191 Gia tri nho nha't cua S bang ——; Khi 16
Bai 2.35: Xet tinh chan le ciia ham so "'''^
a) y = x + v/x x ^ - x ^ + x ^ - l a) TXD: D = R\}
Suy ra Vx e D => -x e D -x +
b) y = Huang dan gidi
y • !
x - l | - | l + x|
f ( - x ) = . x + ^x
Vay ham soda cho la le
b) TXD: D = [-1;1]\{0} I ,4 Suy ra Vx e D - X G D
4 2-^ = -Hx), V x e D
f ( - x ) = + X
| - x - l | - | l + (-x)| |x-l|-|] + x|
Vay ham so da cho la chan.
= f ( x ) , V x e D
X c i
~UiyTNHHWll V UVVH Knang^vjer
. r, 'If, • Xac dinh a va b de do thi ciia ham so y = ax + b cat tryc hoanh tai pal • ' ,
jjiern CO hoanh do bang 3 va di qua diem A ( - 2 ; l ) .
Huong dan gidi • • ? ; v ' .
y i duong th§ng y = ax + b cat true hoanh tai diem c6 hoanh tai diem c6 hoanhdo nen: 3a + b = 0 (1) , , _
Vi A ( - 2 ; 1) thuoc duong thang y = ax + b nen: -2a + b = 1 (2) [ 3a + b = 0
Tu'(l)va(2)tac6hept: <! . . . o -2a + b = 1
a = — 5
5 > 1.1 XlHlh • iM
Bai 2.37: Cho ham so y = -x^ + mx - 4 (m la tham so) ) 0 ^ a) Vai m = 5, hay ve do thj ham so tren. ? r- + M H .
b) Tim m sao cho do thj ciia ham so noi tren la parabol nhan duong thang x = 2 lam true doi xiing.
Humgd^ngidi 6 j [ o d 6 ô i .
a) Voi m = 5 ta c6 ham so: y = -x^ + 5x - 4 , Vedothi: Dinh I
Bang gia trj
(5 9]
l 2 ' 4 j . True doi xiing: - 2 •
X 1 2 5 3 4
2
y 0 2 9 2 0
4
b) De duong thang x = 2 lam true doi xung thi- - m = 2 ô m = 4
-2.(-l)
Bai 2.39: Day truyen do nen cau treo c6 dang Parabol ACB nhu hinh ve. Dau cuoi ciia day dugc g3n chat vao diem A va B tren true A A ' va BB' voi do eao
30m. ChieudainhipA'B' = 200m. R A E>9 cao ngan nhat eiia day
truyen tren nen cau la OC = 5m . Xac djnh chieu dai cac day cap treo (thanh thang '^U'ng no! nen cau voi day truyen)?
Huang dan gidi Chpn true Oy trung voi EL
true do'i xiing ciia Parabol, true Ox nam tren nen cau nhu H i n h ve. K h i do ta c6A(100; 30), C(0; 5 ) , ta tim phuong trinh cua Parabol c6 dang
y = ax^ + bx + c . XA'
Parabol c6 dinh la C va di qua A nen ta eo h§ phuong trinh 2a = 0
a.O + b.O + c = 5 a.lOO^ + b.lOO + c = 30
a=- 1 400 b=0 c=5
m I t ,
' — ' - " - f
i i I
Suy ra Parabol c6 phuong trinh y = + 5 . Bai toan dua vifc xac dinh chieu dai cac day cap treo se la rinh tung dp nhiing diem M j , M j , M 3 ciia Parabol. Ta de dang tinh dupe tung dp cac diem eo cac hoanh do
X j = 25, X2 = 50, X3 = 75 Ian lupt la
y i =6,56 ( m ) , y j =11,25 ( m ) , yg =19,06 ( m ) . Do chinh la dp dai cac day cap treo can tinh.
Bai 2.40 Cho f la ham so le va dong bien tren R . a,b,c la cae so thyc thoa man a + b + e = 0 . Chung m i n h rang f(a).f(b) + f(b).f(c) + f(c).f(a) < 0.
Huang dan gidi
Trong ba so a,b,c phai c6 hai so khong trai dau them mta do f ham so le ' nen f(a).f(b) + f(b).f(c) + f(c).f(a) = f(-a).f(-b) + f(-b).f(-e) + f(-c).f(-a)
va (-a) + ( - b ) + (-c) = 0. Do do khong mat tinh tong quat ta c6 the gia s u a > 0 ; b ^ O . V i f l a ham sole nen f(c) = f ( - a - b ) = - f ( a + b)
Do do bat dang thuc can chung m i n h tuong duong v o i (f(a) + f(b))f(a + b)>f(a)f(b) (*)
M a t k h a c 0 = f(0)<f(a), 0 = f(0)<f(b), f ( b ) < f ( a + b) do f la ham so dong bien tren R. | j ^ -if. ..nX
Vi vay theo bat d i n g thuc cosi ta c6 2f(a + b) > f(a) + f(b) > 27f(a)f(b) T u do: (f(a) + f(b))f(a + b) > 2Vf(^)f(b)7f(i)f(b) = 2f(a)f(b) > f(a)f(b)
Do do (*) diing suy ra dieu phai chung minh. ../^^u.
^hmngS PHUONG TRINH VA HE PHUONG TRINH
§1. D A I C U O N G V E P H U O N G T R I N H
T O M T A T L Y T H U Y E T ^ u r l q -rnvh J M U n n n ; u i i ^ • • jj)inhnghia.
Cho hai ham so y = f (x) va y = g(x) c6 tap xac dinh Ian lupt la Df va Dg . E)§t D = Df n D j , . M?nh de chua bien " f ( x ) = g ( x ) " dupe gpi la phuong trinh mot an ; x dupe gpi la an so'(hay ah) va D gpi la tap xac d i n h ciia phuong trinh. ,. .
XQ e D gpi la mot nghiem cua phuong trinh f (x) = g (x) ,
neu "f (xo) = g(xo)" la m^nh de dung. , Chii y: Cae nghiem cua phuong trinh f (x) = g ( x ) la cae hoanh dp giao diem
do thi hai ham so y = f (x) va y = g ( x ) . . 2. 'Phmmg trinh tuxmg duang, phuong trinh he qua.
a) Phuong t r i n h t u o n g duong: Hai phuong trinh f i ( x ) = g i ( x ) va (x) = g 2 (x) dupe gpi la tuong duong ne'u ehiing c6 cung tap nghiem. K i hieu la fi (x) = g i (x) o f2 (x) = g 2 ( x ) .
• Phep bien doi khong lam thay doi tap nghiem ciia p h u o n g trinh gpi la phep bien doi tuong duong.
h) Phuong t r i n h h f qua: (2 (x) = g 2 (x) gpi la phuong trinh h# qua cua phuong trinh f i ( x ) = g i ( x ) neu tap nghiem cua no chua tap nghiem ciia phuong
tt n h f i ( x ) = g i ( x ) . '-'-^-y
i h i ^ U la fi (x) = g i ( X ) = ^ f2 ( X ) = g 2 (x) - -nr- c) Cac d i n h ly: >
D i n h ly 1: Cho phuong trinh f ( x ) = g(x) c6 tap xac dinh D; y = h ( x ) la ham so xac dinh tren D. Khi do tren D, phuong trinh da cho tuong duong voi phuong trinh sau
1) f ( x ) + h ( x ) = g ( x ) + h ( x ) : m x r . - . , 2) f ( x ) . h ( x ) = g ( x ) . h ( x ) ne'u h ( x ) ^ 0 voi mpi X G D
D i n h ly 2: K h i binh phuong hai ve'ciia mot phuong trinh, ta dupe phuong :inh h ^ qua ciia phuong trinh da cho.
I f ( x ) = g ( x) = ^ f 2( x ) = g 2( x ) .
-mi y: K h i giai phuong trinh ta can ehii y
101
• Dat dieu kien xac dinh(dkxd) ciia phuang trinh va khi tim duoc nghiem cq^
phuong trinh phai doi chie'u voi dieu kien xac dinh.
• Neu hai ve'ciia phuong trinh ludn cung dau thi binh phuong hai ve ciia ta thu dugc phuong trinh tuong duong.
• K h i bien doi phuong trinh thu dup-c phuong trinh he qua thi k h i t i m duoQ nghif m cua phuong trinh he qua phai thu lai phuong trinh ban dau de lo^j bo nghiem ngoai lai.
B. C A C D A N G T O A N V A P H l / Q N G P H A P G I A I .
DANG TOAN 1: TIM DIEU KIEN XAC DINH CUA PHUONG TRINH.
'Phuang phdp gidi. , ' . ,
- Dieu kien xac djnh cua phuong trinh bao gom cac dieu kien de gia trj cua f ( x ) , g ( x ) ciing duoc xac djnh va cac dieu kien khac (neu c6 yeu cau trong de bai)
- Dieu kien de bieu thuc ,
• 7 f ( x ) x a c d m h la f ( x ) > 0 -'•^ I
Oi f(x)
i 1
xac dinh la f ( x ) 0 xac dinh la f (x) > 0
£ a i . C A C V l D U MINH HOA
V i 1: T i m dieu kien xac djnh cua phuong trinh sau:
a) 1 + ^ J 2 ^ = i iô'f^^v b) = x + 1
x'^-3x + 2 Lai gidi
a) Dieu kien xac djnh ciia phuong trinh la
b) Dieu kien xac dinh ciia phuong trinh la
2 x - 3 > 0
3 x - 2 > 0 X . 2 2 3
4 - 2 x > 0 x< 2
^^^^ [ x ^ - 3 x + 2 ^ 0 | ( x - l ) ( x ^ + x - 2 ) ^ 0 rx<2
x < :
<=> I
( x - i r ( x - 2 ) ^ 0
<=> X v t l <=>
x * 2
x < 2 x ^ l
mo
y ['^72i ^Tim dieu kien xac djnh cua phuong trinh sau roi suy ra tap nghiem ciia no: r - M > ; ., 4x + V 4 ^ = 2 V 3 ^ + 3 b) 7- x ^ + 6 x - 9 + x-'=27
3) Loi gidi
[ 4 x- 3 > 0 Oieu kien xac djnh cua phuong trinh 4^ > Q ^ '
3
3
X < —
4 Thu vao phuong trinh thay x = - thoa man
Vay tap nghiep cua phuong trinh la S = b) Dieu ki^n xac dinh ciia phuong trinh la -x^ + 6x - 9 > 0 <=> - ( x - 3)^ > 0 ằ X = 3 Thay x = 3 vao thay thoa man phuong trinh Vay tap nghiep cua phuong trinh la S = {3}
m 2. BAI T A P L U Y ^ N T A P
Bai 3.0: T i m dieu kien xac djnh cvia phuong trinh sau:
a) 5
x ^ - x - 1
c) 1 + V 2 x- 4 = V 2 - 4 X
b) 1 + V x ^ = V x ^
X + 1
d ) V 2 ^ = ^ Huang dan gidi
x^ - 3 x + 2
a) DKXD: x ^ - x - l ^ O o x ^ l ^ b) D K X D :
c) D K X D : 2 x - 4 > 0
2 - 4 x > 0 <=> x e 0 d) D K X D :
x - l> 0 x- 2 > 0
2 x - 6 > 0 x^ - 3x + 2 ^ 0
x > 3 x ^ l x^ 2 Bai 3.1: T i m dieu ki?n xac djnh cua phuong trinh sau roi suy ra tap nghiem cua
no:
a) 4x + 2V4x - 3 = 2 V 4 x - 3 + 3 c) V2^ + = V 2 ^ + 2 :J
b) ^|-x^ +x-l + x = l
"' d) V x^ - 4 x 2 + 5 x- 2 + X = ^f2^
Huang dan gidi
^) D K X D : X > - . De thay x = - la nghiem ciia phuong trinh.
4 4
103
b) D K X D : -x^ + x - 1 > 0 ô - X - - - - > O < : : > x e 0 4
! Vay tap nghiep cua phuang trinh la S = 0 . x > 0
c) D K X D : x-2>0<=>x = 2 2 - x > 0 ,i
T h u lai phuang trinh thay x = 2 thoa man Vay tap nghiem cua phuang trinh la S = {2}.
d) D K X D : x^ - 4 x 2 + 5 x- 2 > 0 J( x - l f ( x - 2 ) > 0 2 - x > 0 x < 2 <=>
x = l x = 2
Thay vao phuang trinh ta c6 tap nghiem cua phuang trinh la S = { 1 } .
D A N G T O A N 2: GIAI PHUONG TRINH BANG PHEP BIEN DOI TUONG DUONG VAHEQUA
'Phmmg phdp gidi. , i , :
De giai phuang trinh ta thuc hi^n cac phep bien doi de dua ve phuang trinh tuang duong voi phuang trinh da cho don gian hon trong viec giai no. Mot so'phep bien doi thuang su dung
• Cong (tru) ca hai v e c u a phuong trinh ma khong lam thay doi dieu ki|n xac dinh ciia phuang trinh ta thu duoc phuang trinh tuang duong phuang trinh da cho.
• Nhan (chia) vao hai ve vai mot bieu thuc khac khong va khong lam thay doi dieu kien xac djnh ciia phuang trinh ta thu duoc phuong trinh tuang duong voi phuang trinh da cho.
• Binh phuang hai ve'ciia phuang trinh ta thu dug-c phuang trinh hf qua ciia phuong trinh da cho.
• Binh phuong hai ve ciia phuang trinh (hai ve luon cung dau) ta thu dugc phuang trinh tuang duong voi phuong trinh da cho.