Ebook phương trình bất phương trình hữu tỉ, vô tỉ, mũ, logarit phần 2
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ChUcfng 3.
PHtfCfNG TRINH, BAT PHlJdNG TRINH MU
A. Tom tit ly thuylt
1. Cac cong thijfc bien doi dai so:
Duai day la cac cong thiic h'lkn d6i ca ban \h luy thua, ap dung de bien doi cac
phuang trinh mu:
, in oam,^
<
a
a'
a
a'"-b'"=iabr,^
=
vai a , 6 > 0 , m 6 E .
'
1 I—
a"* = _ , ^ a ' " = a " vai a > 0,6 e ! , / « , « e Z \
a
2. Cac npi dung ve giai tick:
•
• -
Tap xac dinh va tap gia tri:
Ham so luy thua y = x'' c6 tap xac dinh nhu sau: Neu a e
thi x € M,
neu a e Z " u { 0 } thi X G ] R \ { 0 } , ngu a Z thi x e
,^
vox Q < a ^ \i tap xac dinh la R va tap gia tri la M"".
Dao ham cua cac ham s6 (x") = x""',{a'')' = a'\na.
•
Ham s6 mvi y ^
-
V a i a>\,
-
Giai han cua ham so:
•
N6u 0 < a < 1 thi cac ham s6 y = a'' nghich bi6n.
-
Neu a > 1 thi cac ham SO
-
Tinh dan dieu cua ham s6:
•
,
= a"" dong bien.
tac6 cac giai han sau lim a" = +oo, lim a" = 0.
JC->+<»
-
X->-00
V o i 0 < a < 1 ta CO cac giai han sau lim a" = 0, lim a"" = +oo.
182
3. Cdc dgingphirfftig trlnh ca ban:
5^,;;
; -n,
, ? / :i)
• Phuongtrinh a" =m vai ni>0, luonc6nghiemduy nhat la x = log„m.
• Phuong trinh a^^"^ = a^'-"^ <=> / ( x ) = g{x) do tinh don dieu cua ham s6 mii.
• Phuong trinh a^*""' =ft^*""^voi a^b thi tinh logarit ca s6 a true tiSp hai
ve, ta duge
-
~
log4a^^^') = log4M<^>)«/(x) = g(x)-Iog,6.
-
Day la cae dang eo ban va noi ehung hau het cae phuong trinh mu va
logarit tir ea ban den nang cao deu phai thong qua. Trong goc nhin nay, eon
mot so dang pho bien nhu phat hien tinh doi xung de chia va dat §n phu, biSn
doi ap dung eong thuc thich hop de dua ve eung ca so, ...
4. Cdc dfing bat phuang trlnh mu ca ban:
Tuang ung voi cae dang cua phuong trinh, bat phuong trinh ciing eo eac dang
cabannhusau:
•
Dang a'' > 6 vai 0 < o 7i 1 : ta c6 2 truang hop nhu sau:
"
-
N6u 6 < 0 thi bit phuong trinh nghiem dung vai moi x.
-
Ngu 6 > 0 thi ta CO x> log„h neu a > 1 va A:< log„h neu a<\.
•
Dang a"
-
Neu * < 0 thi bat phuong trinh v6 nghiem.
-
NSu i > 0 thi ta CO jc> log^ h nlu o < 1 va x < log„ h neu a > 1.
•
Dang a^*''^ > a^'^''^ vai 0 < a
'
.
1 thi f(x) > g{x) ngu o > 1 va
/ ( ^ ) < ^(^) rieu 0 < o < 1.
-
Tuy nhien, ro rang trong vice giai cae bat phuong trinh thi doi khi danh gia
true tidp la khong kha thi va do khong phai la mot dang thuc nen viec bien doi
cung khong the linh boat duge. Trong phan nay, ehung ta se nhac lai mot dinh
w
*
w
*
li lien quan den viec xet dau cua ham so tren mot mien lien tue nhu sau:
Cho ham s6 f (x) lien tuc tren mien D. Gid sir phuomg trinh / ( x ) = 0 c6
dung n nghiem phan biet la x^,x^,x^,...,x^^ D. Khi do, giira cdc khodng
183
= 1,«-1 thi ddu cua ham so f{x) khong doi vd triing vai ddu cua
mot diem bdt ki thuoc khodng do.
Ta ihky rSng k^t qua nay cung tuong tu nhu viec lap bang xet diu cho mot
ham so, mot cong viec kha quen thuoc. Khi do, nlu xet mot bit phuomg trinh
CO dang / ( x ) < 0 hoac / ( x ) > 0 , ta chi can xet phuong trinh tuong ung la
/ ( x ) = 0 roi giai no ra de tim cac nghiem, xet ddu cua ham s6 giua cac khoang
do (thay cac gia tri cu thI vao) r6i so sanh voi yeu cSu 6k bai la hoan tit. Cach
nay giup ta tan dung dugc cac bien doi dudi dang dang thuc cua mot phuong
trinh cung nhu han che dugc viec lam dung cac tinh chat cua ham s6 va tinh
toan phuc tap.
Ngoai ra, ta ciing c6 cac danh gia sau (dang rut ggn cua cac danh gia da neu
d tren):
Voi 0 < a,6 ?t 1 va jc e R thi a"-b" cung diu voi
-
Voi Q
-
(a-\)(x-y).
x{a-b).
B. PhLrcyng phap giai
1) Phuang phap bien doi ve cung ca so hoac lay logarit hai ve
Nhan xet: Phucmg phdp nay da duac giai thieu trong noi dung ly thuyet a
tren. O day, ta se sir dung thudn tuy cdc bien doi dai so lien quan cung nhu
tinh chdt ca bdn cua hai ham nguac nhau la f (x) = a" vd g{x) = log^ x thong
qua dang thuc
f{g{x)) = gif{x)) = x.
Vi du 1. Giai phuong trinh 16
=
v
-1024^.
2112
Lai giai *) Dieu kien xac dinh x > 0, x
/3x+2>
4
12X+8
J = 2"3 .2'°^ «
l . Phuong trinh da cho tuong duong voi
n
4
= 2>°^ «
2"^^3
, o
^
1 ^ + ^ = 1OV^
x-1
3
Khu mau va nit ggn, ta dugc 4x + 2 = 3(x - l)\/x.
184
Dat/ =
0,taCO
't = 2
3/^+2/ + l = 0
'
De thay phiromg trinh thu hai v6 nghiem nen phuang trinh tren c6 nghiem la
t = 2.
Vai t = 2 thi X = 4 nen phuang trinh tren c6 nghiem la x = 4.
Vi du 2. Giai phuang trinh 2"' = 1 6 ' ' .
L&igidL Ta c6
4-2'
2'*' = 16'' <=> 2'" = T''
_
^2-"
-.2"^
« 2'" = 2 ' " « •>4x
2'^ = T2X + 2
<=>4x = x + 2<=>jc = - .
3
Vay phuang trinh da cho c6 nghiem duy nhat la ^ = " j •
2x-3
V i du 3. Giai phuang trinh 3^"' • 4 ^ = 18.
LoigidL *) Dieu kien xac dinh jc 9^ o . Ta c6
2x-3
3^^-'-4^
4x-6
=18«3^'-'-2^
6-3x
= 2 - 3 ' « 3 ^ ' - " =2"
Lay logarit ca so 3 hai ve, ta dugc
2
x'-4
,
=
6-3x
log, 2 « ( x - 2)(x + 2)x + 3(x - 2) logj 2 = 0
+ 2) + 3 log3 2] = 0
(x - 2)
x-2
x=2
=0
x(x + 2) + 31og32 = 0
(x + 1)' +31og3 2 - l = 0
D I thiy phuang trinh thu hai v6 nghiem nen phuang trinh ban dau c6 nghiem
la
X
= 2.
V i du 4. Giai phuang trinh sau 2^"' • 3" = 4^' • 3 6 ^ .
L&igidL *) Dieu kien xac dinh:
x^-l.
185
2x
Taco 2^^'-3'=4^ • 36^^' o 4 • 6" = 4 ' •6^^'o4'-^
=6'^'.
Lay logarit ca so 4 hai v l , ta duQfc
x+1
"l-x = 0 ,
log4 6 « (1 - x) [(x +1) - X log4 6 = 0
<=>
x=l
(x +1)' - x l o g ^ 6 = 0 " [ x ' + x(2 -log4 6) + 1 = 0
D I th4y phuomg trinh thu hai v6 nghiem nen phuomg trinh ban dhu c6 nghiem
la x = l .
V i du 5. Giai phuong trinh sau ( 5 ^ " ' + ( s ^ =
[l"''-
6 • (2"*
J.
Z,^i^/fl£ Phuomg trinh da cho tuomg duomg voi
O 26 • 5^
=26-r
« 5'
=r
o (x' + 3x) log^ 5 = x' + 6x
« x[(x + 3) log2 5 - (x + 6)] = 0
"x = 0
125
logjS-l
64
6-31og,5
,
'
--i^^ -
-- •
Vay phuomg trinh da cho c6 hai nghiem phan biet la x = 0,x = log.
64
125
V i du 6. Giai bdt phuomg trinh sau 3^^''^' >
(DH Bach khoa Ha Noi 1997)
L&i gidL * ) DiSu kien xac dinh: x^ - 2x > 0 <»
Ta
CO
3^^ >
/ I
X >
2
J- ,
x<0'
. !'
V
(*)
-X
1
Neu | x - l | - x < 0 < : > | x - l | < x < = > - x < x - l <
luon dung.
^ ,
> - thi bat phuomg trinh (*)
_
186
NSu \x-\\-x>0^x<-
thi M t phuong trinh (*) tra thanh ylx^-2x
binh phuong 2 vS, ta dugc
- 2x > 1 - 4x +
<=>
>\-2x,
- 2x +1 < 0, v6 nghiem.
Do do, nghiem cua bat phuong trinh da cho la x > 2.
:
.
4" +2x-4
Vf du 7. Giai bat phuong trinh sau
< 2.
x-1
(DH Van Hoa Ha Noi 1997)
Led giai *) Dieu kien xac dinh x*\. Ta c6
x-\
Ta CO 2 truong hcfp:
- Neu
-Neu <
4^-2<0
4^<2
x
x-l>0
x>l
x>l
4^-2>0
f4^>2
x-l<0
[x
2 , khong thoa man.
1
x>log42 = 1
2 o-
x
Vay nghiem cua bat phuong trinh da cho la x e
V
?5X
L\
Bai tap phan 1
1. Giai phuong trinh sau 5^+5^*' +5^"' = 3^+3^^' +3^"'
2. Giai phuong trinh sau 5' • 8 ^ = 500
(DHKinh
ti quSc dan 1998)
3. Giai phuong trinh sau 5"^ - 1 + 5^"'-125 = 4-5
4. Giai phuong trinh sau (41 + 29^12)"^ = (V2 -1)
,sin4j:
5. Giai phuong trinh sau (10 + 6V3)'''"^ = V(V3+1)
6. Giai b i t phuong trinh sau 343 • 1"'''^ > 2,2-1
7. Giai b i t phuong trinh sau ————— < 1
(DH SP Ha Noi 2001)
187
Huong din bai tap phan 1
1. Xet phuomg trinh 5" + 5""' + 5^"' = 3" + 3""' + 3""'. Ta c6
5^(1 + 5 + 5') = 3^(1 + 3 + 3') « 5^-31 = 3^-31
«5^=3^«
.3j
= 1 < » x = 0.
Vay phuong trinh da cho c6 nghiem la x = 0.
2. Xet phuomg trinh 5" • 8 ^ = 500. D i l u kien x ^ 0.
Ta
CO
3(x-l)
5^-2 '
2-
3x-3
=2'-5'<=>5^-'=2'" -
3-x
«5^-'=2-
Lay logarit co so 2 hai vS, ta c6
(X-3)
log, 5 =
3-x
o(x-3)
1
log2 5 + - = 0<::>
y
X
X =
3
x = -log5 2
xj
Vay phuomg trinh da cho c6 2 nghiem la x = 3 hoac jc = - log, 2.
3. Xet phuomg trinh sau Is"-ll + ls"^^-125
=4-5'*\
Taco 5 ^ - l > 0 » x > 0 va 5 ^ * ^ - 1 2 5 > 0 « 5 ^ > 5 « x > 1.
Ta xet cac truomg hop sau:
-NSu x<0 thi 1 - 5 ^ + 1 2 5 - 5 ^ ^ ' = 4 - 5 " ' « 1 2 6 = 1 2 6 - 5 ' o 5 ^ = l < : > x = 0,
thoa man.
-Ngu x > l thi 5"-1 + 5^*'-125 = 4 - 5 ^ " ' « 74-5^=-126, v6 nghiem.
-Neu 0 < j c < l thi 5 ' - 1 + 1 2 5 - 5 ^ ^ ' = 4 - 5 " ^ ' « 1 2 4 - 5 " = 1 2 4 5 " =1
<=>x = 0, loai.
Vay phuomg trinh da cho c6 nghiem duy nhit la x = 0.
4. Xet phuomg trinh (41 + 2972)""' = (V2-1)"'^"\
Ta CO 41 + 29V2 = (V2 +1)', V2 - 1 = (V2 +1)-'
nen lay logarit ca s6 V2 +1 hai v8, ta dugc
188
1
x=l
5(jc-2) = - ( x ^ + x + 3 ) « x ^ + 6 x - 7 = 0<=>
x = -7
Vay phuomg trinh da cho c6 2 nghiem \a x = \,x = - 7 .
5. Ta CO 10 + 6V3 = (Vs +1)^ nen phuomg trinh da cho tuomg duomg voi
(73 +1)'^'"^ = (73 + if""'' « 6 sin X = 2 sin X • cos X • cos 2x
» sin x(cos X • cos 2x - 3) = 0
Do cos X • cos 2x - 3 < 0, Vx nen phuomg trinh nay chi c6 nghiem la
sinX = 0 <=>X =
kn:,keZ.
6. Xet bit phuomg trinh 243 • 2"'"' > 32 • 7 \
Ta CO 7' • 2"'-' >2' -r <^ 1"-^ < 2"'"'.
Lay logarit 2 ve theo ca so 2, ta dugc
( x - 3 ) l o g 2 7 < x ' - 9 < » ( x - 3 ) ( x + 3-log2 7 ) > 0
<=>
"x>3
X < -3 + log2 7
Vay nghiem cua bit phuomg trinh da cho la x G (-00; -3 + log^ 7) u (3; +00).
1. Xet bat phuomg trinh
'^^
< 1. DiSu kien xac dinh 3"^ - 2^ ^ 0 » x ;t 2.
^
. 2-3^-4-2^-(3^-2^) ^
3^-3-2^ „
^<0<=>
<0.
Taco
:
3^-2^
3'-r
Ta xet 2 truong hop:
.3V
f 3 ' - 3 - 2 ^ < 0 <=> < .2
3^-2" > 0
x>0
3^-3-2^ > 0
3^-2^ < 0
2)
<3
.
<=> 0 < X < logj 3. _
2
^, khong thoa man.
x<0
Vay nghiem cua bat phuomg trinh da cho la x e 0;log3 3
2 J
189
2) Phuong phap dat an phy.
Nhan xet: Trong phucmg phdp nay, ta se gap cdc phucmg trinh c6 su xudt
hien cua mot bieu thirc ndo do nhieu Idn vd dieu ndy dinh huang cho ta dat
bieu thirc do 1dm an phu de bdi todn duac dan gidn di. Chit y rdng khi dat dn
phu nhu the, cdn tim mien gid tri cua bieu thirc do de tuang irng suy ra dieu
kien xdc dinh ciia phuang trinh vai biin mai.
Trong phdn ndy, ta c6 cdc dang dn phu chu yeu nhu:
• /(gix))
dat t =
= 0 thi dat t = g{x), dua vi phuang trinh f (t) = 0.
, dua
• A • fl'^+ B •
At^ -\- Bt + C = 0, trong do a^b.
+C •
'
= 0, chia 2 vi cho a'«'^* > 0.
',duavg ^r^+5/ + C = 0 .
1
Vi du 1. Giai phucmg trinh 27^ = 9 ^ + 3" - 3.
LM gial Ta bien doi nhu sau: (3' )^ = 3 • (3' )^ + 3" - 3. Dat ^ = 3"^ > 0 thi
= 3/' + / - 3 «
- 3)(/ -1)(/ +1) = 0
3^=1
t =\
>=3
/=3
'x = \
x =0
Vay phuofng trinh da cho c6 2 nghiem la x = 0, x = 1.
x-^
jc-3
3x+l
Vi dy 2. Giai phuomg trinh 8^*^ + 2^^^^ = 2
_ 2.
L^/^/ai *) Dieu kien jf 7 i - 2 . Ta CO
1 - ^
.
2 ^+2+2
2 ^"^2
o4
x+2
<»8
3--^
^2
'
_2
+ 2-2
_2
=8-2
-3-2 -2 = _ i
190
Dat y^l
-—
1
> o , t a c 6 4 / - 3 ; ; =-1«(:i; + l)(2>'-l)'= 0 7 = - .
^ -1 o
= - l « x = 3.
2 da xcho
+ 2c6 nghiem la jc = 3.
Vay phuomg trinh
Do do 2
Vi du 3. Tim m dS phuomg trinh sau c6 hai nghiem trai diu:
(/M + 3)16"+(2w-l)4'^+7M + l - 0 .
/
Lei gidl Dat / = 4"^ > 0. Ta c6 phuomg trinh {m + 3)/^ + {2m -1)/ + w +1 = 0.
Ta can tim m sao cho phuomg trinh nay c6 2 nghiem t^,t^ thoa man
0 < /, < 1 < ^2 .
Tmoc het, ta can tim dieu kien de phuomg trinh nay c6 2 nghiem deu duomg,
tucla
(2m-l)^-4(w + l)(m + 3)>0
A>0
11_
S > 0 <=>-i 2m-\
m + 7>>0
20
P>Q
m+1
>0
l/n + 3
Dat tilp M = r -1 <=> / = M +1 thi ta dua \k phuong trinh
(m + 3)(« +1)^ + {2m - 1)(M + l) + m + l = 0<»(w + 3)M^ + (4m + 5)M + 4/w + 3 = 0.
Tiep theo, ta can tim m sao cho phuomg trinh tren c6 2 nghiem u trai diu,
dieu nay tuomg duomg vai
3
^ , .
(w + 3)(4w + 3)<0«-3' '
4
Ket hcfp cac dieu kien lai, ta c6 -1 < w < - —.
Vi du 4. Tim m dl phuomg trinh sau c6 nghiem duy nhat thuoc [0;1]
22-2^ _
+m+2=0
191
L&igidL
Phuang trinh da cho tuong duong vol
1
( 2X--'K) 2
1
^x-\
+ m + 2 = 0.
1
Dat / = — > 0 , ta thay xe[0;1]<=>/e[1;2]. Ta can tim m sao cho phuong
trinh da cho c6 dung 1 nghiem thuoc midn [1;2]:
-t + m + 2 = 0.
1
Dat / ( 0 - r - / + 2 vol ^ e M , t a c 6 / ' ( 0 = 2 / - l hay/'(O = 0 o r = | .
Ngoai ra, /(1) = 2 , / ( 2 ) = 4 nen lap bang bifin thien, d l dang suy ra / ( 0 = -m
CO nghiem duy nhit thuoc [1;2] khi -4
-2.
Vay dieu kien can tim cua m la -4 < m < - 2 .
Vi du 5. Giai phuomg trinh 4 ^ ^ ' + 2• 2 ^ " =2-4".
L&igidL
Phuomg trinh da cho tuong ducmg voi
4 . 2^^^^^ + 2 • 2^"^'^" - 2 • 2'" - 0 o 4 • 2'
Dat / = 2^'^'-" > 0 , ta dugc 4 r + 2 / - 2 = 0
+ 2.2-"^'-' - 2 = 0.
2 / ' =
"/ = - 1
0 «
1 •
t= 2
Ta chi nhan nghiem r = ^ va khi do
2^'-'
=-<^l[I+5-x
= -\<:>l!7T5-x = -\<^x + 5 = {x-Xf
< » x ^ - 3 x ^ + 2 x - 6 = 0 « ( x - 3 ) ( x ^ + 2 ) = 0<»;c = 3
Vay phuomg trinh da cho c6 nghiem la x = 3.
V i du 6. Giai bat phuomg trinh 36^'^""'+64"'^"''> 100-48"'^"-*.
L&i gidL Bat phuomg trinh da cho tuong duong voi
36 • ( 6 ^ ' ^ ^ - ' + 64 • (8^'"^-')'> 100 • 6^'""-'• 8^'^^-'
/
«36
-100
_
\
+J:-6
+ 64 > 0
v4.
192
« 9
-25-
3
+ 16>0
4)
Dat / =
>0 thi 9/^-25/ + 1 6 > 0 < = > ( / - l ) ( 9 / - 1 6 ) > 0 < »
4,
- Vai r < l , t a c 6
x^+x-6>0<:>
x>2
16
9 .
/<1
t>
'/'^-^
16
-Vai / > —,tac6
v4j
-i-Vi7 ^
<^x^+x-6<-2<;^x^+x-4<0<^
4j
Ket hop lai, ta dugc x e (-oo; -3
-i-7i7
V i du 7. Giai phuang trinh (2 +
-i+Vn
u[2;+oo).
4
+ (2 - V3)
^2-V^L&igidL
Phuang trinh da cho tuong duang vai
(2-fV^P
« (2 +
Dat ^ = (2 + V
3
N/3
( 2 - V 3 ) + (2-V3p"-'(2-V^) = 4
+ (2 - V
3
= 4
> 0 thi 1 = (2 - > / 3 v a ta dugc
t + - = 4<=>t^-4t + \ 0^
t
- V o i / = 2-73 « ( 2 + >/3r'-'^ =2-73 » x ' - 2 x = - l « x = l .
- V a i t = 2 + S<^{2^SY-^'
=2 + S<^x^-2x
= \<^x = \±y[2.
Vay phuong trinh da cho c6 nghiem la x = 1, x = 1 ± V2.
-1 +
Bai tap phan 2
1. Giai phuomg trinh 9^'"'"' -10- 3^'^^"' + 1 = 0
2. Giai phuomg trinh 2'
_22+--•
=3
(Dy bi DH khSi B 2006)
(DH khoi D 2003)
(DHSP Ha Noi 2000)
4. Giai bat phuong trinh 3'' - 8 • 3"^"^ - 9 • 9 ^ > 0
(DH Y Ha Noi 2000)
3. Giai phucmg trinh 2'^-6-2^-^3i^ + p = l
5. Giai bit phuomg trinh 27'' +12"^ > 2 • 8"^
6. Giai phuomg trinh ^Jl + ^l-l^"
= (l + 2^1-2^^)• 2"
7. Giai bat phuong trinh V9 + 8 - 3 ^ - 9 ^ + 3 ^ > 5 (DH SP Ha Noi 2012)
8. Giai phuomg trinh ( 6 - + (6 + s[\\y
zx-2
=9A-5'
9. Giai bit phuomg trinh Vl5-2''^'+l > 2^^ -1 + 2"^'
(DubikhdiA2003)
Huong dan giai bai tap phan 2
1. Xet phuong trinh 9'''^"-' -10-3"'^"-' +1=0.
Dat / = 3''""^ > 0 thi phuong trinh da cho viSt lai thanh
't = \
9 r - 1 0 / + l = 0<=>
-Vdd / = 1 thi 3 ^ ' ^ ' ' - ' = l « x ' + x - 2 = 0 o
1 •
9
x=l
x = -2"
x=0
^ - V o i t = - thi 3 ' ' " ^ - ' = - « x ' + x = 0 «
9
9
x =- l •
Vay phuomg trinh da cho c6 4 nghiem la x = -2, x = -1, x = 0, x = 1.
2. Xet phuomg trinh 2 ' ' - 2'^""^' = 3.
Dat / = 2''
.
, ^,. v
>0,tac6 / - y = 3 < » / ^ - 3 / - 4 = 0 «
'••:w
;
,
^V + £ - ^
/=4
Ta chi nhan nghiem t = 4>0 va khi do 2''
x=2
= 4 <=> x" - x = 2 <=>
x=- l
Vay phuong trinh da cho c6 nghiem la x = - l , x = 2.
3. Xet phuang trinh 2'"-6-T
+ ^ = 1.
2^
Dat / = 2">0 taco
2
^ 3 _ 6 / - l + l ^ = l « / ^ - 3 . / ^ . - + 3-/-
V
=1
t
2^
t — =
= l<=>r^-/-2 = 0-»
Ta chi nhan nghiem t = 2>0 va khi do x = log^ 2
r =- l
/ =2
1.
Vay phuang trinh da cho c6 nghiem duy nhit la x = 1.
4. Xet b i t phuang trinh 3'" - 8 • 3 " ^ ^ - 9 • 9 ^
> 0.
1.
Dieu kien xac dinh: x > -4.
Bat phucmg trinh da cho tuang duang voi
> 0 « 32--2^/^ _ 8 . 3 - - V ^ - 9 > 0.
3^' - 8 - 3 ' .2^-9-3^^
Dat / = 3 " - ^ > 0 thi
- 8 / - 9 > 0 <=> (/ + l ) ( / - 9 ) > 0 » / > 9.
- Vai / >9<=>x-Vx + 4 >logj9 = 2 o x - 2 > Vx + 4 .
Dat dieu kien x > 2, binh phucmg 2
ta dugc
x^-4x + 4 > x + 4<=>x^-5x>0<=>x>5.
Ket hap lai, ta thay nghiem cua bat phuang trinh da cho la x e (5;+QO).
5. Xet bat phuang trinh 27^ +12^ > 2 • 8 \
Ta bien doi nhu sau
Dat / =
/ 2 f
v2.
> 0 thi ta CO
>2.
2)
+ / > 2 <^ (/ - l){t^ + / + 2) > 0 <=> / > 1.
v2y
Dodo
>l«x>0.
195
Vay nghiem cua bat phuang trinh da cho la jc e (0; + 0 0 ) .
6. Xet phuang trinh
+
= (l + 2^1-2'^) • 2'.
DiSu kien 1 - 2^"^ > 0 <:> X < 0.
Dat t = yjl-l^'
thi 0 < / < 1, suy ra 2^' =\-t^ <^2'
=
yjx-t^
nen phuang trinh
da cho tuang duang vai
r=0
« 1 = (1 + 2tf (1 - 0 «t{At^ - 3) = 0 «
- V a i / = 0, tadugc V l - 2 ' " = 0 o 2 ' ' = l o x = 0.
-Vai
= - , t a d u a c 2"" =\-- = -<^2x
4
•
4 4
= -2<^x = -\.
Vay phuang trinh da cho c6 2 nghiem la x = 0, x = - 1 .
7. Xet b i t phuang trinh V9 + 8 - 3 ^ - 9 ^ + 3 ^ > 5.
x
x<4
4-x>0
9 + 8 - 3 ^ - 9 ^ >0
(3^-9)(3^+l)<0^|3^<3
<:>0
Dat ^ = 3 ^ > 0 t h i t a c o A/9 + 8/-?'
> 5 « ^ 9 + 8/-^' > 5 - / (*)
- Neu / > 5 <=> 5 - / < 0 thi (*) nghiem dung.
- N8u / < 5 <» 5 - / > 0 thi (*) tuang duang vai
9 + 8 / - / ' > (5-0^ « 9 + 8 / - / ' > 25-10/+
< » / ' - 9 / + 8< 0 <:>!
Trong truang hap nay, ta dugc 1 < / < 5.
Ket hop lai, ta c6 nghiem la / > 1 hay 3 ^ > 1 o
V4-Jc
>
0
x<4.
Vay nghiem cua b i t phuang trinh da cho la x e [O; 4 ) .
6-vriY r6+vrT
8. Ta bien doi nhu sau
A
94
25
5
196
Chu y r&ig
(6
+ y/U
6-vrT
= 1 nen neu dat / =
6+ViT
>0 thi
\
'6-VrT
t
Ta CO phuang trinh r + - = — « 25?^-94t + 25 = 0^t = "^"^ -^^VlT
25
t 25
^ 47+i2Vn ^
Vai r =
,
, ta C O
25
'
(6+VrT 47 + 12^/^T <»x
25
= 2.
[ 5
Vai. = ^ Z ^ , t a c 6 ^ ^ ^ ^
25
47-12Vri
25
<» X=
Vay phuang trinh da cho c6 2 nghiem \a x = ±2.
9. Dat / = 2^ >0,tac6 V30/ + 1 > | / - l | + 2/.
-2.
'
-
-
-
Ta CO 2 truang hap sau:
- N g u / > l , t a c 6 V307+T>3f-1. Chiiy rang 3 / - l > 0 vai / > ! nentaco ^ferf
30t + \>i3t-\f
^9t^-36t<0<^0
Do do 1 < 2^ < 4 <:> 0 < X < 2.
- N8u 0 < / < 1, ta CO V307TT > f +1. Chu y rang ^ +1 > 0 nen ta c6
30/ + l>(? + l ) ' « / ' - 2 8 ^ < 0 « 0 < / < 2 8 .
Do do, trong truang hop nay, ta c6 / < 1 <» 2"" < 1 « x < 0.
. j - \•
Ket hap lai, ta duac x < 2 la nghiem cua bat phuang trinh da cho.
3) Phuang phap phan tich
Nhan xet: Trong nhieu tinh huong, ta gap phdi cdc phuang trinh khd rdc roi
vd ddi hoi cdn phdi phdn tich bieu thirc tuomg ung thdnh nhdn tic de Idm dan
gidn hoa phuang trinh han roi xu ly rieng le timg phuang trinh nhdn tu. Ta
Cling CO the coi x hogc bieu thuc ham mu lien quan den x Id bien, thdnh phdn
con lai Id tham sd de tinh delta vd de bien ddi han.
197
V i du 1. Giai phuong trinh 25 • 2^ -10^ + 5^ = 25.
L&igidL TabiSn d6i nhu sau
25-2^-2^-5'+5"-25 = 0
« 2^(25 - 5^) + (5^ - 25) = 0 » (2' -1)(5^ - 25) = 0
2^ = 1
<=>
y
= 25
x =0
x=2
Vay phuong trinh da cho c6 nghiem la jc = 0, x = 2.
V i du 2. Giai phuong trinh sau 4x' + 2x.2''^' +3.2"' = x'.V' + 8x +12.
Lcfi gidu Phuong trinh da cho tuong duong voi
x'(2"'-4) + 2JC(4-2"') + 3(4-2"') = 0
•
(2"' - 4)(x' - 2x - 3) = 0 <=> 2"' - 4 = 0 V
-
2x - 3 = 0
<=>x = ± \ ^ , x = - l , x = 3
Vay phuong trinh da cho c6 cac nghiem la x = ±V2, x = -1, x = 3.
V i du 3. Giai phuong trinh ( V 3 - l ) " " ' % x ( V 3 = x ' +1.
L&igidL
f r- -
*) Dieu kienxac dinh: X > 0.
Phuong trinh da cho tuong duong voi (nhan them 2 v6 cho (yfs+lf'^''' > 0 ) :
(V^+if"[(V3-i)'""+x(V^+if'^]
\
-
= (V^+i)'"^^(x^+i)
«2'-'+x(V3+lp"=(V^-flp(x^+l)
O x + x(V^ + l p "
" ;
.
=(V5 + l ) ' " " ( x ^ + l )
s
/
1—
\21og2J:
21og2Jt
+ 1 = V3 + 1
<=> X
(x^+l)o
= -^^
Ta thay dang thuc
o a b + b = ab + a
{a-b)(ab-1)
= 0 <=>
'a = b
ab = \
a
198
Ta CO hai tnrcmg hop sau
.-N
; -
- N6u X = (V3 +1)'°^^'' thi lay logarit co s6 2 hai v l , ta dugc
logj jc = log2(>/3 + l).Iog2 X <=> logj X = 0 <=> X = 1, thoa man dieu kien xac dinh.
- Neu x(\f3 +1)^°^'" = 1 thi cung lay logarit co s6 2 hai vg, ta duac
log2X
+ log2xlog2(>/3 + l ) = 0 < » l o g 2 X l + log2(>/3+l)
<=> log2 X = 0 «> X = 1.
=0
ma'!
Vay phuang trinh da cho co nghiem duy nhat la x = 1.
Vi du 4. Giai phuang trinh 4" - 2"^' + 2(2" -1) sin(2" +
-1) + 2 = 0.
(Du bi khSi D 2006)
L&igidL Phuang trinh da cho tuang duomg voi
(2'" - 2.2" +1) + 2(2" -1) sin(2" +
-1) +1 = 0
o (2" - 1 ) ' + 2(2" -1) sin(2" + > ' - ! ) + sin' (2" +
2" - 1 + sin(2" +y-1)J'
<=>
-l2
-1) + cos' (2" +
-1) = 0
+ cos'(2" + 7 - 1 ) = 0
2"+sin(2"+>'-l) = l
_
cos(2"+>'-l) = 0
Do cos(2" +
-1) = 0
sin(2" +
-1)
± 1 . Ta CO hai truong hop sau
-Ndu sin(2" + >; -1) - 1 thi 2" = 0, v6 nghiem.
-N6u sin(2"+>;-l) = - l thi 2" = 2 < : ^ x = l .
Suy ra sin(j + l) =
• "
TT
= -—-\ k27r,k e Z .
Vay phuang trinh da cho c6 nghiem la x = l,>' = - — - 1 + kin,k
e
Vi du 5. Giai bat phuang trinh
2"^'+ (5x'+11) • 2'-" - x ' < 24 - X[1 - ( x ' - 9) • 2^"".
LcigiaL
Dat r = 2",r > 0, bdt phuang trinh da cho tra thanh:
,
10x'+22
2t +
2 ^/
^'-9x
x'<24-x +
199
« 2 / ^ - [ x ^ - x + 24)t + 22 + 9x + \0x^-x^
I
<0.
Taxem 2/^ -[x^ -x + 24)t+ 22 + 9x + l0x^ -x^ =0 laphuomg trinh i n /. Tac6
A = (x^-x
+ 24f - 8 ( 2 2 + 9x + lOx' - x ^ )
= [x' + 3xf - 40 ( x ' + 3x) + 400 = ( x ' + 3x - 2 0 ) ' .
1
2
1
Suy ra, phuomg trinh c6 hai nghiem phan biet: t = -x^ + - X
2
2
/ = -x + ll
,
+1
Do do, bat phuong trinh da cho tuong duong:
4
(2/-x^-x-2)(/ + x-ll)<0
2-2^-x'-x-2>0
2r-x'-x-2>0
<=>
r+ x-ll<0
2'+x-ll<0
<=>
2-2^-x'-x-2<0'
2/-x'-x-2<0
r+ x-ll>0
Lb' ' + x - l l > 0
Xet ham s6 / ( x ) = 2" + x -11 thi / ' ( x ) = 2Mn 2 +1 > 0 nen no dong biSn tren R.
Mat khac, / ( 3 ) = 0 nen ta c6 / ( x ) > 0 « x > 3 va / ( x ) < 0 » x < 3.
Xet ham s6 g(x) = 2 - 2 " - x ' - x - 2 tren M. Taco
g\x) = 2 • 2^ • In 2 - 2x ii> g'Cx) = 2-2^ (In 2f-2=^
g"'(x) = 2 • 2^ (In 2)' > 0
Mat khac g(0) = g(l) = g(2) = 0 nen phuomg trinh g(x) = 0 c6 ba nghiem la
X
= 0,x = l,x = 2.
Lap bang xet ddu ciia ham s6 g(x) tren cac khoang (-oo;0),(0;l);(l;2);(2;+oo),
ta c6:
• g(x)>0«xe(0;l)u(2;+oo).
.
g(x)<0«XG(-oo;0)u(l;2).
Do do, bdt phuomg trinh da cho tucmg ducmg vai:
200
\g(x)>0
\fix)<0
0
<=>
3
0
g(x)<0
x<0vl
f(x)>0
x>3
2
Vay tap nghiem cua b i t phucmg trinh da cho la (0;l)u(2;3).
Vi du 6. Cho bdt phuong trinh xyllx-x^
<
- ax2" + al^yjlx-x",
tham s6.
voi a la
- r V '
•
a) Giai bat phuong trinh khi a = - 1 .
^
.^
b) Tim a de bat phuong trinh c6 nghiem x > 1.
LoigidL
a) *) Qihu kien 2x - x' > 0 « 0 < x < 2.
Voi a = - l , taco
xV2x - x' < x' + x2' - 2" V2x - x'
S-^^CH^
» V 2 x - x ' ( x + 2^)
yjlx-x^
Vay nghiem cua b i t phuong trinh la x e (0; 1).
v^
b) Ta bien doi nhu sau: xV2x - x^ < x^ - 0x2"" + al^-Jlx
ulM -
- x^
<:>x(V2x-x^ - x j < a 2 ^ (V2x-x^ - x j « ( V 2 x - x ^ - x j ( x - a 2 ' ^ ) < 0
Do X > 1 nen x-x^ <0<=> 2x-x^
>0^a<
- x < 0 , suy ra
X
Xethams6 / ( x ) = — , x e [ 1 ; 2 ] thi / ' ( x ) =
2^-x2Mn2 ,
- va
1
/ ' ( x ) = 0 <z> 2 " - x 2 M n 2 = 0 « X = — e [1;2].
In2 ^ ^
Khao sat ham s6 / ( x ) tren mien [1;2], ta thdy / ( x ) > /(1) = ^ va suy ra a < ^ .
201
V i du 7. Giai phucmg trinh
16' + 48^ + 96^ + IT + 8 r = 18" + 64^ + 24" +54^+108'.
LMgidL Dat ^ = 3"^ > 0, ta CO
-
,
2'" + 2 ' ' / + 2 ' ' / + / ' + / ' = 2 ' / ' + 2 * ' + 2'"r + 2 ' / ' + 2 ' ' / \
Ro rang nSu thay t = T thi c6 ding thiic nen phuong trinh tren c6 nghiem
/ = 2"^. Taphan tich nhu sau
:
•
- -^,
• (2'" - 2 ' " / ) + (2'"/-2'"e) + ( 2 ' " t - 2 ' " ) + [t' -2U^)-^[t'
» 2'" ( 2 ' -t) + 2''t(2'
-1) (2" +t)- 2'" (2" -t)-t'
<^[2''-t)(2''+2'U
o (2" -1) [[2'^
o (2" - 1 ] [2'' -
+ 2^U^-2''-t^-t')
+1[2"' -t')-
-
-2U') = 0
[2' -t)-^
(T -t) = 0
=0
2'^ [2''
'
=0
mmA -^s.
) ( l + / - 2'") = 0 o ( 2 ' - 3") (8' - 9") ( l + 3" - 4 ' ) = 0 .
Ta CO cac truong hop:
^
*
-Neu 2 ^ - 3 ' = 0 o ^2Y
3.
-Neu 8"-9'=0<=>
8
- Neu l + 3 ' - 4 " = 0 o
= 1 <=>x = 0.
3
v4y
4;
= 1. Ve trai giam theo jc nen phuong trinh
CO khong qua 1 nghiem. Ta lai thay x = l thoa man nen phuong trinh c6
nghiem duy nhdt la x = 1.
Vay phuong trinh da cho c6 2 nghiem la JC = 0, x = 1.
*"
H « ^' i
B a i tap p h a n 3
1. Giai phuong trinh 4x^ + x • 3' + 3'"' = 2x' • 3' + 2x + 6
2. Giai phuong trinh
52-^-3^-2+5^^+3x^2
^ 625 125"' +1
3. Giai phuong trinh 8 • 3" + 3 • 2" = 24 + 6"
4. Giai phuong trinh
(DH QuSc gia Ha Noi 2000)
• T +r{2-7>x) = 6x^-x'-\\x
+6
202
5. Giai bat phuang trinh 4x'+x-2"'^'
+ 3• 2"" >
\log,4r
/
—
• 2 + 8x +12
> logj-t
Ino. r
6. Giai bat phuang trinh (^S + dY'" - ( 3 V 5 - 6 )
T2
C
3
7. Tim a sao cho phuang trinh sau c6 dung 2 nghiem
^
• 3^ + o x ' + 2 • 3^ + 2 a x ' = X • 3^"'+ 3a • Jc'
Huang dan giai bai tap phan 3
l.Xet phuang trinh 4 x ' + x• 3^ + 3^"' = 2 x ' - 3 " + 2 x + 6.
Ta bien doi nhu sau
x ' ( 4 - 2 - 3 ^ ) + x ( 3 ^ - 2 ) + 3^"'-6 = 0
o - 2 x ' (3" - 2) + x(3^ - 2) + 3(3^ - 2) = 0 <=> (3^ - 2)(2x' - x - 3) = 0
3^ = 2
2x'-x-3 = 0
X = logj 2
<=>
,
3
x = -l,x = -
Vay phuang trinh da cho c6 3 nghiem la x = logj 2, x = - 1 , x = —.
2. Xet phuang trinh 5
= 625 • 125" + 1 . Ta b i l n d6i nhu sau
-2x'-3x+2
2x'-3x + 2 = 0
y+3x +2= 0
-1 = 0
Phuang trinh v6 nghiem
x = - l , x = -2
Vay phuang trinh da cho c6 2 nghiem la x = -2,x = - 1 .
r*
3. Xet phuang trinh 8 • 3"^ + 3 • 2"^ = 24 + 6 \a bi6n d6i nhu sau
8(3" - 3 ) + 2^(3 - 3") = 0 « (3" - 3)(2" - 8) = 0 »
3" = 3
x=l
2" = 8
x=3
Vay phuang trinh da cho c6 2 nghiem la x = l,x = 3.
4. Xet phuang trinh x' • 2" + 2" (2 - 3x) = 6x' - x' - 1 Ix + 6.
203
Ta bien doi nhu sau
2" (jc' - 3x + 2) = -{x - \){x - 2){x - 3)
« 2^(x - l)(x - 2) + (x -
- 2)(x - 3) = 0
X
^ ( x - \)ix - 2)(2" + X - 3) = 0 «
= l,x = 2
2'^ + X - 3 = 0
Ptiuomg trinh cuoi c6 ve trai d6ng biSn nen c6 khong qua 1 nghiem. Ta nhdm
thiy x = 1 thoa man nen phuang trinh da cho c6 2 nghiem la x = l,x = 2.
5. Xet bat phuang trinh 4x' + 2x • 2"'"' + 3 • 2"' > x' • 2"' + 8x +12.
Ta bien doi nhu sau
x ' (4 - 2^') + 2x(2^' - 4) + 3(2'' - 4) > 0
<:> (T'
-
4)(x' - 2x - 3) < 0 « (2^' - 4)(x + l)(x - 3) < 0
Chii y r i n g 2 " ' - 4 - 0 o x^ = 2 » X - ±V2 .
Lap bang xet d i u cua b i l u thuc, ta thu dugc nghiem cua hk phuang trinh la
.- X G ( - > ^ ; - 1 ) U ( V 2 ; 3 ) .
6. Xet b i t phuang trinh (3V5 + 6)'°''' -(3>/5 - 6 ) ' ° " ' > x ' - 1 .
Dieu kien x>0. Ta thay x = 1 thoa man bat phuang trinh. Xet
x^l.
Ta CO (3N/5 + 6)(3V5 - 6) = 45 - 36 = 9. Suy ra
2 _
log,((3^/5+6){3^/5-6)) _
logJ(3^/5+6)+log,(3^y5-6) _
log,(375+6)
Iog,(3,/5-6)
Do do, ta bien doi nhu sau
log,(3^/5+6) _
X
logj(3V5-6)
X
^
\ogy{i45+6)
^ X
1083(375+6) _
N—»^ X
'
«
log3(3r^-6)
1
* X
log,(3V5+6)_
log3(3v/5-6)
X
* X
log,(3x^-6)
^ X
_
1
^106,(375+6) L _ ^log,(3V5-6) \ ^ l o g , ( 3 V i - 6 ) _ ^
(^log3(375-6) _ ^ ) ( ^ l o g 3 ( 3 / 5 + 6 ) _ ^ ^ ) ^ ^ ^
^-06,(3^5-6) _ ^ ^
^
Do logj ( 3 V 5 - 6 ) < 0 nen X > 1.
Vay nghiem cua b i t phuang trinh da cho la x > 1.
204
7. Xet phuong trinh
• 3" + ox' + 2 • 3' + 2ax' = x • 3'"'
+3a-x\
Ta phan tich nhu sau
3"(x' - 3x + 2) + ax\x^ - 3x + 2) = 0
« ( x ' - 3x + 2)(3' + a x ' ) = 0 «
x ' - 3x + 2 = 0
3^+ax' = 0
Phuong trinh x^ - 3x + 2 = 0 da c6 2 nghiem la x = 1, x = 2.
Neu fl > 0 thi de thay phuong trinh thuhai v6 nghiem.
Dat / ( x ) - 3 " + a x ' voi a<0.
D l thiy
lim / ( x ) = -oo va / ( 0 ) = 1>0 va / ( x ) lien tuc tren (-oo;0) nen
x--*-co
phuong trinh c6 it nhat mot nghiem am, khac voi hai nghiem x = l , x = 2 o
tren; do do, khong thoa man.
Vay dieu kien can tim la a>0.
' ' - > >
-r^
'
-
i i . , > - t . H - ^ T J H D 3.icf
tiX
4i
4) Phuong phap danh gia
Nhan xet: Trong mot so truang hap, ta c6 the sic dung bat dang thirc de chtrng
minh phicang trinh v6 nghiem hoac c6 khong qua mot so nghiem nao do. Cu
the la ta se chon mot dai luomg trung gian nao do de lam ca sa ddnh gid, so
sdnh hai ve.
;
V i du 1. Giai cac phuong trinh sau
a)i2 +
yf3y+(2-yf3y=4\
h){2+y[3y+{2Sy=r.
L&igidL a) Ta bien doi phuong trinh da cho thanh
2 + V3
+
2-V3
= 1.
2 + ^/3
2 — V3
,
< 1 nen ve trai cua phuong trinh la ham nghich
Chii y rang 0 <
4
4
bien va do do, no c6 khong qua 1 nghiem.
Hon nua, x = 1 thoa man nen phuong trinh da cho c6 nghiem duy nhat la x = 1.
205
b) Ta bien doi phuang trinh da cho thanh
„, , , J
2 + V3 , 2-yf3
Chu y rSng
>1 >
- NIU X > 0 thi
V
2 + V3
2
~
y
+
3iaX A
= 1.
. ^ ' r , . ,
nen ta co cac truong hop sau:
> 2 + V3Y = 1, vaJ2-S' > 0 nen ve trai ion hon 1.
V
2-V^
- Neu x<0 thi
2
,
2-S
V
= 1 va
2
/
2 + V3
> 0 nen ve trai ciing
lomhom 1.
Do do, phuomg trinh da cho v6 nghiem.
Vi du 2. Giai b i t phuang trinh sau 2^"' + 3"' < 3'^"' + 2^^^'.
LM gidi
< 3'^"' + 2'^"'.
Xet bat phuang trinh 2'^^ +
Ta thay rSng n^u x = \i hai \h cua b i t phuang trinh bSng nhau. Ta xet cac
truong hap sau
-N6u X > 1 => X + 2 < 2x +1 thi 2"' < 2'^"',3^"' < 3'-^"' ^ T^^ + 3^"' < 2'^"' + 3'^"',
thoa man.
_ ^
-Neu X < 1 thi bat dang thuc a tren doi chiSu va khong thoa man 6k bai.
Vay b i t phuang trinh da cho c6 nghiem la ;c > 1.
r^"' + 2^'"'' = 2*-''"' + 2'"'*-'
V i du 3. Giai phuang trinh
„
, i-:,.,
(MSG TP. Ha Noi 2005)
LotigidL Theo bat dang thuc Cauchy cho 2 s6 duang thi:
-3
9-4
+ 2 UJ>2V2^'^^
-2 ^^^=2-V2^'^
^"^ >2V2^=4.
Ta CO 3=°^'' + 2'"'^ < 3'°^'^ + 3""''. Ta se chung minh rSng s^"^'"^ +3™'^ < 4 .
That vay, dat / - 3'°^'". Do 0 < cos^ x < 1 nen 1 < r < 3 . B i t ding thuc c i n chung
minh tuong duang v a i / + - < 4 <» ( / - ! ) ( / - 3 ) < 0 , dung.
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