TÌM GIÁ TRỊ LỚN NHẤT NHỎ NHẤT

[Pages:18] - T?I LIU HC TP MIN PH?

T?M GI? TR LN NHT NH NHT

PP 1: S dng ph?p bin i ng nht v? t?nh cht ca h?m s lng gi?c.

V? d 1. T?m gi? tr ln nht, gi? tr nh nht ca c?c h?m s sau.

1. y 4 sin x cos x 1

2. y 4 3sin2 2x

1 Ta c? y 2 sin 2x 1 .

Li gii:

Do 1 sin 2x 1 2 2 sin 2x 2 1 2 sin 2x 1 3 1 y 3 .

*

y

1 sin 2x

1

2x

k2

x

k.

2

4

* y 3 sin 2x 1 x k .

4

Vy gi? tr ln nht ca h?m s bng 3 , gi? tr nh nht bng 1 .

2. Ta c?: 0 sin2 x 1 1 4 3sin2 x 4

*

y 1 sin2 x 1 cos x 0 x

k.

2

* y 4 sin2 x 0 x k .

Vy gi? tr ln nht ca h?m s bng 4 , gi? tr nh nht bng 1 . V? d 2. T?m gi? tr ln nht ca h?m s sau y sin x 1 trong khong 0 x

sin x

Li gii:

1 V? 0 x n?n 0 sin x 1,do ? sin x

sin x

Group:

 - T?I LIU HC TP MIN PH?

Vy h?m s t gi? tr , ln nht l? 0 ti sin x 1 x .

2

PP2: C?c b?i to?n s dng bt ng thc ? bit t?m gi? tr ln nht, gi? tr nh nht

V? d 1. T?m gi? tr ln nht ca h?m s y 1 1 cos2x 1 5 2sin2 x

2

2

Li gii:

Ta c? y 1 1 cos2x 1 5 2sin2 x y 1 1 cos2x 5 1 sin2 x

2

2

2

42

?p dng bt ng thc Bunyakopvsky cho 4 s: 1; 1; 1 1 cos2x ;

2

5 1 sin2 x ta c?:

42

1. 1 1 cos2 x 1. 5 1 sin2 x 12 12 . 1 1 cos2 x 5 1 sin2 x

91 2.

22

2

42

2

42

4 2.1 2

22

Hay y

2

Du

bng

xy

ra

khi

1

1

cos2 x

5

1

sin2

x

x

k , k

2

42

6

V? d 2.

T?m g?a tr nh nht ca h?m s

y

1

1

2 cos x 1 cos x

vi

x

0;

2

.

Li gii:

Ta thy

2 cos x 0,x R

v?

1

cos

x

0,

x

0;

2

.

Suy

ra

1 2 cos x

v?

1 1 cos x

l?

hai s dng. ?p dng bt ng thc AM- GM cho hai s dng ta c?

1

1

2

2 cos x 1 cos x 2 cos x1 cos x

Mt kh?c tip tc ?p dng bt ng thc AM-GM ta c?

Group:

 - T?I LIU HC TP MIN PH?

2 cos x 1 cos x 3

2 cos x1 cos x

2

2

2

4

y

2 cos x1 cos x 3

Vy

4

min y , du bng xy ra khi

0;

2

3

1

cos x x

2

3

v?

x

0;

2

.

PP3: C?c b?i to?n s dng h?m s, t?nh cht h?m s, c bit t?nh ng bin, nghch bin ca h?m s.

V? d 1. T?m tp gi? tr ln nht, gi? tr nh nht ca c?c h?m s sau.

1. y 6 cos2 x cos2 2x

2. y (4 sin x 3 cos x)2 4(4 sin x 3 cos x) 1

Li gii: 1. Ta c?: y 6 cos2 x (2 cos2 x 1)2 4 cos4 x 2 cos2 x 1

t t cos2 x t 0;1 . Khi ? y 4t2 2t 1 f (t)

t 0 1

f (t)

7

1

Vy min y 1 t c khi cos x 0 x k

2

max y 1 t c khi cos2 x 1 x k 2. t t 4 sin x 3 cos x 5 t 5 x

Group:

 - T?I LIU HC TP MIN PH?

Khi ?: y t2 4t 1 (t 2)2 3 V? t 5; 5 7 t 2 3 0 (t 2)2 49 Do ? 3 y 46 Vy min y 3; max y 46 . PP 4: C?c b?i to?n li?n quan n a cos x bsin x c

V? d 1. T?m gi? tr ln nht, gi? tr nh nht ca h?m s: y 2 cos2 x 2 3 sin x cos x 1

Li gii:

Ta c? y 2 cos2 x 2 3 sin x cos x 1 2 cos2 x 1 3 sin 2x 2 cos 2x 3 sin 2x 2*

1

2

2

cos

2

x

3 2

sin

2x

2

2

cos

2x

3

2

Mt

kh?c

1

2 cos

2x

3

2

4, x

R

0

y 4, x R

.

Vy gi? tr ln nht ca h?m s l? 4 x k

6

V? gi? tr nh nht ca h?m s l? 0 x k

3

V? d 2. T?m gtln v? gtnn ca c?c h?m sau :

sin x 2 cos x 1 y

sin x cos x 2

Li gii:

Do sin x cos x 2 0 x h?m s x?c nh vi x

X?t

phng tr?nh

:

sin y

x 2 cos x 1

sin x cos x 2

(1 y)sin x (2 y)cos x 1 2y 0

Group:

 - T?I LIU HC TP MIN PH?

Phng tr?nh c? nghim (1 y)2 (2 y)2 (1 2y)2

y2 y 2 0 2 y 1

Vy min y 2; max y 1.

B?I TP TNG HP MIN MAX

B?i 1:

T?m gi? tr ln nht gi? tr nh nht ca c?c h?m s sau:

a). y f x 2 3 sin 2x cos 2x

b). y f x sin x cos x2 2 cos 2x 3 sin x cos x c). y f x sin x 2 cos x2 sin x cos x 1 d). y 4sin2 x 3 3 sin 2x 2cos2 x

LI GII

a). ?p dng bt ng thc: ac bd . a2 b2 c2 d2

2

Ta c? 2 3 sin 2x cos 2x 2 3 1 2 3 sin 2x cos 2x 2 2 3 2 2 3 2 3 sin 2x cos 2x 2 2 3

Vy min f x 2 2 3 , max f x 2 3 . b). y f x sin x cos x2 2 cos 2x 3 sin x cos x f x 1 1 sin 2x 2 cos 2x

2

Group:

 - T?I LIU HC TP MIN PH?

Ta c?

1 sin 2x 2 cos 2x

1

1

4 sin 2x 2 cos 2x

17

2

4

2

4

17 1

17

sin 2x 2 cos 2x

22

2

1

17 1 1 sin 2x 2 cos 2x 1

17

(cng th?m 1 v?o ba v)

2

2

2

1 17 f x 1 17

2

2

Vy min f x 1 17 , max f x 1 17 .

2

2

c). Khai trin v? r?t gn f(x)

f x sin x 2 cos x 2 sin x cos x 1 2 sin2 x 3 sin x cos x 2 cos2 x 1

1 2

cos2 x sin2 x

3

3

sin 2x 1 cos 2x sin 2x

2

2

Ta c?

3 sin 2x 2 cos 2x

9

3

5

4 sin 2x 2 cos 2x

2

4

2

2

53

5

sin 2x 2 cos 2x

22

2

5 3 sin 2x 2 cos 2x 5 (nh?n th?m (-1) v?o ba v )

22

2

5

3

5

1 1 sin 2x 2 cos 2x 1

(cng th?m (-1) v?o ba v)

2

2

2

3 fx 7

2

2

Vy min f x 7 , max f x 3 .

2

2

d).

1 cos 2x

y 4.

3

2

1 cos 2x

3 sin 2 x 2

2

3

3 sin 2x cos 2x

1 6

3

1

2

sin

2x

2

cos

2x

1

6

sin

2x 6

1 5 y 7

Group:

 - T?I LIU HC TP MIN PH?

(V?

1

sin

2x 6

1,x

. Gi? tr ln nht y 7 khi sin 2x 6 1 x 3 k

. Gi? tr nh nht y 5 khi sin 2x 6 1 x 6 k

B?i 2:

T?m gi? tr ln nht v? gi? tr nh nht ca c?c h?m s sau: a). y a sin x b cos x (a v? b l? hng s, a2 b2 0 ). b). y sin2 x sin x cos x 3cos2 x c). y a sin2 x bsin xcos x c cos2 x ( a, b, c hng s). LI GII a). Ta c? a sin x b cos x a2 b2 a2 b2 a sin x b cos x a2 b2

Kt lun max y a2 b2 , min y a2 b2

y sin2 x sin x cos x 3 cos2 x 1 cos 2x 1 sin 2x 3 1 cos 2x

2

2

2

1 2 sin 2x sin 2x

2

Ta

c?:

1 sin 2x cos 2x

1

1

1 sin 2x cos 2x

5

2

4

2

2

51

5

5

1

5

sin 2x cos 2x 2 2 sin 2x cos 2x 2

22

2

2

2

2

5

5

2 y2

2

2

Kt lun max y 2

5 , min y 2

5.

2

2

Group:

 - T?I LIU HC TP MIN PH?

c). y a sin2 x b sin x cos x c cos2 x

1 cos 2x b

1 cos 2x a c b

ca

y a

sin 2x c

sin 2x cos 2x

2

2

2

22

2

Ta c?

b

ca

sin 2x cos 2x

b2 (c a)2

2

2

44

b2 (c a)2 b

ca

b2 (c a)2

sin 2x cos 2x

442

2

44

a c b2 (c a)2 a c b

ca

a c b2 (c a)2

sin 2x cos 2x

2

4

4

22

2

2

4

4

a c b2 (c a)2

a c b2 (c a)2

y

2

4

4

2

4

4

Kt

lun:

ac max y

b2 (c a)2

ac

, min y

b2 (c a)2

2

4

4

2

4

4

B?i 3:

T?m gi? tr ln nht v? gi? tr nh nht ca c?c h?m s sau: a). y 3sin x 4 cos x 7 b). y sin2 x 4 sin x cos x 5 c). y 2 sin2 x sin 2x cos2 x 3 d). y sin x 3 cos x 5

sin x cos x 2

LI GII a). Ta c? 3 sin x 4 cos x 32 (4)2 3sin x 4 cos x 5

7 5 7 3sin x 4 cos x 7 5 2 y 12

Kt lun: max y 12, min y 2

Group:

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