng md be ngodi tuang chieng khdng cd lien quan gi din TTH, tudn theo qui trinh: Lua chgn npi dung — Hinh thdnh y tu&ng - Thuc hiin hogt dpng.
Theo [1] "Dd hge dugc radt KN, HS can biet chftng ta frdng chd d edc em phdi cd khd nang lara gi, va ldra nhu thd nao (ldra chi tiet). Cdc era phdi bidt vi sao lam cdch do la tdt nhat, cung vdi nhiing thdng tin phft hgp (giai thich).
Cdc era phai cd co hdi thuc hanh (sft dgng), dugc kiem fra vd hieu chinh ddi vdi vide thuc hdnh do". Ndi each khac, frong vide hinh thdnh KN can tuan theo theo qui trinh "Lua ehgn npi dung - Hinh thdnh y tuang — Thtrc hien hogt dpng" Ve Iga chgn ndi dung GV can tap trung vdi nhfing dang todn loai 2 (co thd tft ldp 10), dd hinh thdnh y tudng GV nen tap HS tg frd ldi cdc dgng cdu hdi nhu: Cd the gidi BT theo hudng van dgng TTH dugc khdng? Mudn the can sft dgng nhiing TTH nao? Cdn viec thuc hien boat ddng thi can xay dung ragt he thdng cau hdi thich hgp de ggi y cho HS nhdm phdt hien ra nhung TTH an chfta frong BT va van dung nd de cd the gidi dugc BT. Cac cau hdi hogc do GV ddt ra, HS thao luan tta ldi hay cd su hudng din cua GV, hogc HS tg dat ra vd tu trd ldi. Sa dd va mdi lien he cua quy trinh nhu sau:
104
TAP CHi KHOA HQC - BAI HQC BONG NAI, SO 03 - 2016 ISSN 2354-1482 Ni dui (YDiy. Gidi PT:
> / n ^ - . ^ ^ = V 9 J + 7 - r / ^ (1)
• Ndi dung: Day dgng todn thudng gap d ldp 10, HS cd thd gidi theo each thdng thudng, kdt qud x = 2.
• Y tiidng: Dung TTH dd giai BT theo cdch khac (HS l&p 10 dd hgc dinh nghta PTvd tgp xdc dinh cua PT).
• Hogt dgng: Lieu cd nhirng TTH nao dugc hge ed lien quan den BT? Cd the xem ve frdi vd phai cua (1) la cac ham sd duge khdng?... Tft nhiing ggi y dd de HS xem (1) dudi dang f(x) = g(x).
Khi da quan niera mdi ve la ragt hara sd thi ggi cho ta viec dau tien can lara la xet tap xac dinh cua chiing. Tft do hudng HS tdi each giai: xet
f(x) = ylnx+3- -X va
g(x) = f9x+7-ylx-2, do D^nDg= {2}, ndn (1) cd nghiem duy
nhat X = 2.
VD2: GidibdtPT: •Jx^-4x^<'& (1)
• Ngi dung: Day la ragt BT frong SGK todn 10 hien nay, bdi nay HS cd ttid giai theo cdch giai thdng thudng, kdt qua X > 6 .
• Y ttrdng: Dung TTH dd gidi BT theo each khac (HS l&p 10 dd hgc tinh dan dieu cua hdm sd
y= yfx.y= -Jx+bf
• Hoat dgng: Lieu cd nhung TTH ndo dugc hgc cd lien quan ddn BT? O day cln cd thdra budc hogt dgng chudn bi dd HS cd dugc tti thftc vd tinh dong bidn cua ham sd dang: y
Tft dd, ggi y cho HS idra sao xudt hien dugc dgng tdng cfta hai can bac hai frong BT? Dd hudng HS tdi (1)
• » 7 x - 2 + V x - 6 > - (2). Cd thd 2 ' xem vd frdi cfta (2) Id mgt hdra sd dugc khdng? ... , tft nhiing ggi y dd, dmh hudng HS tdi each gidi: Dat f(x)
•Jx-l + 'Jx-e (Df = [6; +00)), do f ddng bien fren D nen Vx > 6
< = > / ( x ) > / ( 6 ) = 2 > - . Vay nghidm cfta bdt PT la X > 6 . ^
Ldi gidi sau eua BT cd the khdng ngan ggn bang ldi giai ddu, tuy nhidn dieu dd cd the giup HS phat tridn KN gidi todn dd la van dung tinh don dieu cua hdm so di gidi bdt PT. Ndu HS dugc tap luyen nhieu bang nhiing BT tuong tu thi KN dd trd thdnh thudn thuc, luc dd HS cd thd giai dugc nhftng BT khd han, ehang ban; Gidi bdt PT
-Jx-2 +V2JC-I-3 +V3X-I-7 >8.
VD3: Cho ba sd duang a, b, i,, trong do a > c, b > c. Chung minh rdng yje(a ~c)+ yje(b -c)< 4ab .
• Ndi dung: Ddy Id BT chftng minh bat dang thftc, nhin bd ngodi khd cd thd nhdn thdy radi lien he vdi nhiing TTH.
Thdng thudng HS se gidi theo hudng van dgng bdt ddng thftc Cd — Si hay Bunhiacdpski.
• Y tirdng: Dftng TTH dd giai BT theo each khac (HS l&p 10 dd hge vecta, bieu thicc tich vd hu&ng cua hai vecta, cd kit qud M.V< « . vh.
• Hogt ddng; Ddy la BT rdt khd dd tim ra mdi quan he giua TTH vdi nhiing gia thiet da cho. Tuy nhien cd
TAP CHI KHOA HQC - BAI HQC BONG NAI. SO 03 - 2016 ISSN 2354-1482 the lgi dgng su tuong ftng giita cap sd
thuc (x;y) v&i mpt vecta. giiea mpt so thuc ndo do v&i tieh vo hu&ng eua hai vecta, giica mpt so thuc ndo dd v&i dp l&n ciia mpt vecta. Tft hinh thuc cua BT, do M.V< M.V, lidu cd the ddt nhflng tuong ftng giua nhftng cap sd frong cac sd a, b, c v6i w va v sao cho M.V la ve frai, edn M . v Id ve phai cfta BDT? Y tiidng dd cd tiid giftp HS tim ra: M = ( - J a - c ; yjcy, v = ( V c ; y j b - e ) , dd H . i'
..Jc(a -e)+ .^c(b - e) vd lul.lvl=
Qua cdc VD tren (tft rauc do de ddn khd cfta su xudt hien TTH frong BT), cho ta thdy hai budc: Hinh thanh y tudng (van dung nhflng TTH nao) vd thgc hidn boat ddng (tta ldi nhiing cdu hdi nao de tim ra radi quan he gifta y tudng va ngi dung BT) la quan frgng nhat. Tft do chung tdi cho rang vdi HS, ngoai viee dugc frang bi rapt sd TTH co ban, thi dieu mdu chdt la tu HS (hoge cd su ggi y eua GV) tim ra dugc nhung TTH an chfta frong BT tft do cd the tira ra cdng cu thich hgp lien quan den TTH dd gidi. Cac ky thudt: "Phdt hiin hodc thiit lap cdng nhu nghiin cieu vd lgi dung nhirng su tuang img, biin thiin.
phu thupc " giua cac yeu td frong BT la rat can thiet. De lam dugc didu nay HS phai dat ra cho rainh nhiing cdu hdi dudi ddy: V&i moi phdn tic cua tap hgp ndy ed chdng mpt phdn tu tuang ieng duy nhdt lien hi vai no thupe tap hgp kia? Lieu cd the bieu dien su tuang ung dd phdt hien duge bdng edng thiec cua nhirng hdm sd quen thupc khdng? Lieu
CO thi tan dung cdc tinh chdt eua hdm sd nhu dan diiu. dd thi, gid tii l&n nhdt, nhd nhdt,... di gidi BT dugc khdng?
V&i nhirng BT cd chiea nhieu dgi lugng biin thien, lieu ed thi tdn dung gid thiet vd cdc tinh chdt ehung cua no de qui ve mpt biin? Lieu ed thi chuyen hda npi dung vd hinh thuc BT vi BT m&i cd lien quan den TTH (dgt dn phu, "phiin dfch" theo ngdn ngir hdm,...)? Viec tu tta ldi cae eau hdi tten hy vpng giup HS nhdn ra edc yeu td hara frong BT.
Dinh tairdng 2: Xdy dung mpt quy trinh gidi phu hgp, tiong dd chii trgng xdy dung vd truyin thu tri thuc phuang phdp.
Thdng thudng cd ba giai doan dd gidi BT tiieo hudng van dgng TTH.
Giai dogn 1: Nhin nhdn BT ban ddu theo hudng van dung TTH, rdi "phien dich" thdnh BT radi raang mdu sac
"ngdn ngu hdra". Trong do chu frgng phdn ket ludn cua BT ban ddu phai
"phien dich" sao cho sdt vdi nhirng TTH nhara tgo thudn ludn lgi eho hudng tira Idi gidi BT mdi. Giai dogn 2: Su dung cdc TTH dd gidi BT radi.
Giai dogn 3: "phien dich" ngugc lai, tftc la frd ldi nhung yeu eau cfta BT ban ddu.
Vide binh thdnh cho HS nhung BT ralu cung nhu dd ra dugc nhflng PP cu the cho timg logi BT Id vide lara rdt can thidt, cd nhu thd HS mdi cd the "bat chude" do gidi cac BT tuong tu. GV phai phdn tich kheo leo de bat ra dugc y tu&ng, xdy dung cdc cdu hdi thich hgp di hu&ng HS van dung TTH vdo gidi BT. Cdn chu y den vide hudng ddn HS
TAP CHi KHOA HOC - BAI HOC BONG NAI, SO 03 - 2016 ISSN 2354-1482 tu chuyin hda npi dung vd hinh thicc BT
nhdm tim ra cdng cu thieh hgp lien quan din TTH di gidi todn. Cd tiid kd mdt sd dang chuydn ddi BT thudng ggp;
Chuyen ddi BT bdng phuang phdp (PP) dgt dn phu; Chuyen BT chimg minh BDT thdnh BT iim gid tri l&n nhdt, nhd nhdt cua hdm sd; Chuyen BT tim gid tri l&n nhdt vd gid tri nhd nhdt ciia hdm sd vi BT chung minh BDT;...
Xdy dtfng vd truyin thu nhirng tri thuc PP- Viec xdy dung cdc tri thuc PP cd the xuat phdt tft nhiing tri thftc frong gid hgc ly thuydt (nhdn dang tti thftc radi), ciing ed thd tiidng qua cac hoat ddng gidi todn (cd thd giai mgt BT cu the). GV ed tiid xdy dgng va tinydn thu tri thuc PP bang cdch xay dung cac thuat gidi (dftng ldi hay dung so dd), sau do didu chinh thudt giai de HS ndra sau hon bdng cdch phdn tieh va cho VD ddn ddt HS Idem chftng, phat hien ra nhiing sai ldm va giup cae era vugt qua nhiing khd khan dd. Chang hgn, vdi gidi vd bien luan PT, Bit PT, he PT, ta cd luge do giai:
- Bidn ddi PT(Blt PT) vd dgng f(x)
= g(m) hay f(x) > g(my,f(x) < g(m) - Khdo sdt su bien thidn cfta ham sd f(x) fren tap xdc dinh D, tim gia tri ldn nhat, nhd nhat
- Tft bdng bidn thien suy ra edc gid tri ra can tira
Ndu he PT cd dang cd the xet hdm y =
\fM=f(y) f(t) (thudng la ham sd lien tgc frdn tap xdc dinh cfta nd)
- Ndu hara f(t) dan dieu thi ta suy ra X = y vd giai tidp
- Neu hdra f(t) cd mdt cue tri tai x = to, luc dd ta cd X ^ y hogc x, y ndm vd hai phia cua to.
Dd rainh hpa ro net hon, chftng ta xet mdt sd BT frong ky thi tuydn sinh Dai hgc va Cao dang nam 2013 rad khi giai nd cd van dung den TTH.
Bai 1 (CD - 2013): 7?m m di bdt phucmg tiinh
(x-2-m)-Jx—l < m - 4 ( l ) cdnghiim.
LM giai:
(1) « • f{x) = ' , < m .Ydu 1 + V ^ - l
cdu bdi todn < ^ / ( x ) < m cd nghidra Vxe[l;+oo). Dat t^^x-l. t > 0 Iftc dd f(x) frd thdnh g(t) = -/+4
t + l khdo sat hara sd g(t) ta cd m>2.
Bai 2: (Khoi B - 2013).CAo a, b, c la edc sd thuc duang. Tim gi tii l&n nhdt (GTLN) cua bieu thiec
4?+b
LMgiai: Tacd
(a+b)^(a+2e)ib + 2c) <(a+b) a+b+4e
<2(fl^+6^+cO
= •Ja^+b^+c^+4 , vdi t > 2. Lfte do 4 9 . „ P<f(t)^
t 2(r-4)
107
TAP CHi KHOA HOC - BAI HQC BONO NAI, SO 03 - 2016 ISSN 2354-1482
^,/^ -(t-4)(4t'+7t'-4t-l6\ „ 1 I 7 + 10,/?
r (r - 4 ) ^ '4J • " • ' -"^4' 30 4 ( ' - f 7 ( ' - 4 ( - 1 6 = 4(<=-4) + ( ( 7 ; - 4 ) > 0 7 + 10^/5 k h i t > 2 . „ e n f ( t ) = 0 k h i . = 4 . D u a "^n GTLN cua P la / - = Z ± i ^ , kh, vao bang biSn thien f(t) = > / • < - . x = —v = 2
8 2 •
Khi a = b = c = 2 thi /> = ^ . Vay GTLN "**' " J ^ ^ ' '^ - ^ • ' " ) - G'di Aeptocmg
>/r+T-I-</Jc^ - 7 / + 2 = y(l) x' +2x(y-l) + y^ -6y+l = 0(2) Bai 3 (Kh6i D - 2013). Chox, y la cdc LM giai: DiSu kien j : > l . Ttr (1) ta co
^ / I + T - ^ < / ^ = 7 ( / + l ) + l + ^ ( / + l ) - l (•). Dat / ( r ) = ,/F+T-i-</r-r thi f d6ng bi^n V/e[l;-Ko), tir (*) ta co f('^) = f(.y'+i)<^x = y'+'l, thj vao (2) ta tim duoc nghiem cua he la (1;0);
(2;l)
B4i 5 (Khoi A - 2003;.- Cho x, y. z la ba sd duang vd x+y + z<l, chtrng
minh rdng SO thifc ducmg thda man diiu kien
xy<y~l. Tim GTLN cua biiu thicc p _ x + y x-2y
^x^-xy+3y^ 6(x+y) Ldi giai: Ta cd
w~-
± + 3 6 ( - + l ) y yx v < y - l » - < j = - ( — - y + - y y y y 2 4 y 4 = - = > 0 < / < - .
y 4 LM giai: Dat
u = (x;-);v = (y;-);w = (z;'-) x y z
f\t)
^t'-t + 3 6(t + l)
2,1 + 1 1 fit), tet CO
'2^(t'-t + 3)' 2{t + \f' -3t + 7 ^ 8 ^ 5
> i ' = M -I- V-t-U> H-f-v-t-\M hay
-f'
X y zV r e ( 0 ; - ] = 4
2^(1'-t+3f 27
•-ff^^^iS^
(xyzf1 Dat ( = lj(xyzf
2(t+\f 2
f (t) > 0 vi the fl;t) la ham dong bien > 0 < r < ( ^ ^ ^ ^ ^ ) ' < i . Xet ham 3 9
TAP CHi KHOA HOC - BAI HOC BONG NAI, SO 03 - 2016 ISSN 2354-1482 f(t) = 9t + -;Vte(0;~]do t(t) nghich
bi6n V / £ ( 0 ; i ] ^ / ( r ) > / ( i ) = 8 2 nen P>.Js2 (dpcm).
3. Mdt so dang loan lien quan 3.1. H^ hoan vi
Gia su CO he ; fM = g(x,) giai he f(xj = g{xj) dang nay ta dua vao tinh chit (TC) sau:
TCI; Neu f(x) va g(x) la cac ham ciing tang hoac cung giam tr6n tap xac dinh va (xj, X2,...,x^) la nghi?m cua he tren tap xac dinh thi x, = x., = ... = x . 1 z n TC2: Neu f(x) va g(x) khac tinh don dieu tren tap xac dinh va (x^,X2,.-,x^) la nghiem cua h& tr8n t3p xac dinh thi X, = X., — ... = x„ n8unle;
n-1 , VD4: Giai he:
•Jx' -2x + 6 log3(6 - y) = V y ' - 2 y + 61og3(6-z) = sjz' - 2 z + 61og3(6-a;) =
l0E3(6-y).
log3(6-i)- 4z''-2z*6
Trong do
/ ( i ) = l o g 3 ( 6 - i ) ; g{t)
•it^ -2t + 6 voi t e ( ^ » ; 6 ) ; Ta c6 f{t\ la ham nghich bien va g{t\ c6
9'{t)-
V i e (-co; 6) => 5(i) la hara ddng bien.
z^ x = y = z .thay vao he ta cd l o g 3 ( 6 - x ) = PT ndy co
Vx^ - 2x + 6 nghiem duy nhdt a; = 3
=>a; = T/ = z - 3 . 3.2. IJng dung ham l^i TCI (Bdt ddng thiec tiep tuyin) Cho ham so y = f(x) lien tuc va cd dao ham ddn cdp hai fren [a;b] .
a) Ndu / "(a;) > 0 Va; e [a; b] thi f{x)>f\x^){x-XQ) + f{xQ) Vxo€[a;6]
b) Ndu / "(a;) < 0 Va; e [o; 6] thi f(x) < f'(xQ){x - XQ) + fix^) VXQ e[a;&]
TC2 (Bdt ddng thiec cdt tuyin) Cho hdm so y = f(x) lien tuc vd c6 dao hdra den cap hai fren [a;b] .
a) Ndu / "(x) > 0 Va; e [o; b] thi / ( a : ) > M z M ( 3 ; _ a ) + / ( a ) yx^e[a;b]
a-b
b) N6u / " ( l ) < 0 V i e ( a ; 6 ] thi
/(:,) < MzM (^ _ „) + /(„) y ^ [ J]
a-h \) '
109
TAP CHI KHOA HOC - BAI HQC BONO NAI, SO 03 - 2016 ISSN 2354-1482 DSng thirc trong cac BDT tren co khi va BDT c6 dang sau; Cho chi khi a: = a hoac 1 = 6. a , e D c f i , i = u i thoa man VD5 (Vd dich Todn Ba Lan 1996): Cho "
3 2 . . s ( ' ' , ) i ( i ) n g ( ' n ) vcri m thuOc D, a,i},c> va a + b + c = l. Chiing '='
minhrSng: chiing mmh ring j ; / ( a , - ) > ( < ) n / ( m ) .
1=1
a 6 c 9 Be giai loai todn nay, ta di tira cdc sd
~^ "*" ~^ "*' "^ ~ 7^ • thuc a, b sao cho dd thi hara sd y = f(x) a + I 0 + 1 c + l ^jgp y^^^ ^^ ^g ^ j j j ^ so y = ag(x) + b HDG: Ta thay ddng thue xay ra khi tgi XQ = m. Sau dd ta chftng minh dd thi
\ ndy nara dudi do thi kia frong mdt a = h = c^-\a BDT da cho cd dang: khoang hay dogn ndo dd. Cd tbd chia ra
mdt sd dang eg the nhu sau;
/W-^/W-^/(c)<^
^ 2 + 1 4 2 Tiep myen cua do thi ham so y = f{x) tai diem c6 hoanh do 1
363;-I-3 50 Taco:
361-f 3 50 f(x)-
50
trong do £,^„g y. g^j ,g^„ ^^ gj^ ,;„-^, i^„g j ; „ j phuang cdc bien bdng hdng sd:
VD6. Cho cac s6 a,b,c duong thoa man a^ +b^ +c^ =1, chimg minh rSng
a b c ^3^
la b'+c' a'+c' b'+a'~ 2 HDG; Do a,b,c s ( 0 ; l ) . Ta co
a b c _ a b c b' +c''^a' +c''^b'+a' ~T^ T^ 1-c' Tli do ta tun hai so m, n sao cho do thi ham s6 y = mx^ +n nam phia duoi do
: 3 5, 50(3:^+1) * 4 2
thi ham sd y = j frong l-x
(0;I)va tiep xue nhau tgi XQ = — . Tft Vay:
, 36(a + 6 + c) + S 9 . - l-;c
= —didu kidn tidp xue i , 10 ' ^ \ x^+\
= 2mx
dpcm. [(1-x^) 3.3. ITng dung do thi Js , ^ 3^I3
nghiem x^ = — , ta tun dugc m = ——
Neu dd thi y = f(x) va y = g(x) tiep xuc 3 2 nhau tai xo thi tdn tai mdt khodng (a;b) va n = 0 . Ta chftng minh
chua Xo sao cho fren khoang dd, do thi x 3 - ^ > ^ ,„ , , . . , -_- J .o; .*- .1,- 1- - ' >——X , V x e ( 0 ; l ) , tu do ta CO nay nara duoi do thi laa nen l - x^ 2 v ' ^ >
f(x)>(<)g(x),VxG(a-,b).' Sft dung ^^^^
TC nay ta cd thd chftng minh radt sd UO
TAg^CHi KHOA HOC - BAI HQC PONG NAI, SO 03 - 2016 ISSN 2354-1482 Dgng 2: Bdi todn co gid thiet tich cac VD8; Cho cac s6 a,b,c duong , chiing
bien bdng hang so: minh ring VD7: Cho cac s6 a,b,c duong thoa man " ' - * ' ^ b'-<:' ^ c'-a' ^ ^
abc=l, chiing minh rSng ; a + 3b b + 3c c + ia"
.ll + a V1 + * irang
c ^3^
.J\ + c~ 2 HDG; Tvr abe = 1, ta c6 Ina -^ hib -l-lnc
= 0. Ta tim hai sd m, n sao cho d6 thi ham so y = m\nx+n nSm phia duoi do thi ham so y= trong khoang
Vl-l-;c
(0;-f«))va tiep xiic nhau tai jc„ = 1 . Tir
= m\nx+n
HDG: Ta tun hai so m, n sao eho do thi h^m so y = mx^ + n nSm phia dudi d6
; c ' - l
thi ham so y = trong khoang x-\-3
dieu kien tiep xuc
2y](l+x)' X CO nghiem x„ = 1, ta tim duoc
Jl + x 2 + x
(0;+co)vd tidp xftc
dieu kign tidp xue • nhau tgi x^
x'-l
= mx 2 x ' + 9 x ' +
(;c + 3)^
cd nghidra x^ = 1, ta tim dugc m
« = — . Ta chftng rainh - 3
= 1. Tft + n
^2mx
= - va 3 , 8
~ 4yl2
3 1 va « = —= . Ta ehung rainh
•v2 X 3 1 - j ^ = > — i - h i x + - = , V x G ( 0 ; + o o ) b Ti+x 4V2 yfl ang cdch xet hara sd
X - 1 3
- > - ( x ^ - l ) , V x e ( 0 ; + o o ) , t f t d d t a x + 3 8
cd dpcra.
4. Mdt so bai todn vdn dung Bai 1(HSG QG -1996 ): Hay bidn ludn sd nghiera thuc x,y ciia he phuang f{x) = —f= j = h i x — j = , V x e ( 0 ; + c o ) frinh sau theo thara sd a,fe:
Vl+;c 4V2 V2
ta ed bdng bidn thidn J x"y -y' = a"
X^y + 2xy'^ + y"*
Bdi 2 (HGS QG - 2005 ): Cho cdc sd thuc a,b,c. Chung rainh rang X
f(x) f(x)
0 1 +«>
0 +
^ ~ ^ 0 -"^^
Nen f(x) > / ( I ) = 0 , tii do ta CO dpcm.
Dgng 3: Bdi todn bdt ddng thuc ddng bgc CO hieu giiia bgc ciia tit vd hgc cita mdu khdc I:
6{a + ii + c)(ii^-H(i^+c^)<27ak + 10(a^+62+c2)2 Bai 3 (flSG QG - 2007 ) : Hay xac dinh so nghiem ciia h$ phuong ttinh sau;
l o g j i . l o g , i/ = l
111
TAP CHl KHOA HQC - BAI HQC BONG NAI, SO 03 - 2016 ISSN 2354-1482 BM 4 (Albania 2002): C h o a , 6 , c > 0 . B41 7:
Chiing minh rang: ^j Cho x, y, z duong thoa man ' ^ ' ^ ( j ' l i ' ic')(^ I ' I ' ) ; a i n ; i - / i ' n ' I J J ^ ' + Z + z ' ' ! , chiing minh ring
3^/3
- + - + --(x + y+z)>2S X y z
Bai 5 (Olympic Toan Nhat Bin 1997).
Cho a , 6 , c > 0 . Chiing minh rang; b) Cho x, y, z ducmg thoa man (b + c-a)' {c + a-h)' (a + b-cY ^ 3
(b*cf-K? (c + af+l? {a + bf+,? 5
xyz = I , chiing minh rang x^ ^ / ^ z^ ^ 3
\-vyz \+xz \-k-yx~ 2 c) Cho cdc sd a, b, c duang, chftng minh Bai 6 {Trung Quoe 2005): Cho ^dng:
a , & , c > 0 va a + 6 + c = l . Chftng
minh rdng: a"^ b^ c* a^+b^+e^
3 + 46 b+4c c + 4a~ 5 lOCa^+ft-^+c^)-
TAI LIEU THAM KHAO
1. Geoffrey Petly (1998), Teaching today, Stanley Thomes Publishers, United Kingdom
2. Nguydn Bd Kira (2008), Phuangphdp dc^ hgc mdn Todn, Nxb. Dgi hgc su pham 3. Dinh Quang Minh, (2004), Tri thtrc vi hdm vdi nkirng Id ndng gidi mpt sd logi todn o-/op 70 TTiPT; T?p chi Thdng tin Khoa hgc gido dgc, Hd Ndi sd 108/2004, fr 24-28
4. Ld Hd Quy (2012), Sie dung dgo hdm di gidi mpt so logi todn, Todn hgc &
Tudi tte, sd 423/2012
5. Nguyen Tudn Nggc (2014), Diing phucmg phdp do thf di chirng minh bdt ddng thiec vd tim gid tri l&n nkdt, gid tii nhd nhdt, Toan hge & Tudi fre sd 442/2014 6. Nguydn Tdt Thu, Trdn Van Thuang (2010), Phuang phdp hdm sd trong cdc bdi todn Dgi sd, Nxb. Dai hgc Qude gia TP. Hd Chi Minh
TAP CHl KHOA HOC - BAI HQC PONG NAI, SO 03 - 2016 ISSN 2354-1482 APPYING KNOWLEDGE OF ARITHMETIC FUNCTION TO SOLVE
SOME HIGH SHOOL MATHMATIC PROBLEMS
ABSTRACT
The subject of arithmetic function appears throughout high school math syllabus. Thus, applying knowledge of arithmetic function to solve mathematic problems, through wkick students ean practice skill of solving mathematic problems is considered necessary. The writing aims at not only showing that some high school mathematic forms can be solved thanks to applying knowledge of arithmetic fiinetion but also suggesting some pedagogic orientations wkick help teachers guide students in mathematic practice in order that they ean form a certain mathematic solving skills througk this application. Tkose forms of mathematics often appear in curriculum of grade 10, II, 12 and also commonly in university entiance exam papers.
Keywords: knowledge of arithmetic function, mathematic solving skill, content, idea, activity
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