Chuyên đề bồi dưỡng học sinh giỏi giá trị lớn nhất, giá trị nhỏ nhất phan huy khải (phần 6)

Chuyfin cirBDHSG Tojn gia trj I6n nhS't v.i gia tri nh6 nliat - Phan Huy K I T S T GIA TRj idiN NHAT VA NHO NHAT CUA HAM SO §1... Chuyen de BDHSG Toan gia trj I6n nhjt vk gia tr| nhi nh

Chuyfin cirBDHSG Tojn gia trj I6n nhS't v.i gia tri nh6 nliat - Phan Huy K I T S T

GIA TRj idiN NHAT VA NHO NHAT CUA HAM SO

§1 LfNG DgNG G I A TRj L6N NHAT V A NHO NHAT DE GIAI

PHl/ONG TRINH V A BAT P H J O N G TRJNH K H O N G C O THAM S 6

Ta sur dung cac kct qua hicn nhicn sau day de giai mot so Idp phiTdng trinh

va bat phu'dng trinh c6 cau triic dac bict difdc xct den trong muc nay

M e n h d e 1: Xet phifdng trinh f(x) = g(x), vdi x e D (1)

Neu nhiTvdi phtfdng trinh (1) ta c6 dieu kien m a x f ( x ) < m i n g ( x ) thi (1) v o

x e D X € D J

nghiem

Menh de 2: Xet phtfdng trinh f(x) = g(x), vdi x e D (2)

Ne'u nhif ta c6 m a x f ( x ) = ming(x) = a thi (2) < »

x e D x e D

f(x) = a g(x) = a

Xet cac bai toan minh hoa sau day ,

Bai 1 Giai phiTdng trinh V x - 2 + V 4 - x = x^ - 6 x + 1 L , , , ,

Vay X = 3 la nghiem duy nha't cua phufdng trinh

Nhqn xet: Xet each giai khac sau day:

D a t u = V x - 2 + V 4 - x => u^ = 2 + 2 V ( x - 2 ) ( 4 - x ) = 2 + 2 V - x ^ + 6 x - 8

Cty TMHH M!V DVVH KhanglZiH"

hay u^ = 2 + 2 V 3 - ( x ^ - 6 x + l l ) Ttf do ta di den phU'dng trinh he qua sau: ; ^ ,

= 2 + 2 N / 3 - U <=> U^ - 2 = 2 ^ 3 - u ' "

\J2<M<7, ^ J > ^ < u < 3

^ [ u ' * - 4 u ^ + 4 - 1 2 - 4 u [ u ' * - 4 u ^ + 4 u - 8 = 0

> / 2 < u < 3 "

( u - 2 ) ( u ^ + 2 u ^ + 4 ) = 0 o u = 2 •i-.J^j r'^-'~ ^s: ''-it ^- h"^ i<ni]

Ttf do dan den phifdng trinh V x - 2 + N / 4 - X = 2 o 2 + 2V^t^ + 6 x - 8 = 4

o -x^ + 6x - 9 = 0 o X = 3

Thuf lai X = 3 vao phifdng trinh da cho ta tha'y thoa man, nen do la nghiem

duy nha't cua phifdng trinh da cho

Ta thu lai ket qua tren!

Cac ban c6 nhan xet gi ve tinh hieu qua cua moi each giai tren! J

Bai 2 Giai cac phifdng trinh sau:

Chuyen de BDHSG Toan gia trj I6n nhjt vk gia tr| nhi nha'l - Phan Huy Khii

2 Xet phurdng trinh Vsx^ + 6x + 7 + Vsx^ + lOx + 14 = 4 - 2x - ' ' (2)

Tijr do ta CO X = - 1 la nghiem duy nhat cua (2)

Bai 3 Giai phi/Ong trinh 2^"+' + 2^"^^ =

Vay X = ^ la nghiem duy nhat cua phi/dng trinh da cho

Cty TNHH MTV D W H Khang Vigt

p^i 4 Giai phUOng trinh 2^^ ' - 2" " = (x -1)^

Nhir the' (1) CO nghiem duy nhat x = 1

m 5 Giai phifdng trinh Sx" - 4x^ = 1 - Vo + x ^

Chuyen dg BDHSG ToAn gii tr| Idn nhaft va gii trj nh6 nhS't - Phan Huy Kh^i

V a y minr(x) = () o x = 0

x e l

Ro rang phiMng Irinh da cho co the vict duTdi dang f(x) = 0

Ttr do suy ra phu'(tng trinh da cho c6 dang min f(x) = 0 o x = 0 ,,

Vay X = 0 lii nghicm duy nhal can tim

Chuy: ^

Dat biet vdi phUdng trjnh dang f(x) = a, x e D

ma thoa man dicn kicn maxl"(x) = a (hoac minf(x) = a ) ,

Do sin^x + cos^x = 1 sin'^x + cos'^x = (sin^x)' + (cos^)'' > —

Cty TNHH MTV DWH Khang Vigt

Isin'x + cos'x = 32(sin"x + cos''x) (1)

^ f(x) = g(x)

Ttf cac ket qua tren suy ra , ' s » ,

•K

f(x) = l g(x) = l <=> \

x - k - , k e Z

x = - + k - , k e Z (3)

4 2

Ro rang he (2) (3) v6 nghiem (1) v6 nghiem

Chu y: (*) chtfug minh nhiT sau:

Xet ham so h(x) = x" + (1 - x)" vcti 0 < x < 1

Nhu- vay do 0 < a < 1 => h(a) > h

1^ o a " + ( l - a ) " > - i ^ h a y a " + b " > - i ^ •a'u b^ng xay ra<=>a=:^<=>a = b = ^ = > dpcm!

x e R

Ap dung bat ding thuTc Bunhiacopski, ta c6:

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Chuyen dg BDHSG Toan gia tri Idn nha't va gia tri nh6 nhaft - Phan Huy KhSi

cos^ 3x + (2 - cos^ 3x) (1 +1) > |cos3x + V2-cos^3x

Tit do suy ra f(x) = cos3x + yjl-cos^ 3x < 2 Vx e

f(x) = 2 <=> cos3x 1 -Jl-cos^ 1 3x

o cos3x = V 2 - ; o s ^ 3x fcosSx >0

cos^ 3x = 2 - cos^ 3x ocos3x = 1

Vay X = k27t, k e Z la nghiem cua phiitfng trinh da cho

^'i'H>-Bai 8 Giai phtfdng trinh Vx^+ x - l + \/x-x^+ 1 = x^ - x + 2

Theo ba't d^ng thiJc Cosi Vx e D, ta c6

Cty TNMH MTV DVVH Khang Vigt

fir suy ra \ / x ' + x - I + V x - x ^ + 1 <x +1 hay f(x) < 0 Vx e D

I vay X = 1 la nghiem duy nha't cua phiTctng trinh d5 cho

Bai 9 Giai phU'dng trinh

1

COS^ X + +1 sin^ X + ^

sin^ X y COS \J

ChuySn de BDHSG Toan gia tri I6n nhat va gia tri nli6 nhat Phan iiuy Kh^i

f(x) =l2-<=> sin'2x = 1 c=> cos2x = ( ) « x = - + n ^ , n € Z

2 4 2

V i phifdng trinh da cho c6 dang f(x) - g(y) x, y e M (3)

Tijf cac lap luan Iron siiy ra (3) o

l-(x) = 1 2

-g(y) = i 2 ^

<::>x = —+ n— ; v = — + k27t, n va k 6 Z

4 2 2

B a i 10 Giai phiTdng Irinh (sin3x + cos2x)' = 5 - sinx

HUdng dan giai

D a l f(x) = (sin3x + cos2x)\(x) = 5 - sinx, x e R

(1)

f(x) - 4 o sin3x + cos2x = 2 o

sin3x = - l cos2x = - l

g(x) = 4 (4)

T i r ( l ) ( 2 ) s u y r a ( 3 ) ( 4 ) « x = - + kn, k e Z

Vay X = ^ + kn, k e Z la nghiOm ciia phi/dng Irinh da cho

Nhanxet:

Kho CO each giai nao khac gon gang hdn each giai trcn

Cac ban ciJ thijr ti/dng lu"dng sau khi siir dung cong thuTc sinSx = 3sinx - 4sin

cos2x = 1 - 2sin^x, la difa phiTi^ng trinh da cho ve dang:

304

Cty TNHH IVITV DWH Khang Vl§t

( 4 s i n \ 2sin^x - Bsinx - 1)^ = 5 - sinx

<r> 16sinS + 16sin'*x - 20sin''x - 20sin\ 5 s i n \ 7sinx - 4 = 0

o (sinx - l)(16sin'x + 32sin''x + 12sin''x - 8 s i n \ 3sinx + 4) = 0 Den day mdi cac ban giai tiep!! Cac ban thay the' nao?

—^ = m ( l ) c o n g h i e m

x ^ + 2 x + 10 • ' , i „• •

Do x^ + 2x + 10 ^ 0 Vx ( V I x^ + 2x + 10 > 0), nSn (l)<::>4x^ + I 4 x + 46 = mx^ + 2 m x + 10m

Vay X = 2 la nghiem duy nhat can tim - ; A^Aa« jcef: X e t each giai khac sau day - • l «

Chuyfin dg BDHSG Toan gia trj I6n nhat vi gia tr| nh6 nhflt - Phan Huy KhAi

Vay minP = 0 o x = k7t;y = 0 , k 6 Z

Tir do suy ra nghiem cua (1) la x = kK, y = 0 vdi k e Z

Bai 13 (De thi tuyen sinh Dai hoc Cao ddn^ khoi A)

\ y-yfty =3

Vx + i + 7y + i = 4

Hudng dan giai

x + y - V x y= 3 (1)

Giai he phifdng trinh

X€i he phi/dng trinh

Vi the theo bat ding thtfc Cosi, ta c6 tijr (1): x + y = 3 + 7xy ^ 3 +

Cty TNHH IMIV DVVH Kliang Vi$t

Tur do suy ra (2) o X = y = 3 va x = y =3 cung thoa man (1)

Vay he (1) (2) c6 nghicMn duy nha'l x = y = 3

Nhan xet: Neu khong sit diing phiTdng phap tim gia trj U^n nha't ci'ia hiim so de

danh gia hai ve', ta c6 the giai ihuan tuy he phtfdng trinh trcn nhU'sau:

Tiir(2)c6x + 1 + y + 1 + 27(x + I)(y + 1) = 16

<=>2 + x + y + 27xy + (x + y) + ] =16 ,0)' t ' ^ ff Dat t = 7xy > 0, thi tuf (1) CO X + y = 3 + t -

Thay lai vao (7) va c6 3 + t + 27t^ + 3 + t + l = 14

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Chuygn 6& BDHSG ToJin gii tri Idn nh9"t gia tri nh6 nhS't - Phan Huy Kh^i

§ 2 G I A TR! L 6 N N H A T V A N H O N H A T C U A

H A M SO PHg T H U Q C T H A M SO

Cho ho ham so vdi x e D Vcli moi gia tri cua tham so m, ta xct bai

toan tim gia trj Idn nhat, nho nha't cua ham so F^Cx) vdi x e D NhU vay dT

nhien gia trj Idn nhat, nho nha't cua ham so F J x ) Ircn D la cac dai lifdng p h u

thuoc tham so m Tiay theo gia trj cua m hay khao sat cac linh chsi't cua cac

gia tri Idn nhat, nho nhat cua ho ham so Fn,(x) theo mot lieu chi nao do Do

la noi dung cua bai loan chiing ta xct trong muc nay

A Bien luan theo tham so' gid tri Idn nhat va nho nhat cua ho ham so

phu thuoc tham so

B a i 1 Cho ho ham so f^Cx) = tan^(x+ a ) + t a n ' ( x - a ) , vdi tham so a e

T i m gia tri nho nhat cua faCx) va bien luan theo a

HUfing ddn gidi Bien ddi r„(x) ve dang sau day {*'\ >^ 4 / n %

^ ^ s i n ^ x + a ) s i n ' ( x - a ) •• i »f (J) Kil idi ,C1 i xi\r f

Ta CO Vx e D ^ , (d day D„ lii mien xac dinh cua ham so f a ( x ) ) /^-j ^, j y,,

0 < sin'2x + sin'2tt < sin^2a (2)

Dau bang trong (2) xay ra <=> sin2x = 0

Do cos2a > 0, nen (cos2x + cos2a)^ < (1 + cos2a)^ (3) " ' M

Dau bang trong (3) xay ra <=>'cos2x = 1

V I he •' ' " " ^ ^•\^••{^^ (.(•, nghiem, ch^ng han x = 0, v i the trong truTdng

( - l + c o s 2 a r 4sin'^a Nhi/ vay theo tham so' a, ta c6 ket luan sau:

fn,(x) = sin'*x + cos'*x + msinxcosx, x e R Bien luan theo m j

t l HUo'ng ddn gidi

1

- 1

m Viet lai fm(x) di/di dang sau: f,„(x) = 1 - - s i n 2x + — s i n 2 x !! I "-f

Co ba kha nang sau: : ;

^Ne'u m > 2, khi do ^ - ^ cd bang bien thien sau

Join gia trj I6n nha't va gii trj nhd nhlft - Phan Huy Khji

TiT do suy ra max F,,,(t) = F„,(l) = m + I

ncu m < - 2 : ,6!)' W;| S, < m

m - + 8

8

1 + m , ncu - 2 < m < 2 ; , n c u m > 2

1 + m

2

l - m , ncu m < 0 , n c u m > 0

^ min F,„(l):

) Cho ham so f(x) = Vacos' x + bsin" x + c + Vasin^ x + bcos^ x + c

Tim a, b, c de ham so' xac dinh vc'li m o i x e R

2 Gia suf f(x) la ham so xac djnh tren R T i m gia tri Idn nha't cija ham so

F(x) = f(x) + msin2x, vdi x e

Bicn luan iheo m

1 Viet lai f(x) difdi dang sau:

l + cos2x , l - c o s 2 x f(x) = , a + b + c + a + b + c 1 - c o s 2 x l + cos2x

b - a min

|i|<i Khi do (4) (5) i ± ^ ± ^ + ^ _ l > 0 o c + b > 0

ra max F(x) = max f(x) + max g(x) = J2(a + b + 2c) +

312

Cty TNHH MTV DVVH Khang Vi?t

; J 4,Tim gia trj Ic'Jn nhaft va nho nhat cua ham so':

' ' " l " (x) = mx^ + (m + l ) x + m + 2, Iren mien D = { x : l < x < 2 }

Hiidng dan gidi ,

fa ihay l()(x) = 1 > 0, nen f()(x) la ham dong bie'n khi x e f 1; 2]

ri>minr()(x) = io(l) = 3; maxf,)(x) = f,)(2) = 4 ^

x e D x e D

, Khi m > 0 Tif f|„(x) = 2mx + m + 1 , va do m > 0, nen

m + 1 2m < 0, nen c6

ba kha nang sau: i ;

a Neu < i m < - - Liic nay ta c6 bang bien thien

Chuygn di BDHSG loAn gia Iri I6n nhgt va gia Iri nUd nh3't - Phan Huy KhSi

Tir do suy ra max f,„ (x) = f„

x e D

m + l 2m

3 m 2 + 6 m - l '-^ii'-i^

4m rmnf,„(x) = m i n { f , „ ( l ) ; f „ , ( 2 ) }

m i n f „ ( x ) =

X6D

1 3m + 3, ncu m > —

^ ; maxf,„(x) = 7m + 4, neu m < —

4 '.^-.x -"•

7m + 4 , ne'u m > - —

~ 4 3m + 3, neu m < —

4

B Cdc ling dung cua vi^c khdo sdt gid tri Idn nhd't vd nhd nhd't cua cdc

ham so'phif thuQC tham so

Khi do tiir (2) suy ra a + 2 > 0 <=> a > - 2 Loai kha n5ng nay v i khong thda man a < - 4

^'Bay g i d tir (2) suy ra 2 - a > 0 o a < 2

Loai kha nang nay v i khong thoa man a > 4

a 3- Nc'u - 1 < - - < 1 (turc la k h i - 4 < a < 4) Bay g i d cd bang bien thien:

Tif (2) ta t o : —a >0^ a- < 0 <=> a = 0

Tom lai a = b = 0 la cac gia Iri can tim cua tham so' a va b '*

Bai 2 Tim m de phiCdng trinh sau:

(cos4x - cos2x)^ = ( m ^ + 4 m + 3 ) ( m ^ + 4 m + 6) + 7 + sin3x c6 nghiem

Hudng ddn gidi

Dal f(x) - (cos4x - cos2x)\e R

Khi do ro rang do |cos4x — cos2x| < 2 Vx 6 K

= : > f ( x ) < 4 V x e E

V a f ( x ) = 4 <;=>|cos4x-cos2x| = 2

cos4x = 1 cos2x = — 1

Tirdo suy ra: maxf(x) = 4 <=>cos2x = - 1 (1)

Cty TNHH MTV DVVH Khang Vi?t

b Neu m = - 2 , Ihi (4) <=> m a x l " ( x ) = m i n g _ , ( x ) = 4

xeR xeR

cos2x = —1 (5)

s i n 3 x = = - l (6)

Ro rang he (5) (6) c6 nghiem (thi du x = — thoa man (5) (6))

0ai 3 Tim m de phtfctng trinh sau:

317

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oiiuyuii ue u u i i j u lujii L)id in luii iiiui vd g u ill iiliu ililjl - Hlljll HUy MIUI

a Ncu m ?^ 2, Ihi max K x ) < miiig,,,(x) : o I'hiAfng Irinh da c h o \

Do he (*) M"> nghicm => Phifdng trinh da cho cung v6 nghicm khi m = T

Tom lai vc'li moi m, phU'dng trinh da cho v6 nghicm

Bai 4 Tim m sao cho moi nghicm cua bat phiTdng trinh x' - 3x + 2 < 0 cung i;,

nghicm cua bat phifdng trinh mx' + (m + 1 )x + m + 2 > 0

Hitdng ddn gidi

V i x' - 3x + 2 < 0 o I < X < 2

Do do bai toan da trcl thanh:

Tim m dc ba't phU'dng trinh: ijj.) = mx^ + (m + 1 )x + m + 2 > 0 ( 1 ) diing vdi

moi 1 < X < 2

o t , u ^ - t M, 'Ui i '{iw fsj £«3 Ukn m ;

Dieu nay xay ra khi va chi khi: min 1"„ (x) > 0

Vay m > — la cac gia tri can tim cua tham so m

Bai 5 Tim m dc bat phiTdng trinh sau: '

gai toan trcf thanh: Bicn luan theo m gia trj nho nhaft cua ham so fm(x)

Lap bang sau day:

X

Tirbang bic'n thien, xct cac kha nang sau day:

Neu m > 1 Khi do ta co bang bien thien sau:

2 Neu m < - 1 Khi do ta co bang bicn thien sau:

Chuyen BDHSG Toan gii tri I6n nha't vA gia trj nh6 nhat - Phan Huy Khdi

- 1 < r n < 1

TCr do di den x c l he sau

- l < m < l 2m^ + m - 1 < 0

Tif do suy ra xet cac kha nang sau:

1 Neu — > 3 (tuTc la neu m > 6) Liic do ta c6 bang bien thien sau:

it-Luc nay taco: min f ^ ( x ) = : f n i ( 3 ) = 9 - 2 m

TiJf do dan den vice xet he sau: m > 6

Cty TNHH MTV DWH Khang Vijt

Luc nay ta c6: min f„,(x) = !„

Nhqn xet: Hay so sdnh cdch i^idi nay vt'/i cdch fiidi trinh hay tnmi^ bdi 6 phdn

B, §3 chu<rn}> 7 cuon sdch nay Cdc ban thdy cdch nao thich h(fp \'<)i ban hfn?

Bai 7 Cho ba't phu-cJng trinh: sin3x + msin2x + 3sinx > 0

Tim m dc ba't phm^ng trinh diing vdi moi x €

' Hifdng ddii gidi

DiTa ba't phiTtfng trinh da cho ve dang tU'dng diTcfng sau:

3sinx - 4 s i n \ 2msinxcosx + 3sinx > 0

<=» 2sinx(-2sin^ x + mcosx + 3) > 0 2sinx(2cos^ x + mcosx + 1 ) > 0

Chuyen ai BDHSG Join gii tr| Ifln nhaft g\& tr| nhd nha't - Phan Huy Khai

Dat t = cosx Khi X e 0 ; ^

2 , t h i l e [0; 1] J Bai toan da cho trd thanh:

Tim m de bat phiTdng trinh: Im(l) = 2t^ + mt + 1 > 0 dting vdi moi t e [0; 1]

Dieu nay xay ra khi va chi khi: min f„,(t) > 0 - (3)

o<t<i £ i

Taco: 4(t) = 4 t + m =>4(l) = 0 <^ t = TOd6 x^t cac kha nang sau:

1 Ne'u —— > 1 (tu'c la khi m < - 4 ) Liic nay ta c6 bang bic'n thicn sau:

m > - 3 he vo nghipm TiTdo loai kha nang n;iy

2 Ncu < 0 (tuTc lii khi m > 0) Liic nay ta c6 bang bicn thicn sau:

/

r

/ /

Ket help lai suy ra: m > l a cac gia tri ciin tim cua m

jV/»fl« xet: Ciich giai trcn di/a vao phUdng phap tim gia trj k'ln nhat va nho nhat

ciia ham so phu thupe lham so m

Qic ban hay so sanh each giai tren vdi each giai bai nay bSng phi/dng phap

siir dung gia trj k'ln nhat vii nho nhat ciia mot ham so phan tht?c (khong c6

tham so) da Irinh bay trong bai 5 phan B, §3 chi/dng 7 cuo'n sach nay

Va tinh hicu qua cua tifng phu'dng phap, xin danh quyen blnh luan cho c^c ban

Bai 8 Cho ham so: r„,(x) = 4x" - 4mx + m' - 2m Xct trcn mien -2 < x < 0

m = l + V3 "

„i = l _ 7 ^ 4 ^ m = l + V3

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Tom lai: m = I + V3 va m = - I la hai gia trj can tim ciia lham so m '

Bai 9 Cho he phifitng Irinh: v ' ^ i +7y + 2 = m „ ,

Tim m dc he co nghicm

X + y = 3m

HuYfiif/; dan giai

Dal u = v^x + I ; \ sjyTl , ihi u > 0, v > 0 => x + 1 = va y + 2 = v'

Bai toan da cho tnt lhanh: Tim m dc he:

u + v = m (1)

u" + v^ = 3m (2) CO nghicm

u > 0; v > 0 (3)

T i r ( 3 ) v a ( l ) s u y r a m > ( )

Tir (1) c6: v = m - u Do \ ^ 0 < u < m Thay v = m - u vac (2) ta co:

2u- - 2mu + nr - 3m - 3 = 0 Tir do biii loan da cho lai co dang sau: Tim

Do la cac gia Iri can llm ciia m

Bai 10 Cho bal phiAtng Irinh: n r + 2m(sinx + cosx) + 1 > 0

Tim m dc bat phu'ilng irinh dung vc'^i moi x

Hiding dan gidi

Viet lai bal phuTttng trinh diftKi dang:

Vay m = 0 thoa man yeu cau de bai ' * '

t N e u m > 0 , thi minf„,(x) = - 2 V 2 m (dosinx + cosx <-sl2 Vxe R