Giai:
a) Vi 16x'*+5>0 nen phuong trinh da cho c6 nghiem khi
4x^ + X > 0 <=> x(4x^ + 1) > 0 <=> X > 0 . De y rang khi x = thi VT = VP nen
ta nghi den sir dung bat dMng thuc Co si sao cho dáu bang xay ra khi = • Tir nhiing co so tren ta c6 161 giai nhu sau:
Ta CO 6^4x^ +x = 3^4x{4x^ +1)2 ^ 4x + (4x^ +1) + 2 = 4x2 + 4x + 3
M|it khac ta c6:
16x'' + 5 - (4x2 + 4x + 3) = 2(x'' - 2x^ - 2x +1) = 2(2x - 1)^(2x2 + 2x +1) > 0
. , , . . . . f2(2x-l)2(2x2+2x + l) = 0 1
1 va chi khi •! <z> x = — Dáu bang xay ra khi va chi khi Dáu bang xay ra khi va chi khi
4x = (4x^+1) = 2 1 Tom lai: Phuong trinh c6 nghỉm duy nhát x = —
b) Vi 4x^ + x^ + 3x + 4 > 0 nen phuong trinh da cho co nghiem khi
16x^ + 12x > 0 <=> 4x(4x2 + 3) > 0 « . X > 0 . De y rang khi x = i thi VT = VP
nen ta nghi den sir dung bát dSng thuc Co si sao cho dáu bang xay ra khi
J X = . Tir nhiing co so tren ta co I6i giai nhu sau:
Theo bat dang thuc Co si dang 3\/abc < a + b + c ta co
3\/l6x^+12x = 3^4x(4x2+3) ^
= 3^2.2.x(4x2 + 3) < 2 + 2 + x{4x^ + 3) = 4x^ + 3x + 4
Mat khac ta co:
4x^ H-x^ +3x + 4-(4x^ +3x + 4) = 4x^ -4x^ +x^ =x2(2x-l)^ > 0 . ' - Dáu bang xay ra khi va chi khi x^{2x-lf=0
2 = x(4x2 + 3) <=> X = —
c) Dieu kỉn: x > . De y rang x = - la nghiem ciia phuong trinFi nen ta co loi 8 8
l + l + 4x(8x + l) 32x2+4x + 2
giai nhu sau:
^4x(8x + l)=^1.1.4x(8x + l ) < 3 3 Mat khac ta co I 96x^ 20x I 2 + 4x + 2 ^ 256x^ - 64x + 4 ^ 4(8x -1)^ ^ ^ 3 3 3 • Suy ra 96x^ -20x + 2 + x V 8 x - l -^4x(8x +1) > 0 .
Dáu bang xay ra khi va chi khi x = —
Vi 2: Giai cac phuang trinh sau: a) b) V 2 x - 1 ^ 4 x - 3 X Vx V 2 x - 1 W 4 X - 1 V5X-2 c) ^ + ^ 2 x - l = 4 4 x - l ^ J 5 X - 2 Giai: a) Dieu kif n: > •
Phuang trinh da cho c6 the viet lai nhu sau:
V2X-1 ^ 4 x - 3 = 2.
+ Ta chung minh: > 1. That vay bat dang thiic tuong duang voi
N/2X-1
x^ > 2 x - l o ( x - l ) ^ > 0 . Dieu nay la hien nhien dung.
Dau bang xay ra khi va chi khi x = 1 Ta chung minh:
^ 4 x - 3 > 1. That vay bat dang thuc tuong duong voi x^ > 4x - 3 « x^ - 4x + 3 > 0 (x^ - 2x + l){x^ + 2x + 3) > 0
o ( x - l) 2 ( x 2+ 2 x + 3)>0
Dieu nay la hien nhien diing. Dau bang xay ra khi va chi khi x = 1 Tir do suy ra VT > 2 . Dau bang xay ra khi va chi khi x = 1
b) - 7 ^ + , =yl3
Vx V 2 x - 1 V 4 x - 1 VSx - 2 ,
Tathay rang: 4 x - l = x + x + 2 x - l ; 5 x - 2 = x + 2 x - l + 2 x - l Theo bat dang thuc c6 si ta c6
1 1 Vx V 2 X - 1 Vx V 2 X - 1 1 1 1 - + — ^ + - V ^ + V x + V 2 x - l ) > 9 9 Vx Vx V 2 x - 1 Vx + Vx + V 2 x - 1 Mat khac ta c6 ( V x + V x+ V 2 x - 1) 2 < 3 ( 4 x - l) = > V x + V x+ V 2 x- l < V 3 ( 4 x - l ) (Theo bat
dang thiic Bunhiacopxki cho 3 so)
2 1 ^ 3V3
Tu do suy ra: -j=r + >
Vx V 2 x - 1 V 4 x - 1 . ;
. 1 2 ^ 3V3 , , ,
Tuong ty ta cung co: + . > , :Mnm ^lituti-i yjx v 2 x- l v 5 x - 2
3 3 3 V 3 3V3
Cong hai bat dSng thuc ciing chieu ta c6: + , > , + ,
Vx V 2 x - 1 V 4 x - 1 V 5 x - 2
Vx V 2 x - 1 W 4 X - 1 V 5 x - 2 Dau bang xay ra khi va chi khi x = 1
c) Ap dung bat dang thuc Bunhiacopxki dang:
(ax + by + czf < (â + b^ + c^ )(x^ + y^ + ) ta c6: ( C + N/X+^2X-1J^ < ( l + l + l)(V5^ + V ^ + V 2 x - l ) . c ^ t ^ + V x+ V 2 X - 1 < ^ 3 ( V x + V x+ V 2 x - l ) Lai CO V x + V x+ V 2 x - 1 <73(4x -1) suy ra ^ Vx + ^ + V 2 x - l < ^ 3 j 3( 4 x - l ) = ^ 2 7 ( 4 x - l ) (1) Tuong tu: + ^2x - 1 + ^2x - 1 < ^373(5x - 2) = V27(5x - 2) (2)
Cpng hai bat dkng thuc cung chieu (1), (2) ta c6:
3( Vx + ^2x - 1 ) < ^27(4x -1) + ^27(5x - 2)
Dau bang xay ra khi va chi khi x = 1.
Vay phuang trinh c6 nghiem duy nhat x = 1
Vi dy 3: Giai cat phuang trinh sau:
a) 2x^sinx + x.cosx + V2x + l =x^-x^+x + l \'''
b) V-x^ + 4x + 21 - V-x^ + 3x +10 = V2
G i i i :