gtlt_gtnn ham so

7/28/2019 Gtlt_gtnn Ham So

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3: GTNN, GTLN CA HM S

1. Cc kin thc c bn

nh ngha GTNN, GTLN ca hm s

Cho hm s y=f(x) xc nh trn min D

S M gi l GTLN ca hm s y=f(x) trn D nu : 0 0

f (x) M, x D

x D,f (x ) M

=

S m gi l GTNN ca hm s y=f(x) trn D nu :0 0

f (x) m, x D

x D,f (x ) m

=

2. Cc k nng c bn

K nng tm GTNN, GTLN ca hm s y=f(x) trn mt khong, mt on

Tm GTNN, GTLN ca hm s y=f(x) lin tc trn khong (a;b)

- Tnh o hm f(x).

- Tm cc nghim 1x , 2x , , nx ca f(x) trn (a;b).

- Lp bng bin thin ca f(x) trn (a,b).

Cn c vo bng bin thin suy ra GTLN, GTNN ca f(x) trn (a;b)

Tm GTNN, GTLN ca hm s y=f(x) lin tc trn [a;b]

- Tnh o hm f(x).

- Tm cc nghim 1x , 2x , , nx ca f(x) trn [a;b].

- Tnh f(a) , f(b) , 1f (x ) , , nf (x ) .

Chn s M ln nht trong n+2 s trn x [a;b]

M max f (x)

= .

Chn s m nh nht trong n+2 s trn x [a;b]

m minf (x)

= .

3. H thng bi tp s dng phng php o hm tm GTLN,

GTNN ca hm s.Dng 1. Kho st trc tip

Nu hm s y=f(x) trn min D cho dng n gin , ta c th kho st trc tip hm s

v rt ra kt lun GTNN, GTLN ca hm s.

gii quyt tt cc bi ton dng ny, HS cn c cc k nng sau:

- Tnh f(x) chnh xc.

- Bit cch tm nghim ca phng trnh f(x)=0.

- Bit cch lp bng bin thin ca f(x) trn D rt ra kt lun GTNN, GTLN ca

hm s.Bi 1.Tm GTNN, GTLN ca hm s 2y x 4 x= +


2/18

Li gii

TX D=[-2,2]

2

xy ' 1

4 x=

; y=0 24 x x = 2 2

x 0

4 x x

= x= 2

y(-2)=-2 ; y(2)= 2 ; y( 2 )=2 2

Vy x Dmax f (x) 2 2 = ; x Dmin f (x) 2 =Bi 2.Tm GTNN, GTLN ca hm s

y =2

x+1

x +1trn on [ ]2;1 .

Li gii

Ta c :( )

,

32

-x+1y =

x +1 ,y = 0 x=1.

Do y(-1) = 0, y(1) = 2, y(2) = 53 nn

[ ]1;2max y

= y(1) = 2, [ ]y

2;1min = y(-1) = 0.

Bi 3. Tm GTNN, GTLN ca hm s2

2

x 8x 7y

x 1

+=

+ (x R)

Li gii2

2 2

8x 12x 8y '

(x 1)

=

+; y' 0= x 2= ;

1x

2=

Bng bin thint -

1

2 2 +

y + 0 - 0 +

y

9 1

1 -1

Vy x Rmin y 1 = khi x 2= ; x Rmax y 9 = khi1

x 2=

Bi 4. Tm GTNN, GTLN ca hm s

y 5cos x cos5x= vi x [- ; ]4 4

Li gii

y ' 5sin x 5sin 5x= +k

x5x x k2 2y ' 0 sin5x sin xk5x x k2

x6 3

== + = = = + = +


3/18

*)k

x2

=

Do x4 4

k

4 2 4

1 1k

2 2 k=0 x=0.

*)k

x6 3

= +

Do x4 4

k k 5 k

4 6 3 4 4 6 3 4 6 12 3 12

+

xk 15 1 6kk 04 4

x6

= = = =

y(0) 4= ; y( ) y( ) 3 36 6

= = ; y( ) y( ) 3 2

4 4

= =

Vy Miny=4 ; Maxy = 3 3

Bi 5. Tm GTNN ca 2y x 2x 1= + + ( x R)

Li gii

2

2xy ' 1

2x 1= +

+; 2

2 2

x 0 1y ' 0 2x 1 2x x

2x 1 4x 2

= + = =

+ =

Bng bin thin

x-

1

2 +

y - 0 +

y

+ +

1

2

Vy1

Miny2

= khi1

x2

=

Dng 2. Kho st gin tip

Trong nhiu bi ton tm GTNN, GTLN ca hm s nu ta kho st trc tip c th gp

nhiu kh khn , chng hn nh tm nghim ca f(x), xt du ca f(x). Do thay v kho

st trc tip f(x) ta c th kho st gin tip hm s cho bng cch sau:

- t n ph t, chuyn hm s cho v hm s mi g(t).

- Tm iu kin ca n ph t ( Bng cch kho st hm s, dng bt ng thc)

- Kho st hm s g(t) suy ra GTNN, GTLN ca hm s.

gii quyt tt dng ton ny HS cn phi c nhng k nng sau:

- K nng chn n ph t : Chn n ph t thch hp sao cho hm s ban u c th qui

ht v bin t.


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- K nng tm iu kin ca n ph : tm iu kin ca t, ty theo tng bi ton c

th ta c th dng phng php o hm, dng bt ng thc, nh gi trc tip

Bi 6. Tm GTNN , GTLN ca 8 4S 2sin x cos 2x= + , xR

Li gii

Do

2 1 cos2x

sin x 2

=

nn ta qui S v cos2x

S= 4 41 cos2x

2( ) cos 2x2

+ = 4 4

1(1 cos2x) cos 2x

8 +

t t= cos2x , 1 t 1

Bi ton tr thnh tm GTNN, GTLN ca hm s 4 41

S g(t) (1 t) t8

= = +

vi 1 t 1

Ta c 3 31

g '(t) (1 t) 4t2

= + ; g(t) = 0 3 3(1 t) 8t = 1-t =2t 1

t3

=

g(1) =1 ; g(-1)=3 ; g(1

3)=

1

27

Vy MinS=1

27; MaxS= 3

Bi 7: Tm GTNN, GTLN ca hm s y= 1 sin x 1 cos x+ + + ( x R)

Li gii

Hm s xc nh vi x v y>0 vi x , do y t GTNN, GTLN ng thi vi 2y

t GTNN, GTLN.

Ta c: y2= 2 + sinx+ cosx+ 2 1+ sinx+ cosx+ sinxcosx

t t= sinx+ cosx = 2 sin x+4

= t ( )- 2 t 2

Th y2= f(t) =2t -1

2 + t+ 2 1+ t+2

=2t + 2 t+1

2 + t + 22

= 2 + t+ 2 t+1

Vy 22 t 2(t 1)

y f (t)2 t 2(t 1)

+ += =

+ + +vi

- 2 t -1

-1 t 2

1 2 , ( 2 t 1)f '(t)1 2 , ( 1 t 2 )

= + <

Bng bin thin:


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t 2 1 2 +f(t) - 0 +

f(t)

4 2 2 4 2 2+

1

T bng bin thin ta c

[ 2 ; 2 ]max f( t)

= 4+2 2 ; [ 2 ; 2 ]min f (t)

= 1

x R

max y 4 2 2

= + ;x R

min y 1

=

Bi 8. Tm GTNN ca biu thc 20 20S sin (x) cos (x)= +

Li gii

Nhn xt : Ta quy S v ht 2sin x

Ta c 2 10 2 10S (sin x) (1 sin x)= +

t 2t sin x= (0 t 1) . Yu cu bi ton tr thnh tm GTNN, GTLN ca hm s

10 10S f (t) t (1 t)= = + vi t [0;1]

9 9f '(t) 10t 10(1 t)=

9 9

f '(t) 0 t (1 t)= = 1

t 2=

1 1f (0) 1; f ( ) ; f (1) 1

2 512= = = . Vy

1MinS

512= ; MaxS 1=

Bi luyn tp 1. Tm GTNN , GTLN ca biu thc sau:2012 2012S sin (x) cos (x)= +

Bi 9. Tm GTNN, GTLN ca biu thc

S x 4 4 x 4 (x 4)(4 x) 5= + + + +

Li gii

iu kin 4 x 4

t t x 4 4 x= + + 2t x 4 4 x 2 (x 4)(4 x) = + + + +

2t 8

(x 4)(4 x)2

+ =

Ta c2

2t 8S t 4( ) 5 2t t 212= + = + +

Tm iu ca t:


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Xt hm s g(x) x 4 4 x= + + vi x [ 4;4]

1 1

g'(x)2 x 4 2 4 x

= +

; g '(x) 0= x=0

g( 4) 2 2; g(0) 4; g(4) 2 2 = = =

x [ 4 ;4 ]min g(x) 2 2

=; x [ 4 ;4 ]

max g(x) 4

= t [2 2;4]

S' 4t 1 0 t [2 2;4]= + < S l hm nghch bin trn [2 2;4]

MinS S(4) 7 ; MaxS S(2 2) 5 2 2= = = = +

Bi luyn tp: Tm GTNN, GTLN ca biu thc sau:

S x 1 8 x 4 (x 1)(8 x) 5= + + + + vi x [ 1;8]

Bi 10. Tm GTNN, GTLN ca2010 2011

S sin (x).cos (x)= vi x [0; ]2

Li gii

Nhn xt:

i, S 0 vi mi x [0; ]2

ii, tm GTNN, GTLN ca S ta tm GTNN, GTLN ca 2S (v khi 2S c th quy

ht v2

sin x hoc2

cos x ).Ta c 2 4020 4022S sin (x).cos (x)=

= 2 2010 2 2011(sin x) .(1 sin x)

t 2t sin x= (0 t 1). Khi 2 2010 2011S f (t) t .(1 t)= =

2009 2011 2010 2010f '(t) 2010t (1 t) 2011.t (1 t)=

2009 2010f '(t) t (1 t) [2010 4021t]=

t 0

f '(t) 0 t 1

2010t

4021

=

= =

=

f (0) 0 ;f (1) 0= = ;2010 2011

4021

2010 (2010) .(2011)f ( )

4021 (4021)=

Vy Min S =0 ;2010 2011

4021

(2010) .(2011)MaxS

(4021)=

Bi luyn tp. Tm GTNN, GTLN ca biu thc


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15 20S sin x.cos x= vi x [0; ]2

Bi 11. Tm GTNN, GTLN ca hm s :6 6y sin x cos x 2cos4x sin 2x 5= + + + , vi x R

Li gii

Nhn xt : 6 6 23sin x cos x 1 sin 2x4

+ =

2cos4x 1 2sin 2x=

Do ta a y v ht sin2x

Do y = 23

1 sin 2x4

+2( 21 2sin 2x )+sin2x-5

219

y sin 2x sin 2x 24= +

t t sin 2x= ( 1 t 1) . Yu cu bi ton tr thnh tm GTNN, GTLN ca hm s

219y t t 24

= + vi 1 t 1

19 2y ' t 1 ;y ' 0 t

2 19= + = =

Ta c 31 23 2 37y( 1) ;y(1) ; y( )4 4 19 19

= = =

Do x R

31miny

4= ;

x R

37maxy

19=

Bi luyn tp: Tm GTNN, GTLN ca hm s :6 6 4 4y sin x cos x 4(sin x cos x) 2cos2x 2= + + + , vi x R

Bi 12. Tm GTNN, GTLN ca hm s

1y 2(1 sin 2x.cos4x) (cos 4x cos8x)

2= +

Li gii

Ta c y 2 2sin 2x.cos4x sin 6x.sin 2x= +

2 32 sin 2x.(1 2sin 2x) (3sin 2x 4sin 2x).sin 2x= +

4 3 24sin 2x 4sin 2x 3sin 2x 2sin 2x 2= + +

t t sin 2x= ( 1 t 1)

Yu cu bi ton tr thnh tm GTNN, GTLN ca hm s4 3 2y 4t 4t 3t 2t 2= + + vi t [ 1;1]


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3 2y ' 16t 12t 6t 2= + ;1 1

y' 0 t 1; t ; t2 4

= = = =

y( 1) 5 = ;1

y( ) 12

= ;1 145

y( )4 64

= ; y(1) 1=

Vy min y 1= ; max y 5=

Bi 13. Tm GTNN ca hm s y x(x 2)(x 4)(x 6) 5= + + + + vi x 4 .

Li gii

Ta c 2 2y (x 6x)(x 6x 8) 5= + + + +

t 2t x 6x= +

Khi 2y t 8t 5= + +

Xt hm s 2g(x) x 6x= + vi x 4

g '(x) 2x 6;g '(x) 0 x 3= + = =

x - -4 -3 +g(x) - 0 +

g(x)

-8 +

-9

Suy ra t [ 9; ) +Yu cu bi ton tr thnh tm GTNN ca hm s 2y t 8t 5= + + vi t [ 9; ) + .

Ta c y' 2t 8 ; y ' 0 t 4= + = =

Bng bin thin

t - -9 -4 +

y - 0 +

y

14 +

-11Vy Miny=-11.

Trong nhiu bi ton tm GTNN, GTLN ca hm s khi bi c nhiu hn hai bin ta phi

tm cch qui v mt bin , sau tm GTNN, GTLN ca hm s theo bin s mi.

Sau y l cc bi ton minh ha

Bi 14. Tm GTNN, GTLN ca2 2

2 2

2 2

x xy yS (x y 0)

2x y

+ += + >

+Li gii


9/18

V t s v mu s ca S l cc biu thc ng cp bc hai i x, y nn ta xt TH y=0 v y

0 chia t s v mu s ca S cho 2y , sau chuyn v bin sx

ty

= .

TH1: y= 0 2

2

x 1S

2x 2= =

TH2: y 0 . Chia c t s v mu s ca S cho 2y ta c :

2

2

2

2

x x1

y ySx

2 1y

+ +=

+

tx

ty

= . Khi 2

2

t t 1S

2t 1

+ +=

+2

2 2

2t 2t 1S'

(2t 1)

+=

+; 2

1 3S' 0 2t 2t 1 0 t

2

= + = =

Bng bin thint - 1 3

2

1 3

2

++

S - 0 + 0 -

S1

2

3

2 3 2

3

2 3 2+

1

2

Kt hp TH1 v TH2 ta c : 32 3 2+

S 32 3 2

Vy MinS =3

2 3 2+khi

x 1 3

y 2

= ; Max S =

3

2 3 2khi

x 1 3

y 2

+=

Bi luyn tp : Tm GTNN, GTLN ca cc biu thc sau:

a,2 2

2 2

2 2

x xy yM (x y 0)

3x y

+ = + >

+

b,2 2

2 2

2 2

x xy 2yN (x y 0)

4x y

= + >

+

Bi 15. Cho a.b 0 . Tm GTNN ca4 4 2 2

4 4 2 2

a b a b a by ( )

b a b a b a= + + + +

Li gii

ta b

t

b a

= + . Ta ca b a b

| t | | | | | | | 2

b a b a

= + = + ( Theo C Si )

2 2

2

2 2

a bt 2

b a+ =

4 44 2

4 4

a bt 4t 2

b a+ = +


10/18

4 2 2y t 4t 2 (t 2) t= + + = 4 2t 5t t 4 + +

3y'(t) 4t 10t 1= + 2y ''(t) 12t 10 0= > vi mi t 2

Bng bin thin ca y(t)t - -2 2 +

y(t) + +

y(t)

-11

-

+

13

Suy ra y '(t) 0< vi t 2 ; y '(t) 0> vi t 2

Bng bin thin ca f(t)

t - -2 2 +

y(t) - +

y

-

-2

+

2

Vy Miny=-2 ; Maxy=2 .

Nhn xt

i, ta b

tb a

= + gip ta chuyn y v ht bin t.

ii, xt du ca y ta tnh y , lp bng bin thin ca y, sau suy ra du ca y

trn cc khong ( ; 2] v [2; )+ .

Bi 16. Cho x, y, z > 0 v x +y+z 1. Tm GTNN ca biu thc

2 2 2

3 3 3

3 3 3S x y z

x y z= + + + + +

Li gii

Nhn xt: Ta quy S v x+ y +z

p dng bt ng thc Bunhiacopxki ta c:

2 2 2 2 2 2 2(x y z )(1 1 1 ) (x y z)+ + + + + + 2 2 2 21

x y z (x y z)3

+ + + +

p dng bt ng thc C Si ta c:

33 3 3 3 3 3 3

3

3 3 3 3 3 3 9 9 81

3 . . x y zx y z x y z xyz (x y z)( )3

+ + = =+ + + +


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Vy2

3

(x y z) 81S

3 (x y z)

+ + +

+ +

t t= x+y+z (0 t 1)<

Khi 2

3

t 81S f (t)

3 t = + ;

5

4 4

2t 243 2t 729f '(t) 0 t (0;1]

3 t 3t

= = <

f(t) nghch bin trn (0;1] t (0;1]

244minS min f (t) f (1)

3= = =

Bi 17. Cho A, B, C l 3 gc ca mt tam gic. Tm GTNN ca biu thc

2 2 2A B C A B CP sin sin sin cot cot cot2 2 2 2 2 2

= + + + + +

Li gii

Ta c 2 2 2A B C 1 1 1

P sin sin sin 3A B C2 2 2

sin sin sin2 2 2

= + + + + +

Trong tam gic ABC ta cA B C 3

sin sin sin2 2 2 2

+ +

Ta nh gi cc biu thc theoA B C

t sin sin sin2 2 2

= + + vi3

t (0; ]2

p dng bt ng thc C Si ta c:

2 2 2 22 2 23

1 1 1 3 27A B C A B CA B Csin sin sin (sin sin sin )sin .sin .sin2 2 2 2 2 22 2 2

+ + + +

Vy 227

P t 3 f (t)t

+ =

3

54f '(t) 1 0

t= < vi mi

3t (0; ]

2

Bng bin thint

03

2f(t) -

P=f(t)

+

21

2

Vy

21

MinP 2=

3

t 2 =

Bi 18: Cho 0,0 yx v x + y = 1. Tm GTNN, GTLN ca biu thc

P = 2 x y3 + 3 .


12/18

Li gii

Do x 0, y 0 v x + y = 1 y = 1- x v 0 x 1 Ta c P = 2 x y3 + 3 = 2 x x3

3 +3

.

t t = 3x 1 t 3 . Khi 23

P f (t) tt

= = + vi [ ]t 1;3 .

2

3f '(t) 2t 0

t

= = 33

t =

2

Bng bin thin

t 1 3

3

23 +

f(t) - 0 +

f(t)

4 10

39

34

T bng bin thin ta c maxP = 10 3x = 3 x = 1

y = 0

minP = 34

93 3x = 3

2

3

33

3

3

3x = log

2

3

y =1- log 2

Bi 19: Cho x 0, y 0 v x + y = 1. Tm GTNN, GTLN ca biu thc

P =x y

+y+1 x+1

.

Li gii

Ta c: P =x y

+

y+1 x+1

=( )

2x+ y - 2xy+1

2x+xy

=2-2xy

2+xy

.

t xy = t , v x+ y =1, x 0, y 0 xy 0

1 2 xy

10 xy

4

10 t

4

Khi P = f(t) =2 - 2 t

2 + tvi

10 t

4 .

Do( )

2

-6f '(t) < 0

2 + t= vi

1t 0;

4

nn hm s f(t) lun nghch bin trong on

4

1;0 maxP = f(0) = 1 khi t = xy = 0

x = 0 , y = 1

y = 0 , x = 1


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minP = f(4

1) =3

2khi t =

4

1 x = y =

2

1.

Trong cc k thi chn HS gii thng c bi ton tm GTLN, GTNN ca hm s c nhiu

bin ph thuc ln nhau. gii nhng bi ton dng ny ta c th dng phng php kho

st ln lt tng bin, ngha l : tm GTNN ( hoc GTLN ) ca hm s vi bin th nht v

cc bin cn li coi l tham s , ri tm GTLN (GTNN) ca hm s vi bin th hai v ng

vi gi tr xc nh ca bin th nht m cc bin cn li coi l tham s

Bi 20. Cho min {D (x, y) |0 x 1;0 y 2}=

Tm GTNN ca hm s f (x, y) (1 x)(2 y)(4x 2y)=

Li gii

f(x,y) (1 x)(2 y)[2(2-y)-4(1-x)]=

t u=1-x ; v=1-y u [0;1] ; v [0;2]

f ( x, y ) g(u, v) uv(2v 4u ) = = 2 22uv 4u v=

Coi u l n , v l tham s . Ta c

2g '(u,v) 8uv 2v= +

vg '(u,v) 0 u

4= =

Bng bin thin

u- 0

v

41 +

g'(u,v) + 0 -

g(u,v)

3v

4

0 22v 4v

V 22v 4v 2v(v 2) 0 v [0; 2] = nn 2 2g(u,v) 2v 4v 2(v 1) 2 2 =

Vy Ming(u,v) =-2u 1

v 1

= =

Suy ra Minf(x,y)=-2x 0

y 1

= =

Bi 21. Cho hm s f (x, y,z) xy yz zx 2xyz= + + trn min

{ }D (x, y,z) |0 x, y, zv x y z 1= + + =

Tm GTNN, GTLN ca f(x,y,z)


14/18

Li gii

*) Tm GTNN ca f(x,y,z)

Gi s1

z min{x, y,z} z [0; ]3

=

f (x, y,z) xy (x y)z 2xyz xy(1 2z) z(1 z)= + + = +

23 2(1 z) 1(1 2z) z(1 z) (2z z 1)

4 4

+ =

Xt hm s 3 21

F(z) (2z z 1)4

= vi1

z [0; ]3

21 z(1 3z) 1F'(z) (6z 2z) 0 z [0; ]4 2 3

= = >

z - 0 1 +F(z) +

F(z)

7

27

1

4

Vy Max f(x,y,z) =7

27

1x y z

3 = = =

*) Tm Min f(x,y,z)

f (x, y,z) (1 y z)y yz z(1 y z) 2(1 y z)yz= + +

2 2 2 2y y z zy z 2yz 2y z 2z y= + + +

Xt 2 2 2G(z) z (2y 1) z(1 3y 2y ) y y= + + +

2G '(z) 2(2y 1)z 1 3y 2y

2(2y 1)z (2y 1)(y 1)

= + += +

Gi s1

y min{x, y,z} y [0; ]3

=

1 yG '(z) 0 z

2

= =

Bng bin thin

z-

01 y

2

1 +

G(z) + 0 -

G(z)

1 yG( )

2


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2y y 2y

V 2 2 2y (y y ) 2y y y(2y 1) 0 = =

2G(z) y 0 ;y 0

G(z) 0z 1

== =

Vy Min f(x,y,z)=0x y 0

z 1= = =

Bi luyn tp 1: Cho x, y,z 0 v x+y+z=1. Tm GTLN ca biu thc:

1 1 1 1 1 1S xyz[x( ) y( ) z( )]

y z x z y x= + + + + +

Bi luyn tp 2: Cho x, y, z tha mn 0 x, y,z 1

Tm GTLN ca biu thc3 3 3 2 2 2

P 2(x y z ) (x y y z z x)= + + + +

Bi 22. Cho x, y, z >0 tha mnx y z 4

xyz 2

+ + = =

Tm GTNN, GTLN ca biu thc 4 4 4S x y z= + +

( thi chn HSG QG nm 2004 )

Li gii

t1

2

3

S x y z 4

S xy yz zx

S xyz 2

= + + = = + + = =

Ta biu din S theo 2f ( S ) . Cn c vo bi, tm min bin thin ca 2S . Sau ta kho st

2f ( S ) tm GTNN, GTLN ca S.

*) Ta biu din 2S theo z

2

2

2 2S xy yz zx z(4 z) 4z z

z z= + + = + = +

*) Ta tm min bin thin ca z

Dox y 4 z

2xy

z

+ =

= 2

8(4 z)

z 2(z 2)(z 6z 4) 0 +


16/18

z - 3 5 2 3 5+ +

z-2 - - 0 + +

2z 6z 4 + + 0 - - 0 +

VT - + - +

Vy z [3 5;2] [3 5; ) + +

T gi thit suy ra z (0;4)

Do z [3 5;2]

Xt 222

S 4z zz

= + vi z [3 5;2]

2'

2 2 2

2 (1 z)(2z 2z 2)S 4 2z

z z

= =

Bng bin thin

z- 1 5

2

3 5 1

1 5

2

+2 +

'2S 0 - 0 + 0 -

2S 5 5 1

2

5 5 1

2

5 5

Suy ra 2S5 5 1

[5; ]2

Khi 4 4 4 2 2 2 2 2 2 2 2 2 2x y z (x y z ) 2(x y y z z x )+ + = + + + +

2 2 2

1 2[S 2S ] 2[(xy yz zx) 2xyz(x y z)]= + + + +

= 2 2 21 2 2 1 3[S 2S ] 2[S -2S .S ]

= 2 22 2[16 2S ] 2[S 16]

= 2 22 2 2256 64S 4S 2S 32 + +

= 22 22S 64S 288 + = 2f (S )

2 2f '(S ) 4S 64 0= < 2S5 5 1

[5; ]2

. Bng bin thin


17/18

2S 5

5 5 1

2

f( 2S ) -

f( 2S )

18

383 165 5 Vy MinS=383 165 5 ; MaxS=18.

rn luyn k nng gii cc bi ton dng trn, ta c bi ton sau:

Bi luyn tp: Cho x, y, z >0 tha mnx y z 4

xyz 2

+ + = =

Tm GTNN, GTLN ca cc biu thc sau:

a, 2 2 2M x y z= + +

b, 3 3 3N x y z= + +

c, 3 3 3 3 3 3P x y xy x z xz y z yz= + + + + +

BI TP NGH

Bi 1. Tm GTNN ca hm s:

y x 2 2 x 4 (x 2)(2 x) 5= + + + + ( x [ 2;2] )Bi 2. Tm GTNN, GTLN ca hm s:

2

2

x x 3y

2x 1

=

+

Bi 3. Tm GTNN, GTLN ca hm s:2 2

2 2

x xy yy

2x y

+=

+( 2 2x y 0+ > )

Bi 4. Tm GTNN, GTLN ca hm s:

y (x 2)(x 4)(x 6)(x 8) 5= + + + + + ( x [ 4;2] )

Bi 5. Tm GTNN, GTLN ca hm s:6 6y sin x cos x 2cos4x sin 2x 5= + + + +

Bi 6. Tm GTNN, GTLN ca hm s:x yy 9 3= + vi x, y 0 v x+y=1


y 4cos x cos4x= vi x [ ; ]2 2


18/18

Bi 8. Tm GTNN, GTLN ca hm s:4 2

4 2

3cos x 4sin xy

3sin x 2cos x

+=

+


x x

x x

1 1S 9 4(3 ) 5

3 9

= + + + vi x [ 1;1]

gtlt_gtnn ham so

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