Bt ng thc Bt phng trnh

Trn S Tng

Trn S Tng

Bt ng thc Bt phng trnh

1. Tnh cht

2. Mt s bt ng thc thng dng

a)

.

.

b) Bt ng thc Csi:

+ Vi a, b ( 0, ta c:

.Du "=" xy ra ( a = b.

+ Vi a, b, c ( 0, ta c: . Du "=" xy ra ( a = b = c.

H qu: Nu x, y > 0 c S = x + y khng i th P = xy ln nht ( x = y.

Nu x, y > 0 c P = x y khng i th S = x + y nh nht ( x = y.

c) Bt ng thc v gi tr tuyt i

d) Bt ng thc v cc cnh ca tam gic

Vi a, b, c l di cc cnh ca mt tam gic, ta c:

+ a, b, c > 0.

+ ; ; .

e) Bt ng thc Bunhiacpxki

Vi a, b, x, y ( R, ta c: . Du "=" xy ra ( ay = bx.VN 1: Chng minh BT da vo nh nghia v tnh cht c bn

( chng minh mt BT ta c th s dng cc cch sau:

Bin i BT cn chng minh tng ng vi mt BT bit.

S dng mt BT bit, bin i dn n BT cn chng minh.

( Mt s BT thng dng:

+

+

+ vi A, B ( 0.+

Ch :

Trong qu trnh bin i, ta thng ch n cc hng ng thc.

Khi chng minh BT ta thng tm iu kin du ng thc xy ra. Khi ta c th tm GTLN, GTNN ca biu thc.Bi 1. Cho a, b, c, d, e ( R. Chng minh cc bt ng thc sau:

a)

b)

c)

d)

e)

f)

g)

h)

i) vi a, b, c > 0

k) vi a, b, c ( 0

HD: a) (

b) (

c) (

d) (

e) (

f) (

g) (

h)(

i) (

k) (

Bi 2. Cho a, b, c ( R. Chng minh cc bt ng thc sau:

a) ; vi a, b ( 0

b)

c)

d) , vi a, b, c > 0.

e) ; vi a, b ( 0.

f) ; vi ab ( 1.

g)

h) ; vi ab > 0.

HD: a) (

b) (

c) (

d) S dng hng ng thc .

BT ( .

e) (

f) (

g) (

h) ( .Bi 3. Cho a, b, c, d ( R. Chng minh rng (1). p dng chng minh cc bt ng thc sau:

a)

b)

c)

HD: a) ;

b)

c)

Bi 4. Cho a, b, c, d > 0. Chng minh rng nu th (1). p dng chng minh cc bt ng thc sau:

a)

b)

c)

HD: BT (1) ( (a b)c < 0.

a) S dng (1), ta c: , , .

Cng cc BT v theo v, ta c pcm.

b) S dng tnh cht phn s, ta c:

Tng t,

Cng cc BT v theo v ta c pcm.

c) Chng minh tng t cu b). Ta c:

Cng vi 3 BT tng t, ta suy ra pcm. Bi 5. Cho a, b, c ( R. Chng minh bt ng thc: (1). p dng chng minh cc bt ng thc sau:

a)

b)

c)

d)

e) vi a,b,c>0.f) nu

HD: ( .

a) Khai trin, rt gn, a v (1)

b, c) Vn dng a)

d) S dng (1) hai ln

e) Bnh phng 2 v, s dng (1)

f) S dng d)

Bi 6. Cho a, b ( 0 . Chng minh bt ng thc: (1). p dng chng minh cc bt ng thc sau:

a) ;

vi a, b, c > 0.

b) ; vi a, b, c > 0 v abc = 1.

c) ; vi a, b, c > 0 v abc = 1.

d) ; vi a, b, c ( 0 .

e*) ;vi ABC l mt tam gic.

HD: (1) ( .

a) T (1) ( ( .

Cng vi 2 BT tng t, cng v theo v, ta suy ra pcm.

b, c) S dng a).

d) T (1) ( ( (2).

T : VT (

.

e) Ta c:

.

S dng (2) ta c: .

(

Tng t,

,

Cng cc BT v theo v ta c pcm.

Bi 7. Cho a, b, x, y ( R. Chng minh bt ng thc sau (BT Mincpxki):

(1)

p dng chng minh cc bt ng thc sau:

a) Cho a, b ( 0 tho . Chng minh:

.

b) Tm GTNN ca biu thc P = .

c) Cho x, y, z > 0 tho mn . Chng minh:

.

d) Cho x, y, z > 0 tho mn . Tm GTNN ca biu thc:

P = .

HD: Bnh phng 2 v ta c: (1) ( (*)

( Nu th (*) hin nhin ng.

( Nu th bnh phng 2 v ta c: (*) ( (ng).

a) S dng (1). Ta c: .

b) S dng (1). P (

Ch : (vi a, b > 0).

c) p dng (1) lin tip hai ln ta c:

( .

Ch : (vi x, y, z > 0).

d) Tng t cu c). Ta c: P ( .

Bi 8. Cho a, b, c l di 3 cnh ca mt tam gic. Chng minh:

a)

b)

c)

d)

HD: a) S dng BT tam gic, ta c: .

Cng vi 2 BT tng t, cng v theo v, ta suy ra pcm.

b) Ta c: .

Cng vi 2 BT tng t, cng v theo v, ta suy ra pcm.

c) ( .

d) ( .

a) VN 2: Chng minh BT da vo BT Csi1. Bt ng thc Csi:

+ Vi a, b ( 0, ta c:

.Du "=" xy ra ( a = b.

+ Vi a, b, c ( 0, ta c: . Du "=" xy ra ( a = b = c.2. H qu: +

+

3. ng dng tm GTLN, GTNN:

+ Nu x, y > 0 c S = x + y khng i th P = xy ln nht ( x = y.

+ Nu x, y > 0 c P = x y khng i th S = x + y nh nht ( x = y.Bi 1. Cho a, b, c ( 0. Chng minh cc bt ng thc sau:

a)

b)

c)

d) ; vi a, b, c > 0.

e)

f) ; vi a, b, c > 0.

g) ; vi a, b, c > 0.

HD: a) ( pcm.

b) ( pcm.

c) (

(

(

(

d) , , (pcm

e) VT ( ( .

f) V nn . Tng t: .

(

(v )

g) VT =

= ( .

( Cch khc: t x =b + c, y = c + a, z = a + b.

Khi , VT = ( .

Bi 2. Cho a, b, c > 0. Chng minh cc bt ng thc sau:

a)

b)

c)

HD: a) VT = .

Ch : . Cng vi 2 BT tng t ta suy ra pcm.

b) ( .

Ch : . Cng vi 2 BT tng t ta suy ra pcm.

c) p dng b) ta c: .

D chng minh c: ( pcm.Bi 3. Cho a, b > 0. Chng minh (1). p dng chng minh cc BT sau:

a) ; vi a, b, c > 0.

b) ; vi a, b, c > 0.

c) Cho a, b, c > 0 tho . Chng minh:

d) ; vi a, b, c > 0.

e) Cho x, y, z > 0 tho . Chng minh: .

f) Cho a, b, c l di ba cnh ca mt tam gic, p l na chu vi. Chng minh rng:

.

HD: (1) ( . Hin nhin suy t BT Csi.

a) p dng (1) ba ln ta c: .

Cng cc BT v theo v ta c pcm.

b) Tng t cu a).

c) p dng a) v b) ta c: .

d) Theo (1): ( .

Cng vi cc BT tng t, cng v theo v ta c pcm.

e) p dng cu d) vi a = x, b = 2y, c = 4z th ( pcm.

f) Nhn xt: (p a) + (p b) = 2p (a + b) = c.

p dng (1) ta c: .

Cng vi 2 BT tng t, cng v theo v, ta c pcm.

Bi 4. Cho a, b, c > 0. Chng minh (1). p dng chng minh cc BT sau:

a) .

b) Cho x, y, z > 0 tho . Tm GTLN ca biu thc: P = .

c) Cho a, b, c > 0 tho . Tm GTNN ca biu thc:

P = .

d) Cho a, b, c > 0 tho . Chng minh: .

e*) Cho tam gic ABC. Chng minh: .

HD: Ta c: (1) ( . D dng suy t BT Csi.

a) p dng (1) ta c: .

( VT (

Ch : .

b) p dng (1), ta bin i P nh sau:

P = =

Ta c: . Suy ra: P ( .

Ch : Bi ton trn c th tng qut nh sau:

Cho x, y, z > 0 tho v k l hng s dng cho trc. Tm GTLN

ca biu thc:P = .

c) Ta c: P ( .

d) VT (

=

(

Ch : .

e) p dng (1):

( .

Ch : .Bi 5. p dng BT Csi tm GTNN ca cc biu thc sau:

a) .

b) .

c) .

d)

e)

f)

g)

h)

HD: a) Miny = 6 khi x = 6

b) Miny = khi x = 3

c) Miny = khi x =

d) Miny = khi x =

e) Miny = khi

f) Miny = khi x =

g) Miny = 8 khi x = 2

h) Miny = khi x =

Bi 6. p dng BT Csi tm GTLN ca cc biu thc sau:

a)

b)

c)

d)

e)

f)

g)

HD: a) Maxy = 16 khi x = 1

b) Maxy = 9 khi x = 3

c) Maxy = khi x =

d) Maxy = khi x =

e) Maxy = 9 khi x = 1

f) Maxy = khi x = ()

g) Ta c: ( (

( Maxy = khi x = (1.

a)

VN 3: Chng minh BT da vo BT Bunhiacpxki1. Bt ng thc Bunhiacpxki:(B)

( Vi a, b, x, y ( R, ta c: . Du "=" xy ra ( ay = bx.

( Vi a, b, c, x, y, z ( R, ta c:

H qu:

(

(

Bi 1. Chng minh cc bt ng thc sau:

a) , vi

b) , vi

c) , vi

d) , vi

e) , vi

f)

HD: a) p dng BT (B) cho 4 s .

b) p dng BT (B) cho 4 s

.

c) p dng BT (B) cho 4 s

.

d) p dng BT (B) cho 4 s .

e) p dng BT (B) cho 4 s .

f) t a = x 2y + 1, b = 2x 4y + 5, ta c: 2a b = 3 v BT ( .

p dng BT (B) cho 4 s 2; 1; a; b ta c pcm.

Bi 2. Chng minh cc bt ng thc sau:

a) , vi .

b) , vi .

c) , vi .

d) , vi .

HD: a) ( pcm.

b)

( .

c) ( pcm.

d) ( .

(

Bi 3. Cho x, y, z l ba s dng v . Tm gi tr ln nht ca biu thc:

.

HD: p dng BT (B), ta c: P ( (

Du "=" xy ra ( ( .

Vy Max P = khi .Bi 4. Cho x, y, z l ba s dng v . Chng minh rng:

HD: p dng BT (B), ta c:

(

(1)

Tng t ta c:

(2),

(3)

T (1), (2), (3) suy ra:

P ( =

( ( .

Du "=" xy ra ( .Bi 5. Cho a, b, c ( tho . Chng minh:

.

HD: p dng BT (B) cho 6 s: ( (2).

Ch : . Du "=" xy ra ( x = y = z = 0. T ( (1) Bi 6. Cho x, y > 0. Tm GTNN ca cc biu thc sau:

a) , vi x + y = 1

b) , vi

HD: a) Ch : A = .

p dng BT (B) vi 4 s: ta c:

Du "=" xy ra ( . Vy minA = khi .

b) Ch : .

p dng BT (B) vi 4 s: ta c:

( .

Du "=" xy ra ( . Vy minB =

.

Bi 7. Tm GTLN ca cc biu thc sau:

a) , vi mi x, y tho .

HD: a) Ch : .

A ( ( .

Du "=" xy ra ( .Bi 8. Tm GTLN, GTNN ca cc biu thc sau:

a) , vi 2 ( x ( 7b) , vi 1 ( x ( 3

c) , vi

d) , vi .

HD: a) ( A ( . Du "=" xy ra ( .

( A ( . Du "=" xy ra ( x = 2 hoc x = 7.

( maxA = khi ;

minA = 3 khi x = 2 hoc x = 7.

b)( B ( . Du "=" xy ra ( x = .

( B ( ( . Du "=" xy ra ( x = 3.

( maxB = khi x = ; minB = khi x = 3.

c) Ch : . T : .

(

( ( .

( minC = khi ; maxC = khi .

d) Ch : . T : .

(

( ( .

( minD = 7 khi ;

maxD = 3 khi .

a)

I. BT NG THC

CHNG IV

BT NG THC V BT PHNG TRNH

iu kinNi dung EMBED Equation.DSMT4 a > 0 EMBED Equation.DSMT4 EMBED Equation.DSMT4 EMBED Equation.DSMT4

iu kinNi dunga < b ( a + c < b + c (1)c > 0a < b ( ac < bc (2a)c < 0a < b ( ac > bc (2b)a < b v c < d ( a + c < b + d (3)a > 0, c > 0a < b v c < d ( ac < bd (4)n nguyn dnga < b ( a2n+1 < b2n+1 (5a)0 < a < b ( a2n < b2n (5b)a > 0a < b ( EMBED Equation.DSMT4 (6a)a < b ( EMBED Equation.DSMT4 (6b)

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