A = V (x-I)^+y^ +V (x + l )^+y^ +|y + 2 .
HirOngdSngiai ' Trong mail phdng vdi hO toa dO Oxy. x6t M(x - 1; -y), N(x + 1; y).
Do OM + ON ^ MN ncn:
V (x -l )^+y^ +V (x + l )^+y^ ^yl4 + 4y^ =2Vl + y^
Dod6: A^2.
Vdiy^2=5ằf(y) = 2>/i + y +2-y=>f'(y) = 2y r(y) = 0<*2y= yjl + y^o fy^O
l 4 y 2 = l + y
- 1 .
po d6 ta c6 bang bie'n Ihien:
X - 0 0 1
J3 2
' f(V) - 0 +
f(y)
~ ^ ~ ~ ~ ^ ^ 2 + N /3 ' Vdi y S 2 => f(y) > 2 < 2x/5 > 2 + ^/3 . V2y A > 2 + >/3 vdi moi so Ihi/c x, y.
^ 1
Khi X = 0; y = tiuA= 2 + S ncn gid trj nho nhaft ciia A 1^: A = 2 + ^/3 . Bdi 15. Cho hai so thifc \^0,y ^0 lhay doi va thoa dieu kipn:
(X + y)xy = x^ + y^ - xy. Tim gia tri Idn nhat cua bieu thtfc A = ^ Hu'dng dan gi^i
1 1 1 1 1 Tif gia thiel suy ra: — + — = - J + —T-
X y y'' xy \
Dai — = a . - = b ta c6: a + b = a^ + b^ - ab A = a' + b' = (a + b)(a^ + b^ - ab) = (a + b)^
Tif (1) suy ra: a + b = (a + b)^ - 3ab VI ab<
(1)
a + b nen a + b S (a + b)^ - - ( a + b)^ 4 . > . ' j ' > d - >
• (a + b)^ - 4(a + b) < 0 => 0 < a + b < 4.
1 do?.
Suy ra: A = (a + br ^ 16. Vdi x = y = - thi A = 16.
Kai 16. Cho hai so difdng lhay ddi Ihoa man dieu ki^n x + y ^ 4. Tim gia tri nho nhat cua bidu thtfc: +
4x y-
Httdng dSn giai Ap dung ddng thtfc Co-si, ta c6:
. 3x^+4 2 + y' X 1 ^ A = + — i _ = _ + _ + 2 4x
4 X 1 y y
2 8 8 ii±l.U2.3.i *24
LUyyn null UK IIUW Ky mi Ull J mien IML, Mturtg, j T u i F i 1 Murv-ny^- 1 n u f u i i m n i t w i t y
Da'u " = " xay ra o
X _ j _
4 ~ x 8
<=> X = y = 2 th6a man x + y > 4.
V|y gi^ tri nho nhat cua A la - khi x = y = 2.
Bai 17. T i m gia tri nho nha't cua ham so:
11 L,
y = x + — + . 4 +
^ . 2x V 1 + X ; • ^ > ^ '! Ite;ipm; j*>v €V + S £ /' Hrf(Jng dSn giai S = A frii K :<> = >
Ta c6:
N2
3 + - 3.1 + N/7. ^(9 + 7) 1 + 7 ^
= 16. 1 + 7 ^ ( 7 ^ 1
4. 1 + - T > - 3 + -
\ 4. V X > 2 . Da'u xay ra l = ^ = x
1 V7 11 1
Suyra: + 3 + -X = —+ 2 x + — 9'|
2
9 3 ^ 15
X 2 2
9 , 3 XN/7 -
Dau " = " xay ra o X = - - = —f=r- = x o x = 3 , X 1 v7
Vay ta c6 y„,i„ = y x3y ra khi x = 3.
Bai 18. Gia suf a, b, c, d 1^ so nguycn thay doi thoa man 1 < a < b < c < d < 50. Chu-ng minh bat d^ng thtfc:
i + £. > '^^^^^^^ va tim gia tri nh6 nhat cua bieu thiJc: S = J + . b d 50b , b d
ằ y Hir(}ng d3n giai V I a > 1 . d < 50 va c > b c > b + l ( v i b, c e R)
a c 1 b + 1 b^ + b + 50 „ „ , _
• Do d o - + - > - + = • Dau " = " xay ra o
b d b 50 50b
a = l
d = 50 ^^^y, c = b + l
T i m gia trj nho nha't cua ~ ^ • -s c
a c . b^+b + 50 _b_ J_ _1_
50 b 50
b d 50b
X 6 t h a m s o f ( x ) = + i + _ L ; (2 < x < 48).
50 X 50
1 1
, Bing bien thien:
Jx^ =50
- 2
2 < x < 4 8
5 ^
<=> X
ằ^c/ôy ly iiynn Mivuvvn j\.nang vift
0
50 ^/50
, f(x) dat gia tri nho nha't b^ng — + tai x = N/SO . 50 V50 •
, V i a , b , c, d 6 R v a 7 < V 5 0 < 8 .
S„,i„ =min[f(7);f(8)]=>S„,„ =f(7) = - ^ .
Bai 19. Tim gia trj Idn nha't \k gia tn nho nha't cua ham s^. - iv;, i, y = sin^x + >/3cosx.
Hifdng dSn giai Ta c6: y = sin^ x + ^/3 cosx , v . . f<*.
Tim gia tri Idn nha't: V i jsinxj < 1 nen s i n \ sin^x, Vx
=> y = sin^ X + VJcosx. Vx
Ta chtfng minh: sin"* x + >/3cosx < >/3 (•) That vay: (*) ô 73 (1 - cosx) - (1 - cos^x)^ > 0
•3)
<=>(] - c o s x ) r V 3 - ( l - c o s x ) ^ ( l + cos)2 > 0
C6si ta c6: (1 - cosx)(l + cosx)^ = 2 ~ 2cosx)(l + cosx)(l + cosx)
a 2
2 - 2 c o s x + l + cosx + l + cosx n3 1
< —
2
(-.3
N3 32 fz
Suy ra 73 - (1 - cos x ) ( l + cos x)^ > 0
^ § y ( l - cosx) r 7 3-(l-cosx)(l + cosx)^ > 0 , V x € R .
=> y<73 => yn,^^ =73 k h i c o s x = 1 o x = k 2 7 i , k 6 Z . . f i . T^ra gid tri nho nha't cua y:
Ta c6: y = sin^ x + 73 cosx ^ -sin'* x + 73 cosx ' * ~ * -^ - ' • " < ' '
• . ' i f
Ta churng minh: -sin^ x + \/3 cosx > -yl3 , Vx
<::>(1 + C O S X ) > / 3 - ( l - c o s x ) ^ ( l + cosx) 1
> 0 . Vx
X 6 t ( l + c o s x ) ( l - c o s x ) ^ = - ( 2 + 2 c o s x ) ( l - c o s x ) ( l - c o s x )
^ -1 2
2 + 2COSX + 1- C O S X + 1- C O S X 1
< —
2 . 3 . 27 Suy ra:(1 + cosx) > / 3 - ( l - c o s x ) ( l + cosx)^ >0, Vx
Vay yn,jn = - 7 3 . Dau " = " xay ra khi cosx = - 1 o x = T: + k27i, k e Z . [ x - m y = 2 - 4 n i
Bai 20. Gpi (x. y) Ik nghiem cua phiTdng irinh | ^ ^ ^ y ^ 3 n ^ + i ^ tham so. Tim gia trj Idn nhat cua bicu ihuTc A = x^ + y^ - 2x, khi m lhay doi,
Hif(}ng dSn giai
* • ^ 3m2-3m + 2 Ta c6: < x - m y = 2 - 4 m (I)
mx + y = 3m +1 (2)
X = •
m ^ + 1 y = -
4m^ + m + l m^ + 1
^ A = r + y ^ - 2 x = ^ ^ " ^ ' / " ^ ^ ^ T a p x d c d m h : D = R m^ + 1
^ . ^ 4 m ^ 3 6 m - 4 . ^ . ^ Q ^ ^ ^ . ^ g ^ _ j ^ 0 ^ m, = m2 =
- 9 - V 8 5 2 - 9 + >/85
m - 0 0 m, m 2 +00
A ' + 0 0 +
A 19
A,,,.., =10 + 785 khi m = -9->/85 u 'max • ' - 2
Bai 21. Tim gia irj l<?n nhat, gia trj nho nha'l cua hkm so y = x*" + 4(1 - x^)' ir^"
d o a n [ - l ; IJ.
Hi/(7ng dan giai n.
Ham so y = x*^ + 4(1 - x^)'; vdi - 1 ^ x £ 1. Dai t = x \i - 1 < x < 1 0 < l ^ ' Khi do y = t ' + 4( 1 - t ) ' = - 3 t ^ + 121^ - 121 + 4
khao sat f(l) = -3t^ + 12t^ - 12t + 4; 0 ^ t ^ 1 f.(t) = -9t' + 24t - 12; r(t) = 0 o t = j . t = 2.
Vav maxy = 4 ; miny = —. ^ . / , f. j \
l-i;ir 9
Bdi 22. Chii'ng minh r i n g vdi moi thifc diTdng x, y, z th6a man x(x + y + z) = 3y/
ta c6 ( X + y)^ + (x + z)^ + 3(x + y)(x + z)(y + z) ^ 5(y + z)^
HiTdng dSn giai
Dat a = X + y, b = X + z, c = y + z. Dieu ki?n x(x + y + z)= 3yz trd thknh:
c^ = a^ + b^ - ab . Bat ding thtfc can chtfng minh tifdng difdng:
a' + b^ + 3abc <, 5c^ a; b; c diTdng thoa man dicu ki^n tren.
c^ = a^ + b ^ - a b = (a + b ) ^ - 3 a b > ( a + b)^ - - ( a + b)^ = - ( a + b)^
4 4
=>a + b < 2 c (1)
a* + b ' + 3abc < 5c^ o (a + b)c + 3ab < 5c^
(1) cho ta: (a + b)c < 2c^ 3ab ^ - - (a + b)^ < 3c^ tCf day suy ra dicu phai 4
chuTng minh. Dau bling xay ra khi: a = b = c<::>x = y = z. ;
^^al 23. Cho X , y la cac so thifc thoa man dicu kipn: x^ + xy + y^ < 3. X Chtfng minh r i n g : -4>/3 - 3 < x^ - xy - 3y^ < 4v/3 - 3.
, Hirrfng dSn giai . DatA = x^ + xy + y ^ B = x ^ - x y - 3 y ^ ff ( X i
* Ne'u y = 0 thi theo gia thicl A = x^ < 3; B = x l ' Do do -4>/3 - 3 < B < 4 > / 3 - 3 (dpcm).
* Ne'u y ^ 0. pat t = i i . Ta co: B = ' ' > = A 4 ^
y x + x y + y t +1 + 1 1
Ta tim t4p gid trj cua u = t ^ - t - 3 t ^ + t + l
o ( u - l ) t ^ + ( u + l)t + u + 3 = 0
V i a = u - l v a b = u + l khong dong IhcJi b^ng 0 nen mien gid tri cua u 1^
- 3 - 4 ^ / 3 , ^-3 + 473
A > 0 < ằ - < u < - •. Ta c6: B = A.u va 0 < A < 3. jv 3 3
Dodo - 4 V 3- 3< x ^ - x y- 3 y ^ < 3 < 4 V 3 - 3 (dpcm).
Bai 24. Cho x, y, z Ik ba so thoa man: x + y + z = 0. ChuTng minh r^ng:
73 + 4" +^3 + 4^ + V 3 + 4' >6.
Hi^dng dSn giai
Ap dung ba't d i n g thtfc Co-si cho cdc so di/dng, ta c6:
3 + 4*= 1 - H + 1 +4*> 4 # ' = >N/ 3 + 4'' >2>/VF = 2 . ^ ."TiTdngtir: > 2 > / # ' = 2.^sf4^; N /S ^ > 2 N / ^ = 2 . ^
V$y: ^3 + 4" +^3 + 4^ +V3 + 4'- >2 # ' + ^ + ^4^
Bai 25. Chtfng minh r^ng vdi mpi x, y > 0 ta c6: (1 -H X ) D i n g thtfc xay ra khi nao.
Hif(}ng dSn giai
Ap dung baft d i n g thufc Co-si cho cac so diTcJng, ta c6:
1 + >256.
' * ' - l + x = l + - + - + - > 4 4 3 3 3 ^
. 9 , 3 3 3
V a y d + x )
^ ; 1 + ^ = 1 + — + — + — > 4 4
" 3x 3x 3x ) 3 \ x ' 1 . ' >164
x2
1 + -
I >/yJ
Dau d i n g thiirc xay ra o
>2564 x'' 3**
=256 (dpcm)
^ = 1 3 1 = ^
3x
x = 3 y = 9
. . i " li •a
691 26. Cho a, b, c li so diTdng thoa man: a + b + c = - . ChuTng minh rlnf 4
^ / a + l b + ^ b + 3c + ^c + 3a ^ 3 . Khi n^o d i n g thtfc xay ra.
, each 1: Ta c6: ^(a + 3b)l. 1 < ; = ± ( a + 3b + 2) Hitdng d i n giai
. a + 3b + l + l 1
"3
V ( b T 3 ^ ^ ^ i ^ ^ 4 ( b + 3c + 2)
3/(rr3a)l.l .1 <
3 3 c + 3a + l + l 1 , , ^
= - ( c + 3a + 2)
^a + 3b + ^ b + 3c + ^c + 3a <-5-[4(a + b + c) + 6 ] ^ - ! - 4 . - + 6 4 = 3 Dau " = " x a y ra o a + b + c = —
4 o a = b = c = - . a + 3b b-I-3c = c + 3a = 1 4
* Cach2:Dat x - + 3b => x^ = a + 3b ; y = 7bT3c => y^ = b + 3c;
z = \/cT3a=>z-^=c + 3a=>x-^-hy^+z^=4(a-Hb + c) = 4 . - = 3
4 >
Bat d i n g thiJc can chiJng minh o x + y + z < 3 .
T a c 6 : x ' + 1 + 1 > 3^x^1.1 = 3 x ; y ' + 1 +1 > 3 ^ y \ l . l =3y ; z ^ + l + l > 3 ^ z ^ l . l = 3 z : : ^ 9 > 3 ( x + y + z) ( v i x ' + y^ + z' = 3) V a y x - i - y + z < 3 . hay ^a + 3 b + ^b + 3 c + ^ c + 3a <3
D a f u " = " x a y r a x ' = y^ = z ^ = l v a a + b + c = l
o a + 3b = b-i-3c = c + 3a = l v a a + b + c = - o a = b = c = -
4 4"
B^i 27. Cho a > b > 0. ChuTng minh r^ng:
2') Hi/dng d i n giai Ba't d i n g thtfc can chtfng minh tifdng diTdng vdi:
(1 + 4")" < (1 + 4")" o '"^^-^^'^ < '"^^^4")
a b
X6thams6':f(x)=-l51^ili^ v d i x > 0 .
2 -
Ta c6: f (x) = 4 M n 4 x- ( l + 4- ) l n ( l + 4-) ^ ^ x2(l + 4'')
suy ra f(x) nghjch bien tren khoSng (0; -1-00).
Li^n gUti de truOc 1^ thi DH 3 rnUn Bdc, Trung, Nam ToOn hoc - Ngu_) <-n van 7hang
Do f(x) nghjch bien trSn (0; +oo) a ^ b > 0 n6n f(a) ^ f(b) ttr d6 ta c6 dic^
phii chtfng minh.
Bai 28. Cho x, y, z Ih ba so di/dng \kx + y + z<.l.
Chiirng minh r^ng: Jx^ +-\+ + M +\^yf^
i , 7 ? V X \ V z
Vdi mpi u , V ta c6
U +V
-• —• — -.
U +V u + V
Hi /dng dSn giai
(*)
- 2 - 2 2 2 _ _ -. 2 ^
= u + v + 2 u. v < U + V + 2 u V — u + V
r /
( 0 _ f 0
x ; - , b = y ; - , c = l y)
( v l (
Dat a =
Ap dung bat ding thuTc (*) ta c6:
1 z ; -
b + a + b a + b + c
V a y P =
* Cdch]:
f l 1 if
^(x + y + z r + —+ —+ —
^(x + y + z r +
Ix y , z j
Ta c6: P > ^(x + y + z r +
Vdi t = (^/xyz) = > 0 < t ^
f l—+ 1 —+ lY -
y zj
' x + y + z ^ a .
9 Dat Q(t) = 9t + - Q'(l) = 9 - ^ < 0. Vt 6
t t'' Q(t) giam tren 0;-
l 9 j
Q ( 0 > Q
. 9 , = 82. Vay ? > y[Q(l) > slS2 (DSu " = " x^y ra khi X = y = z = - ) . 1
•J
* Cdch 2: Ta c6: (x + y + zf +
= 81(x + y + z)^ +
>18(x + y + z)
1 1 if
- + —+ - U y z) 1 1 if
—+ —+ - U y z j
—+ —+1 P - Vx y z ;
-80(x + y + z)2
-80(x + y + z)^>162-80 = 82 V$y PSN/82.(Dau"="xayrakhix = y = z = - ) 1
Tim GTLN, GTNN cua bieu thtfc: P = a'*+b^+c'ằ
(a + b + c)'* '
HU^ng dSn giai
Khong maft tinh long qudt, ta gia suf: a + b + c = 4 => abc = 2.
pat t = ab + be + ca. Ta c6:
Q • = (a^ + b H c^)^ - 2(a^b^ + b V + c^a^)
(a + b + c)^-2(ab + bc + ca) ^ - 2 (ab + be + ca)^ -2abc(a + b + c)
= (42 _ 21)2 _ 2(t^ - 16) = 2(t^ - 32t + 144).
2 2 t = ab + be + ca = a(b + c) + be = a(4 - a) + - = -a^ + 4a + -
a a Ma (b + c)^>4bc<:>(4-a)2> - o (a - 2)(a^ - 6a + 4) ^ 0
a
<=>3--j5<a<2 v U 0 < a < 4 ) ; 2 r - ^ - n . 5x/5-l
X6tt = -a^ + 4 a + - , a e 3->/5;2 a L Xet f(l) = 2(t^ - 32t + 144), t e D = 5;
: > 5 < t ^ 2 ta c6 :
maxf(t) = f(5) = 18; minf(t) = f
D D = 383-165>/5
Vay maxP = — dat diTi^c chdng han khi x = 2; y = z = 1.
128
minP = ^ ^ l : i i ^ ^ datdirclcchdnghankhi x = 3->/5; y = z = - ^ ^ .
256 2 2. Cho cae so diTdng a, b, c vdi a + b + c ^ 1. ' ' ,
Tim GTNN cua bicu thiirc: P = 3(a + b + c) + 2
HiTc ^ng dSn giai
Taco: (a + b + c)
(I 1 0
—+ —+ —
U b cj
11 : t Jjj' -
L: f 1 1—+ — + - >3^abc n
/' 1 N
U b c> , VabcJ a b c a + b + c
I
Luy^n gidi di trade kff thi DH 3 miin Bdc, Trung, Nam Todn hoc - NguySn Van Thdng
1 8
Do do: P > 3(a + b + c) +
,^;'f:>+ a + b + c
= 3 6^
t + — = 3f(t) Trong do 0 < I = a + b + c < 1 va f(t) = t + Y .
ih A r t i ) 5SS '-,h.! O i l
Ta c6: f ( t ) = l - — = 6 t
( 0 ; l ] = > f ( t ) > f ( l ) = 7, V t e (0;1
0, Vt 6 (0;I , nen ham so' nghich bie'n ix(.
COng ty TNHH MTV DWH Khang Vm
Vay minP = 21 dat diTdc khi a = b = c = - .
3 I
Bai 3. Cho a, b, c > 0 va a^ + b^ + = 1. Tim GTNN cua bieu thiJc:
P = - + - + - - ( a + b + c ) . ' • t - . a b c
Hif(}ng dSn giai D a t t = a + b + c <73(a^ + b^+c^)=>0<t<N/3.
9 „ 9
, c Cho cdc so' Ihifc khSng am a, b, c th6a mana + b + c = l .
^^-nrn GTNN. GTLN cua F(a. b. c) = (a - b)(b - c)(c - a).
Hif()ng dSn giai
VI F(a, c, b) = (a - c)(c - b)(b - a) = -F(a; b; c) suy ra mien gid tri cfla F Ik t i p doi xtfng vl vay ta chi can tlm GTLN cua F (a; b; c).
f , N^u trong ba so a, b, c c6 hai so b^ng nhau thl F(a; b; c) = 0 ' , Md'u a, b, c doi mot khdc nhau thl khdng m3ft tinh tdng q u i t gi^ sOr a = max{a;
b; c} khi do ne'u b > c thl F(a; b; c) < 0 do vay ta chi c i n x6t a > c > b. Dat x - a + b = > c = l - x . vr,v;r:L. y ri.
Ta c6: F(a; b; c) = (a - b)(c - b)(a - c) ^ (a + b)c(a + b - c) .
= X ( l - X ) ( 2 x - l ) = : h ( x ) . , :.Mi.}^. .'y.K-.:.
1 ' - ^ ' ^ • . X6th(x) = x ( l - x ) ( 2 x - 1 ) . - < x < l , ằ
Tac6: i + i + l >
a b c a + b + c 9
•P> (a + b + c) = — t = f(t).
a + b + c t Xet: f(t) = - - t , t 6 =>t"(t) = - 4- - l < 0 vay hkm so nghich bicn
t \
tren (o.Vs] => f(t) >i[S) = 2S, Vt€(0;V3".
Vay min P = 2N/3 dat diroc khi a = b = c = ^ .
• • 73 Bai 4. Cho 4 so thifc a, b, c, d th6a man: a^ + b^ = 1; c - d = 3.
Tim GTLN cua bieu thiJc: F = ac + bd - cd < 9 + 6V2
Hi^(}ng dSn giai
Tac6: F < V(a^ + b^ ){c^ + d^) - cd = 42<i^ + 6d + 9 - d^ - 3d = f (d)
1 - Tacd f ( d ) = (2d + 3)-
A2
d + 2 j 2d''+6d + 9 V I nen f ( d ) ^ f 3^ 9 + 6>/2
I 2) 4
V | y minF = ^ i ^ ^ dat dir<?c ching han khi d = - - ; c = - ; a = —i=;b = - i ^ 4 2 2 V2 v2
h'(x) = -6x^ + 6 x - l = 0 o x = 3 + >/3 3->/3
V X = • (loai v l X e L$p bSng bie'n thidn ta duTdc: h(x) < h { 3 + V3'
= : ^ , V x . 18
V | y maxF = — dat difdc chlng han khi a = ^ ^ ^ . b = 0.c = ^ — ^ .
18 6 6 minF = -—d a t d i T d c c h i n g h a n k h i a = ^ ^ ! ^ , b = ^ ^ ^ , c = 0
18 6 6 ằ / Bai 6. Cho ba so thifc dtfdng a, b, c thoa: 21ab + 2bc + 8ca < 12.
Tim GTNN cua bieu thuTc: P = i + - + - . . • a b c
Hi/(?ng dSn giai Dat: x = - , y = - , z = - = > x , y , z > 0 .
a b c
Khid6:21ab + 2bc + 8ca<12 ^ 2 2 3 1 2 3
< ^ - + 4.—+ 7.-<2.-.—.-<:=>2x + 4y + 7z<2xyz vaP = x + y + ' z . a b c a b c
Ttlf: 2x + 4y + 7z ^ 2xyz => z(2xy - 7) > 2x + 4y =:> z >
V ^ ^ 2x + 4y
x + y + z > x + y + =^ = x +
2x + 4y
2 x y - 7 7 2x + - ( 2 x y - 7 ) + 2 x y - 7 2x 2x
2 x y - 7
2 x y - 7
= x +
, _ 7 7 2x-H-2(2xy-7) + ^ 14
2 x y - 7 , ^ , - ' x ^ - ^ ^ ^ ^ H ^ 2 x y - 7 ^ ^ 2x 2x 2 x y - 7
, Ap dung ba't ding ihuTc Co-si la c6:
14
2x 2 x y - 7 ^
2x 2x 2xy - 7
2 x y - 7 2 x - ^ y _
= 2.
2x •2xy-7 ^ 1 + - Dod6: P > x + — + 2.
2x \ X
Tac6: f'(x) = l - 11 14 • ' 1
2x^
X
Ta th^fy r ( x ) tang khi x > 0 \h r(3) = 0 = > x + y + z > f ( 3 ) = Y .
Ding thuTc xay ra khi:
x = 3
2 x y - 7 +
14 x = 3 5 ' = 2 2x 2xy - 7
x = 3 5 ' = 2 _ 2x + 4y z = 2
2 x y - 7
1
5 3 '=2
(>,)
VSy minP = ^ .
I I I . B A I T A P T V L U Y $ N C 6 HlTdNG D A N
Bai 1. Tim gid tri Idn nhat. gid trj nho nhat (neu c6) cua h i m s^ sau:
a. y = 2x^ - 3x^ - 12x + 1 tren doan [ - 1 ; 4]
4
b. y = + x^+1 trendoan [ - 1 ; 3]
4
c. y = - ^ ^ i i l iren doan [-3; - 2 ] 3 x - 2
2 x 2 - 3 x + 3 . . . d. y = : irendoan x - 1
3 9 2'2 e. y = 2x - Vx^ +1 tren doan [-2; 5]
x^ +3x
f. y = trenkhoang ( 0 ; + o o ] .
g. y = cos^x.sinx trdn [0;7c]
x^ - 3 x + 2| tr6n [-1;3]
x-^ + 3x2 _ + 90| tren [-5; 5]
sinx + 1 K. y ' 3 _ c o s 2 x
Hi/(}ng dSn giai niaxy=y(4) = 33. miny = y(2) = -19
1-1:41 l-i;4|"
maxy = y(V2) = 2, min = y(3) = - ^ maxy = y(-3) = A ^_™„^y.y(_2) = l
^9^
2.
60 min =y(2) = 5
;L no.
3.9
^2'2j L 2 ' 2 j
e. maxy = y(5) = 1 0 - > ^ , min y = y(-2) = ^ - N/S
|-2;5| |-2;5|
f. min y = yf>/3) = 2V3 , khong ton tai GTLN.
g. min y = y(0) = y(7i) = 0 , max y = dat diTdc khi x = arcsin - ; = .
|0;n| |();n| 9 V3
h. miny = y(l) = o, maxy = y(3) = 20.
I-I;3| |-l;3|
i. n m y = y(-5) = 40O. , ., ^ ^ „ . ,
min y = 0 vdi x„ e (1; 2) Ih nghiym ciia phiTdng trlnh: x^ + 3x^ - 72x + 90 = 0.
miny = 0. m a x y = ^ " ^ ^
' 'KM ifiii^i.
•I • 4
2. Cho hai so thi/c x. y th6a man: j ' ' ^ ^'^^ ^ Tim gid trj nh6 nha't, gid tri [x + y = 3
l<Jn nha't cua bieu thtfc: P = x^ + 2y^ + 3x^ + 4xy - 5x.
Hif<}ng dSn giai Tac6y = 3 - x ^ l = > x ^ 2 = > x e [0;2]
Khi 66: P = x' + 2(3 - x)^ + 3x^ + 4x(3 - x) - 5x = x^ + x^ - 5x + 18 = f(x) X6t hkm so f(x) tren [0; 2] ta c6: r(x) = 3x^ + 2x - 5 => f (x) = 0
*>x = 1 V x = - - (loai v l x G [0; 2])
Vay maxP= max f(x) - f(2) = 20, minP = minf(x) = f(l) = 15
|0;21 |0;2|
B^i 3. Cho X > 0 va Ihifc y th6a man: x^ - x y + 3 = 0
2x + 3 y S l 4 . Tim gi& tri Idn nhat gid tri nho nhat cua bieu thtfc: A = 3x}y - yh - 2x(x^ - 1).
^ ..,.1 Hi^ngdSnglai y =
2x + 3(x'+3) 5x^-14x + 9 ^ 0 o l ^ x ^ - .
^14 Tit gia thiet suy ra
Khidi: A = 3x(x' + 3 ) - ^ ^ ^ - ^ - 2 x ' + 2 x = 5 x - - = f(x) ^
9-1 >'
Tren doan
'••1 hkm so f(x) luon dong bien nen ta c6 max A = maxf(x) = f
.5) = 4, minA = minf(x) = -4 = f(l)
I ; ?
Bai 4. Cho a, b la cac so diTdng thoa man a^ + b^ + c^ ^ - . 4 Tim GTLN cua bieu thtfc: P = (a + b)(b + c)(c + &)+ \ \ \
a-* b'^ c-*
H\i6ng din giai Ttrgid thict - > a^ + b^ + c^ > 3%/a^b^c^ abc < -J- Ap dung BDT Cosi, ta c6: (a + b)(b + c)(c + a) > 8abc
3 , • t y . 1 i 1 3
a-^ b^ c' abc P>8abc + -
abc • Datt = abc=i>0<t< - P^8t + - = f(t) 1 3
8 . t
V
X6t h i m f(t) c6 f (I) = 8 - — > 0 => f(t) > f
.8. = 25. Vte 0;1
V 2 P ^ 25. D i n g thurc xay ra o a = b = c = -
Bai 5. Cho x, y, z > th6a man x + y + z = 1.
Tim gid tri nho nhat cua bieu thtfc P = - + - + i + 2xyz .
X y z
HuT^ng dSn giai
: i- + - + - S- ^ i = = > P ^ - ^ + 2xyz = - + 2t^ =f(t) - >
X y z 3^tyz ^xyz ' t Trongd6,tadat t = 3 / ; ^ ^ 0 < t ^ ^ ^ i ^ = ^ X6t him so f(t). ta c6: f'(t) = et^ - = ^
>f(t)>f ' \ 3
> 0 . Vt G 0:
V§y minP = 2973
4? y i W n , byiiiJj ii) ayjdj nairi srw;,"' ;., ^ B&i 6. Tim ta't ca cac gia irj cua a vji b thoa man dieu ki$n: a > — \k - > 1 sao - 1
2 b
cho bieu thiirc P = 2a^ +1
———- dat giii tri nh6 nha't. Tim gia tri nho nhat d6.
b(a - b)
HtfcJng dSn giai t\i
TO gia thie't, ta suy ra a 0 b(a - b) > 0 r.i^ mal o;' ) .t .
Ta c6: 0 < b(a - b) < — va 2a'+ 1 > 0 nen P ^ ^^^^^Jii^ = f(a).
4 a^
X(St ham so f(a), a > - - c6 f'(a) = "^^^^ => f(a) = 0 o a = 1 ' 2 a
BIng bien thicn
a -1 2 1 )-a
A . +00
+ - 0 +
f(a) -^+00 +00-...^
" ^ 1 2 ^ ^
^'-'B'l Jii>d; iiri
Tir bat ddng thuTc => f(a) > 12 Va > => P ^ 12
E>^ng thti-c xay ra o
VayminP= 12.
a = — 1
3> ,0)'> w. frf.f' hoac
b = — 4
a = l 2
B a i 7. Cho bon so nguyen a, b, c. d ihay ddi th6a: l ^ a < b < c < d ^ 5 0 . T i m gj^
trj nho nhat cua bieu thtfc P = - + £ (Dir bi - 2002). , - , v*,;ô,> Hii^^ng dfin giai • •
V i 1 < a < b < c < d < 50 va a, b, c, d la cac so nguyen n&n c > b + 1.
_ a c ^ 1 b + 1 ^ Suyra: — + — > — + = f ( b ) .
b d b 50 , „ Dc tha'y 2 < b < 48 nen ta xct ha m so: f(x) = - + , x e [2; 48]
X 50
T a c 6 f'(x) = — = > r ( x ) = 0<i>x=5N/2v x = -5yf2 (loai v i x e [ 2 ; 4 8 | )
X 50
L a p bang bicn thien la diTdc niin f(x) = fiSy/l] -~ ^ôằằ<-
|2;4S| V /
l\k 8 la hai s^ nguyen gdn 5^2 nhal v i vay: ^ j j ^ ^g, „,, min f(b) = min { f ( 7 ) ; f(8)} = min I — ; —1 = —
|2;4ii| \^ ' ^ ' I [175 2 0 0 ] 175
V a y G T N N P = — .
175 min
B a i 8. Cho lam gidc A B C khong lii. T i m G T L N cua b i ^ u thtfc:
P = cos2A + 2yf2 (cosB + cosC) ( K h o i A - 2004) Hi/(7ng d S n giai
Ta c6 A < 9 0 cos2A = 2cos^A - 1 <, 2 c o s A - 1 = 1 - 4sin^ — D d n g l h t f c x a y r a o c o s ^ A s c o s A (1). >
c o s B + co.sC= 2 s i n — . c o s ^ — ^ : S 2 s i n — ' 2 2 2
B —C
Ddng ihtfc xay ra o cos 1 (2)
• •*'A''"'^ J 2
D a i l = s i n — = > 0 < t ^ — . T a c o : P ^ - 4 l ^ + 4 > ^ + 1 = f ( t ) 2 2
X6l hhm so f(l), 1 6
2
Lap bang bicn Ihicn la c6: f ( l ) < 1'
, c 6 f ( t ) = - 8 t + 4 N ^ = > r ( t ) = 0 o t = —
v 2 . = 3= > P^ 3 . i
Dilng ihuTc xay ra A = 9 0 "
. V a y m a x P = 3.
B = C = 45"
, Cho c a c so I h i f c d t f d n g x, y. T i m gid i r i Idn n h a t c u a bieu thuTc:
4xy^
P =
x + Vx^+4y^)"
HifiYng d&n giai D a t x = ty t a c 6 P = - 4l
l + x/[^ + 4
- . t > 0 = > f ( l ) =
l ( N / t^ + 4 - 3 l ) Vl^ +4fl-*-Vt^ +4
_>f(l) = 0 o V l ^ + 4 = 3 t < ằ l = - = r .
Lap bang bicn ihicn la di/dc max 1(1) = f 1 I V 2 J 8
Bai 10. Tuy thco gia i r i cua lham so m, hiiy l l m G T N N cua bieu ihtfc:
P = (x + m y- 2 ) ^ +[2x + ( m- 2) y - l f . llu'ô?ng dan giai [x + m y- 2 = 0
X c l h O : < (*)c6 D = - m - 2.
[2x + ( m - 2 ) x - l = 0
• Ncu m 5* - 2 => (*) CO nghiOm d u y nhal (XD; yo) fx = x„
Khi do P > 0, P = 0 o <^ ininP = 0.
iy = y() ^ ^ ' ' i = S £ - J o O = ' y m =- 2 la c6: P = ( X - 2y - 2)-+ {2x - 4y - 1)^
Dat I = X - 2y - 2, la c6: P = l ' + (2l + 3)^ = 5 r + 12l + 9 , g
Suyra m i n P = - .
B ^ i l l . C h o cdc so ihifc m, n, p, a, b > 0 sao cho:
m + n + p = mnp = — a mn + np + pm = —
a Tim G T N N cua: P = 5 a ^ - 3 a h + 2
a ^ ( b - a )
Hif(7ng d S n giai
T a c 6 : ( m + n + p) > 3(mn + np + pm) m + n + p ^ 3^mnp
a a Va
3a a ^
Xet M m so' f ( b ) = 5 a i^ : 3 a b ^ f'(b) = < 0 a ^ ( b - a )
Suy ra f(b) giam tren X6t hkm g(a), vdi a e
I 3a. => f(b) ^ f
a ^ ( b - a) 2
^ 1 ^ 3 ( 5 a^ + l )
3a j 2 =g(a)
a ( l- 3 a 2 ) 0; 1
373
> g(a)> g ^ 1 ^
3^3 j
< 0 , Vae 0;
373
Vay minP = l2>/3 datdiTdckhi ' ' ^ J ^ - b = \/3.
Bai 12. Cho cac so' thi/c a. b. c thoa a^ + b^ + = 1.,^^ ^.^ ^, - j , , - • , Tim GTLN cua bieu IhiJc: P = 2ab + be + ca .
Hi^(7ng dSn giai
Ta c6: |P| <2|ab| + |c||a + b| <a^ + b^ + |c|^2(a^+h^) = I -c^ + |c|72-2c2
= t= > 0< t < l va xet ham s o f ( t ) = l - l ^ + tV2-2t^ , 0< t < i . 2 r 2- 4 t^ - 2 t V 2 - 2 i ^
• f ' ( t ) = - 2 t + V 2 - 2 t ^ -
=>f'(t) = 0 o l- 2 t ^ = t V 2 - 2 r o
Bang bien ihien
7 2 ^ _ 1 _
72 61"-61^+1 = 0
0 < t < 3-7^
t 0 t() 1
f(t) + 0
f(t) ^ -— — •
Di/a v^o bang bien ihicn ta suy ra 1(1) < f(t(,) = 1 + 73
P < P < 1 + 73
D^ng thiJc x5y ra o
a^ + b ^ + c ^ ^ l 2ab + be + ca > 0 a^ + l
a = b
3-S
6 1 + 73
a = b = 3 + 73
' 12 c = 3-73
.q + n + m) 2
:63 <
Vay minP = — ; maxP =
2 2
o n
0di '^^'^ ''^"'''^ dUWng a, b, c th6a a + 2b + 2c = a b c . Tim g j ^ irj "^6 nha't cOa P = a + b + c.
Hi/(}ng dSn giai
=>a + b + c > - - — - + b + c ^ - - + 2 7 b c=-T + 2t = f ( t )
be - 1 be - 1 i2 _ J >
X6t h a m so f(t) v d i t = 7bc > 1. lihn orj iTH) rfnh/ gfibifrk; ,•. ,
f (t) = 0 o to = 72 + 75 (Do t > 1). i^^"^^ ô)A i&i^ ^ ,
U p b a n g b i e n t h i e n => f(t) > f(t„) a + b + c ^ f(to)
Ding thite xay ra o
b = c = to
a = • + b + c) = f(t„) Bai 14. Cho cac so thiTc khong am a, b, c c6 tdng khdc 0.
Tim gid tri Idn nhat, gid tri nho nha't cua bieu thtfc: P = ^"^ + b^+^6c-^
(a + b + c)^
Hirdng dfin giai he i
D?t f(a- b- c) - '^^^ -^^^^^ '* "^''^^ ' ^ ^ S"^*^ S " ' ' (a + b + c)^ •
VI f(ka; kb; kc) = f(a; b; c), Vk > 0 nen khong giam U'nh tdng quat ta gia thiet a + b + c = 1, khi dd f(a; b; c) = a' + b ' + 16c\
Tac6:a^ + b' > i( a + b) ^ = l( l - e ) ^ . ^ V d /
n6nf(a;b;c)>-r(l-c)^+64c^l = l g ( e ) , 0 < c < l ^ ^n-nU^m^^t:
4L J 4
Khao sdt ham g(c), ce [0;1] ta diMc d >v;
g ( c ) > ^ = > f ( a ; b ; c)>i- g ( e)>iy= > m i n P = i y - , i : • Mat khdc: f(a; b; c) = a' + b ' + e' + 15c' ^ (a + b + c ) ' + 15c' < 16 .
Suy ra maxP = 16. •uMiu.t:...^..-^ t ;
PHtfdNG TRINH, Bid PHlJCiNG TldNH VA H6 MJONG TRINH DAI S6
I . T 6 M T A T L i T H U Y E T ' ^
1. Giai phiTdng t r i n h b?c nhat va h? phadng t r i n h b|lc nhat hai a n . a. Phuang trinh dong: ax + b = 0 * , , * •
Neu a 5t 0. phiTcfng trinh (PT) c6 nghi0m d u y nhat x = - - . 3)1 5?; m / Nlfu a = 0, b 0. PT v6 nghi^m
Neu a = 0, b = 0, PT CO v6 so n g h i c m ( m o i so thiTc x t i l y ^ l a mpt n g h i e m ) . b. Phuang phdpgi&ih$ phuang trinh b^cnhdt hai dns6
fax + b y - c .„^. ^ ^ 1 ^ , |i^id,|i,,|d [ a ' x + b ' y = c'
- Phifdng p h d p the. Tif mpt trong h a i PT rut mpt a n theo a n k i a , the v a o PT thi?
h a i , ta diTdc PT b a c nhat mot a n . Giai PT n ^ y d e tlm a n , r o i suy ra a n kia.
Gia s i j r b ' ; i 0 t h l t a c 6 :
, c - a x [ a x + b y = c
la'x + b ' y = c'
y =
b' c'-a'x
b"
Phirdng phdp cong dai so. Nhan (hoSc chia) hai vc cua mot PT vdi mot s6'de lam cho mot an ciia hp d hai PT c6 h? so bllng nhau (hoSc doi nhau). Sau do, trir (hoac cong) theo tifng ve cua hai PT de khijr an 66. Giii PT nhy Urn difdc mot an, roi suy ra an kia. Gia sir a ^ 0, a' # 0 th\a c<3
ax + by = c faa'x + a'by = a'c f(a"b-ab')y = a c - a c [a'x + b'y = c' [aa'x + ab'y = ac' [ax + by = c
+ 0
Neu ab' ^ a'b thi h$ c6 n g h i c m duy nhat (x; y) = Neu ab' = a'b; ac' * a'c thi hp v o nghipm
Neu ab' = a'b; ac' = a'c thi hO c6 v6 so nghipm (x; y) =
^cb'-bc' ac'-a'c^
,ab'-a'b a b ' - a ' b .
m,-c - a x , m tiiy y 2. Giai phi/dng t r i n h b | c hai va phiTcfng t r i n h quy ve b | c hai
- Phtfdng phdp giai phiTdng trinh biic hai ax^ + bx + c = 0 (a ?t 0) Tinh A = b^-4ac (hoSc A' = b'^-ac v d i b = 2b')
Neu A < 0 (hoac A'< 0 ) , PT v6 nghicm
Mau A = 0 (hoac A* = 0). PT c6 nghicm kdp x = - — (hoac x - - — )
2a 2a fjeu A > 0 (hoac A' > 0), phi/rtng trinh c6 hai nghicm phan bi0t , ^ ,
X, 2 = (hoac x,2 = ) - •• ^ . ^ - ' < ' •
• PhU(fng phdp giai phiiOng trinh bqc cao, b^c Idn hOn hai ^ *
Ta thirdng giai b^ng cdch quy vc ciic PT bac nhat v^ PT bac hai dvfa v i o ca(
phep bid'n doi nhtf sau: i <=> <7:~ \ }\
• E)i/a ve PT tich: g(x).h(x) = 0 o g(x) = 0 hoSc h(x) = 0 ^5 ,^| , , ,,,.
ô Dira hai ve ve hai IQy thiTa ciing bac va dp dung ^s^.y , • , ' f ( x ) = g(x)
.f(x) = -g(x)
f2''^'(x) = g2''^'(x)<^f(x) = g(x) ( k e N * ) : ô
• Dira ve PT dang af^(x) + bf(x) + c = 0. Dat X = f(x), roi tlm an phu X.
C h i i ^ :
- Vdi PT (X + a)(x + b)(x + c)(x + d) = mx\g d6 ad = be, ta bien doi ve f2'''(x) = g"'(x)<:*
ad x + — + a + d
X
be ^ x + — + b + c
X = m . roi dat an phu y = x + — ' - Vdi PT (X + a)(x + b)(x + c)(x + d) = m, trong d6 a + d = b + c, ta d^t
y = (X + a)(x + d) hoac y = x^ + (a + d)x + ' t i * rtSH ,^,r , - V d i P T ( x + a)' + (x + b)' = c , t a d 4 t y = x + a + b
* DOng bat d^ng thtfc |A| > A ; A^ > 0, klTt htfp sur dung tinh cha't dong bien va nghjch bien cua h^m s6' trong liTng khoang.
* Boi vdi phi/dng trinh chuTa an d mau thtfc.
LiAi y dat dieu kiC*n ciia an dc mau thuTc khdc 0. Roi cQng si? dung cdc phdp bi^ndoi nhirtren.
^' Giai phiTdng t r i n h c6 an trong dau giii trj tuy?t doi hoSc di/di dau can Phtfdng phdp giai phiWng trinh c6 an trong dSfu gid trj tuy?t doi
PhiTdng phap chung la bien doi di; bo dS'u gid trj tuypt doi. fu S :b.
C h u ^ : f ( , ) _ [ f W . k h i f ( x ) > 0
[ - f ( x ) , k h i r ( x ) < 0 ' ,
"l(x) = a f(x) = -a f(x)| = a ( a > 0 ) o
LuyCn gid: Iruik- kyjhUmj^mjdnBOrrrm^^^ lUfC - Nguyen van I mr^
|f(x)| = |g(x) •f(x) = g(x) f(x) = -g(x)
* PhMiig phdp gidi phaang trinh vd tl (chiia dn diidi ddu cdn) - Phi/dng phdp nang len luy thifa
^ffOO = g(x) o f(x) - g' ( X ) . • •
PhiTdng phdp di/a ve phiTdng irinh chuTa an irong dau gid trj tuy?t doi Chu Si Vdi mpi so nguyen diTdng k thl ^ ' i ^ '
^ / f^ ; 0= f ( x ) ; 2' ' ^ y r ^ ^ = f(x) - PhiTdng phap ddt an phu
Chu y: PhiTdng trinh dang af(x) + b^fOO + c = 0 . ta ddt VfOO (y ^ 0)
- Phi/dng phap bat ding thufc ' ) P i + Neu tap gia trj cua hai ve PT la rcằi nhau, khi 66 PT v6 nghi$m
+ Sur dung tinh dong bien, tinh nghjch bien cua hkm so d moi ve PT de chufng minh PT co nghipm duy nhat.
+ Suf dung dieuki^nxdyrada'u bang d baft ddngthtfckhong chat.
4. Bat phuofng trinh a. Ddu tarn thiic bdc fmi
f(x) = ax^ + bx + c (a 9t 0) A = b^ - 4ac
+ Neu A < 0 thl af(x) > 0 Vx (f(x) cung dau v(3i a) •
+ Neu A = 0 thl af(x) ^ 0 Vx. Dau "=" xay ra khi vk chi khi x = (f(x) cung dau vdi a vdi moi x ^ - — ) D
2a i i f
Neu A > 0 thl f(x) c6 hai nghicm phan bi^t x,, \2 (xi < Xz) xu.- Bang xet dau
X —00
af(x) + 0 - 0 +
f(x) cOng da'u a 0 iraidaua 0 ciingdaua
Cang if TNHH MTVDWH KhmgWt Xit ddu biiu thiic
phSn tich bi^u thuTc cin x6t vc tich cdc thiTa s6' 1^ tarn thtfc bac hai hodc nhi thtfc bac nhat. >.< n . i u - ••..i ..v.ij
^ cm bdtphuang trinh hOu^ ^p^y ^j^^, ,^
Chuyen ta't cd cdc hang ve Cling mpt v^. <) j i i i j i j i - i j *
Riit gon bilu ihtfc c6 dU^c v^ phan tich ra thifa stf
Xdt d^fu bilu thurc. " , DiJa v^o bang xet dau, chon mien nghi$m. • 5. H? phUtfng trinh.
a. H$ddixangloQimOt
. H? doi xuTng hai an: 1^ an kh6ng thay d^i n^'u ta thay d^i vai tr6 x, y C^chgiai:
DSt S = X + y; P = xy: diTa h0 da cho ve h^ mdi vdi hai In S, P
Chii ^ cdc bieu thtfc doi xtfng vdi x, y nhiT: ^" ' x^ + y^ = (x + y)^-2xy = S^-2P Tilil^il M i
x'+ y' = (X + y)' - 3xy(x + y) = S' - 3PS " | , J ' ' x ^ y _ x^ 4- y ^ _ S ^ - 2 P ^ ^ _
y X xy P
Gidi flm S, P. Khi d6. x, y m nghidm ciSa phi/dng trinh - SX + P = 0 Di^u ki^n c6 nghicm: - 4P > 0
Neu (x; y) ^ nghi$m thl (y; x) cung 1^ nghicm. " ằ^ '
- H e doi xufng ba an: =351,0/.! ^
Tim cdc t^ng S = x + y + z, T = xy + yz + zx, P = xyz , t
X, y, z nghicm cua phiTdng trinh bac ba: X ' - SX^ + TX - P = 0 b. H$d6ixiingloQi2:
H$ phtfdng trinh hai an m i khi doi vi tri hai in cho nhau thl phiTdng trinh nky trd thinh phifdng trinh kia goi la hp doi xi?ng loai hai.
Cdchglai: (n'
• Trir ve vdi cdc phifdng trinh da cho.
- Phi/dng trinh diTdc diTa ve dang tich trong dd c6 nghipm x = y
^' ddng cdp d^ng: a,x2+b,xy + c,y2=d,
a2x2 + b2xy + c2y^=d2 _ _
Cdchl: ' ' Giai h$ vdi X = 0
Vdi X ?t 0, dat y = tx ta diTdc h? thco t x. Khuf x difdc phiTdng trinh bSc hai 'heo t.
* Cdch2:
rr!r Khiif (hay x^) ta tinh dvhic y theo x (hoSc x theo y). Thay v i o mpt Irong haj phiTdng trinh cua h ? ta di/dc phiTdng trinh triing phU'dng theo x (hay y).
I I . B A I T A P M I N H H Q A . , , A * . . — a , , ..
B a i 1. G i a i phiTdng trinh (x^ - x + 1 )* - 6x^(x^ - x + 1)^ + 5x* = 0
Hiring dfin gial , .
T a thay x = 0 khong 1^ nghipm cua phifdng trinh ^^^^^ ^^^^ ^ X6t X # 0, chia hai ve cho x**, ta diTdc: x ^ - x + 1
- 6 x ^ - x + 1 N2
+ 5 = 0
D a t y = x ^ - x + 1
(vdi y > 0) ta c6 PT y^ - 6y + 5 = 0. ,. d Giai PT n^y tlm difdc y = 1, y = 5.
V<5i y = 1 thl
Vdi y = 5 thl
x - x + 1 2
= i<=>
I ^ ;
r x ^ - x + i ^ 2
X - x + l = x
X ' - x + l = >/5x
X ' - x + l = -\/5x
x ^ - 2 x + l = 0
x ^ - ( l + V5 )x + l = 0
x^ + ( > / 5 - l ) x + i- 0 o
Timdi/0c X, = — ^ , X2 = ^
Vay tap nghi?m cua phiTdng trinh \h [ 1, X|, X2) B^i 2. Giai phiTdng trinh (x + 3)" + (x + 5)" = 4
HiTitng d i n giai
Dat y = X + 4 thi PT c6 dang (y - l)"* + (y + 1)* = 4, suy ra y" + 6y^ - 1 = 0 Dat z = y^ (z > 0), ta c6 z^ + 6z - 1 = 0, tim diTdc z -3 + VTo (thoa man dieu ki?n) do d6 y = ±V -3 + VIo . Tif d6 ta tim di/dc x = - 4 ± V-sVTo
B a i 3. G i ^ i cdc phiTdng trinh sau y * - a. 3x^ + 21x+ 18+ 27x^+7x + 7 = 2 b. 10Vx^ + l= 3 ( x 2 + 2 )
C. yj\ 2ylx^ +yj\-2yfx^ = 2
UMng dSn giai
a. D5t y = Vx^ +7x + 7 (y > 0). Khi d6 x^ + 7x + 7 = y^ PT da cho c6 dang
COng ty TNHH MTVDWH Khang Vie 3y' + 2y - 5 = 0. tim diTdc y = - - (loai). y = 1 (th6a man)
x = - i Suyra Vx^ + 7x + 7 = l o x ^ + 7 x + 6 = 0 o Vay tap nghi^m cua PT m {-6; - 1 )
^ Dieu ki$n x > - 1. Khi d6 PT cd dang
x = - 6 > .
l07x+T.Vx^-x + l = 3(x2+2) (*)
E)at u = Vx + 1. V = Vx^ - x + 1 (dicu ki$n u > 0, v ^ 0), PT (•) c6 dang lOuv = 3(u^ + v^) o (3u - v)(u - 3v) = 0 <:> u = 3v
3u = V
V d i u = 3v thi >/x+T = 3 V x 2- x + l c>9x^-10x + 8 = 0, P T n a y v6 nghi^m Vdi 3u = V thi 3>/x+T = Vx^ - x + i o9 x + 9 ~
x = 5 - v / 3 3 X= 5 + N/33
= x ^ - x + 1 o x^ - l O x - 8 = 0 0 (thoa man dieu ki^n X > - 1 )
Vay tap nghipm cua phiTdng trinh la js - N/33;5 + >/33 c. D i e u k i c n x > 1. B i c n d d i
\ / x - l + 2> / x ^ + l + V x - l - 2 7 J r n " 7 l =2
H i ' i l l f i ^ ^ i
O V x ^ + i f + ^ ( V x M - i ) ^ = 2 o > / 5 r ^ + i + | v/ 5 r^ - i = 2
Neu X > 2 thi >/rn' + l + >/rn" = 2 o 7 5 r n' - l o x = 2 (khong thoa dicu ki$n)
N6u 1 5 x< 2 t h i VJn" + l- V 7 ^ + l= 2 o 0 x = 0 (lu6ndiing vdi moi x) Vay tap nghipm cua PT la (x e R I 1 < x < 2}
Gi^i phiTdng trinh 2 ^ 3 x - 2 + 3 V 6- 5 x - 8 = 0
Hi^(}ng dSn giai /:
r2u + 3v = 8 (3) 5 u' + 3 v 2 = 8( 4 )
^ h i do dat u = ^ 3 x - 2 , V = V 6- 5 X va ta di den h? sau:
suy ra V = — ^ — . Thay vao (4) roi rut gpn ta c6:
^ ^ j j ! + 4u^ - 32u + 40 = 0 o (u + 2)(15u^ - 26u + 20) = 0 o u + 2 = 0
(do 15u^ - 26u + 20 > 0 Vu) o u = - 2 , v = 4.
Vay ta c6 = -2
yl6-5\4
3 x - 2 = -8
6 - 5 x = 16 o X = - 2 , Vay X = - 2 nghi?m duy nhat cua (1).
•.'i'T qi-.
Bai 5. Q\k\g trinh: x + \/l3-x^ + xVl3 -x^ = 11 (1), Hiy(tng dSn giai r
Dat y = V l 3 ^ ^ 0 . Khi d6 lir (1) tilf c6 h?:
[x + y + xy = l l Jx + y + xy = l l (2)
V- f y 2=13 ^ [ ( x + y)2-2xy = 13 (3) ^
Ttr (2) suy ra xy = 11 - (x + y). Thay vio (3) sau khi rdt gon ta c6:
( X + y)^ + 2(x + y) - 35 = 0 o
Vay ta di den: (2), (3) o
x + y = 5 _x + y = - 7 x + y = 5 xy = 6
y > 0 x = 2;y = 3 x + y = -7 " L ' ' = 3;y = 2 xy = 18