TUYỂN TẬP CÁC CHUYÊN ĐỀ LTĐH MÔN TOÁN HÀM SỐ - TRẦN PHƯƠNG

8/10/2019 TUYN TP CC CHUYN LTH MN TON HM S - TRN PHNG

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Trn Phng

Tuyn tp cc Chuyn L uyn th i i hc mn Ton*/ *

T 1 o

Hm SGm

84 c h u y n v i k h o n g 2000 bi toi

Ti bn ln th 5

N H X U T B N H N I

WWW.FACEBOOK.COM/DAYKEM.QU

WWW.FACEBOOK.COM/BOIDUONGHOAHOCQU

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W.DAYKEMQUYNHON.UCOZ.COM

ng gp PDF bi GV. Nguyn Thanh T


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LI NI U

Trong thi i thng tn bng n, hc sinh hi nay phi truy nhpvo "K

tr thc nhn loi" v tn trong thi gian ngn nht t c hiu qicao nht. V th ngoi vic trau di cc kin thc trong nh trng, CUsng mi buc cc bn, nu khng mun tt hu, phi quan tm n nhiu mkhc nh: Ngoi ng, m hc, Th thao, Truyn hnh, Mng my tnh thng ton cu... Do thi gian vi mi ngi l hu hn cho nn s la chn k rnhng g cn hc v phng php hc tr thnh then ch v mang tquyt nh cho s nghip ca c cuc i. Nhm gip hc sinh PTTH n luyihi vo cc trng i hc thch nghi vi qu trnh t c, t o to; tc gi TriPhng -bin son b sch: TUYN TP CC CHUYN LUYN TII HC MN TON". y cng l vic lm m tc gi ha vi bn (.trong li ni th hai ca cun.:ch: "Ba thp k thi Ton vo cc trng hc Vit Nam". Cc 'Chuyn hm s" l mt tp trong b sch ny. Trorcun sch, cc kin thcv hms c phn thnh 84 chuyn vi kt Cmi chuyn u c phn l thuyt, bi tp mu, bi tp t gii vi tng lrbi tp ln n hn 2000 bi (xem Mc lc). H thng bi tp c a raVnguyn tc t c bn thng ng n nng cao gip bn c rn luyn c nng tnh ton v phng php suy lun ng thi to c hng th trong vi

gii ton. Ngoi cc chuyn truyn thng; X, TGT,... ln u tin mt ichuyn hay thi i hc c phn loi rt chi tit nh cc chuyn : Cc tTip tuyn... c phn thnh 9 chuyn nh tng ng vi 9 loi hm ikhc nhau.

c bit trong phn m u ca chuyn 'Tim cn" tc gi ch1quan nim sa lm v "Tim cn cong" ca rt nhiu sch xut bn. Vn Tip xc v Tip tuyn tc gi cng trnh by theo 2 phng php: iu ki nghim bi v iu kin nh ngha (K o hm) bi v nu b phng phiu kin nghim binth 'rt nhiu bi ton s c ligii phc tp. Mc d V phn cui chuyn 81: "S tip xc" tc gi vn trnh by 1 s bi ton tutuyn, tip xc theo quy nh ca BGD v o to

Trong, qu trnh bin son, tc gi c gng th hin ni dung cuisch t bn thn mnh s xu h nu c nh vit li v s bt hp l tro

kt cu ca cc chuyn . Tuy nhin do phi trnh by mt khi lng thng trt ln nn cun sch vn khng th trnh khi cc thiu xt Rt mong bn ing gp cc kin ln ti bn sau cun sch c tt hn.

Mi kin ng gp xin gi v tc gi

Trn PhngS' 3 p h Hng Tre, Qun Hon Kim - H Ni



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MC LC

Chng/: L THUYT TNG QUAN V HM s ................................. 11

Chuyn 1: Tp xc nh ca hm s..............

........................

..................... 14Chuyn 2: Tp g tr ca hm s ...................... ........................................... 20Chuyn 3:Xt tnh chn, l ca hm s...................................................... 20Chuyn 4; Xt tnh tun hon ca hm s........................ ....................... . 31Chuyn 5: Hm s hp....................... ................................. ............................. 36Chuyn 6: Hm s ngc........................................................ ............ .......... 42Chuyn 7: Phng trinh hm 47

Chng/7: GI HN HM s ....................................................... 51

Chuyn 8: Gii hn dng xc nh................................................................. 52Chuyn 9: Gii hn dng v nh: 0/0.................. .........................................53

Chuyn 1, Gii hn dng v nh: co/oo.............. .......................................... 59Chuyn 11: Gii hn dng v nh: 00- 00................. ..................................... 61Chuyn 12: Gii hn dng v inh: 00 .0.................................................... . 63Chuynd 13: Gii hn dng v nh hm lng gic..................................... 64Chuyn 14: Gii hn dng v nh: 1"........................................................... 71Chuyn 15: Gii hn dng v nh ca hm m v Lgarit....................... 73

Chuyn 16: s dng quy tc Lpitan tm gii hn dng v nh0/0; 00/00............................................................................................ 75

Chng H: HM s LN TC.......................................................... 73

Chuyn 17: Xt tnh lin tc ca hm s.............. :.................... ................... 80Chuyn 18: Chng minh phng trnh c nghim................................ ...... ...83

Chng V: O HM........................................................................87

Chuyn 19: Tnh o hm bng cng thc.................................................... 91Chuyn 20: Qui tc Lepnit - Becnul tnh c hm y = ux)^ ................... 95Chuyn 21: Chng minh ng thc bng o hm.......................i ............ 96Chuyn 22:s dng nh ngha; o hm phi, o hm tri tnh

o .'hm.'.-.................... ....................................................................98Chuyn 23: o hm cp cao........ ................................................................. 101

i



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Chuyn 24: ng thc, phng trnh, bt phng trnh VI cc phpton o hm................................................................................... 105

Chuyn 25\ s dng o hm tnh tng hu hn................................... 108Chuyn 26: s dng o hm tnh gii hn dng v nh 0/0............ 110

Chuyn 27:Vi phn.........................................

...........

........................................ 114Chuyn 28: Cc ng ng nh l Lagrage.................................................. 116

Chng V: KHO ST HM s V CC NG DNG...................... 121

1. Tnh n iu ca hm s............................ ......... ........ 121Chuyn 29: Tm iu kin tham gia hm s n iu........................... 121Chuyn 30: s dng tnh n iu gii phng trnh, gii bt

phng trnh, gii h phang trnh, gii h bt phngirnh... ........ ...7....... ................ ...................... .............. 125

Chuyn 31: s dng tnh n iu. Chng minh: Bt ng thc............. 1292. Cc tri hm s........................................................ - ........ 132

Chuyn 32: Gi tr ln nht, gi tr nh nht ca hm s........................... 136Chuyn 33: s dng GTLN, GTNN ca hm s trong PT, BPT, HPT,

HBPT ....... ...................................... ............................................... 142Chuyn 34: s ng GTLN, GTNN. Chng minh BT............................... 147Chuyn 35: Cc tr hm a thc bc 3...... ...................... ................................ 152Chuyn 36: cc tr hm a thc bc 4........................... ................................. 164Chuyn 37: Cc tr hm phn thc bc 2/bc 1........ ................................... 169Chuyn 38: Cc tr hm phn thc bc 2/bc 2........... ................................ 184Chuyn 39: Cc tr hm a thc bc cao............ ............................................ 189Chuyn 4. Cc tr hm s cha du g tr'tuyt i................................ 194Chuyn 41: Cc tr hm V t............................................................................ 196Chuyn 42: Cctr lm lng gic................... ...................... .......................... 201Chuyn 43: Cc tr hm s m, hm s Lgarit............................................ 204

3. Tep tuyn...................... ......................... ................ ...... 207Chuyn 44: Tip tuyn hm a thc bc 3...................... ............................. 209Chuyn 45: Tip tuyn hm a thc bc 4 .. ........................................ ...... 219Chuyn 46: Tip tuyn hm phn thc bc nht/bc nht....................... 226

Chuyn 47: Tip tuyn hm phn thc bc 2/bc 1....... ............................ 234Chuyn 48: Tip tuyn hm phn thc bc cao ....:............. ...................... 253Chuyn 49: Tip tuyn hm v t ...!............ ................................................... 257Chuyn 5.Tip tuyn hm Siu Vit................................. .............. ............. 261

4. Tnh li, im v im un ca th.............................. 262Chuyn 51: Xc nh rih li, lm v im un ca th........................ 262Chuyn 52: Tm iu kn (C): y = f(x) nhn (a, P) lm im un...... 269Chuyn 53: Chng minh th c 3 im un thng hng, vit phng

trnh ng thng.................................... .7.........T.......... ..............270



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5. Tim cn ca ng cong.............................................................275Chuyn 54: Tim cn hm phn thc hu t................................................. 278Chuyn 55: Tim cn hm v t........................................................................ 285Chuyn 56: Tim cn hm Siu Vit................................................................ 292

6. Kho st v v th hm s................................................... 294Chuyn 57: Kho st hm s bc 3.................................................................. 294Chuyn 58: Kho st hm trng phng................... ..................................... 302Chuyn 59: Kho st hm a thc bc 4................ ........................................ 307Chuyn 60: Kho st hm a thc bc cao (n > 5)...................................... 313Chuyn 61: Kho s hm phn thc.; bc 1/bc 1...................................... 317Chuyn 62: Kho st hm phn thc: bc 2/bc 1............ v......................... 322Chuyn 63: Php bin i th v kho st hm s cha u gi tr

tuyt i............................................................................................. 331

Chuyn 64: Kho st hm phn thc: bc 2/bc 2...................................... 341Chuyn 65\ Kho st hm phn thc hu t bc cao.................................. 34SChuyn 66: Kho st hm V t........................................................................ 353Chuyn 67: Kho st hm s lng gic........................................................ 361Chun 68: Kho st hm s m....................................................................... 364Chuyn 69: Kho st hm s Lgarit............................................................... 367

7. ng dng v tnh cht ca th....... ............................ 384Chuyn 70: Bin iun PT bng th.............................................................. 38Chuyn 71: Bin lun BPT bng th............................................................ 39;

Chuyn 72: Bin lun HPT bng th............................................................ 39Chuyn 73: Bin lun HBPT bng th........................................................ 39Chuyn 74: Cc dng ton tung giao ca hm s bc ba....................... 40;Chuyn 75: PT bc 3 c 3 nghim lp thnh cp s cng, cp s nhn 40*Chuyn 76: PT bc 4 c 4 nghim lp thnh cp s cng, cp s nhn 41 *Chuyn 77: Tuang giao th hm phn thc hu t................... ........... 41:Chuyn 78: Tm i xng v tnh i xng qua 1 im.............................. 41'Chuyn 79: Trc i xng v tnh i xng qua ng thng................... 42'Chuyn 80: Bin lun s th i qua 1 im............................................... 43

Chuyn 81: S tip xc......................... ............................................................... 44'Chuyn 82: im c ta nguyn trn th.............................................. 45:Chuyn 83: Qu tch i s................................................................................. 45Chuyn 84:Khong cch.....................................................................................45'



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T I L I U C C S C H T H A M K H O

1. Bo Ton h c v Tu i tr

2. B.P.MIVIC. B i tp Gii tch ton hc.Nh xu t bn i hv Trung hc chuyn nghip, 1975.

3. Y .Y .L IA SX , A .C.BO IAT R C, IA .G .G A I , G.P.G LV AC . Gitch ton hc v cc v d v cc bi ton.Nh xu t bn i hc vTrun g hc chuyn nghip, 1978.

4. TR N PH NG . Ba p k ii Ton vo cc trng i hVit Nam.N XB i hc Quc gia TP . H C h Minh, 2001.

C H D N M T S K H U V C H V I T T T

1. CM R; c/m ; gt: Chng m inh rng; Chng m i i; Gi iii t.

2. K ;Tx ;T G T : iu kin; Tp xc inh; Tp gi tr.

3. TC ; TC N ; TC X : Tim cn ng; Tim cn ngang; Tin cn xin.4. C ; C T ; V T ; V P: Cc i; Cc tiu; v tri; V"phi.

5. Y C B T; PC M : Yu cu bi ton; iu phi chng m inh.

6. PT ; H P T; BP T; H BP T; B T: Phng trnh; H phng rm h; Bphng trnh ; H bt phng trnh ; Bt ng ic.

7. G FT ; G H PT ; G BPT ; G H BP T; G BL: G ii phng trnh ; G ii hphng trnh; Gii b't phng trnh; Gii h bt phng trnhG ii bin lun .

8. PTT T ; PT TC : Phng trnh tip tuyn; Phng trnh tim cn.

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8/463

Chng! : L thu tng' quan v'hm s

C h v n g

1 m m N G u m v l m m s

NH NGH A HM s V CC KH I NIM L IN QUAN1. nh n g h a: Gi s 0 c R . Mt hm s xc nh trn X l mtqui tc f cho tng ng vi mi phn t X e X xc nh duy nht mtphn t y e R .

* K hiu : f: X RX h-y = f(x)

* Tn gi: X gi l i s'hay bin sy gi l gi tr ca hm s f t i x: y = f(x)

* Tp xc nh . Df = X* Tp gi tr: T f = f(X) = {f(x) xe X }

2 G ian tron g sL mt tp con ca R c 1 trong 9 dng sau y:

[a,b]; (a,b); fa,b); (a,b]; [a,+oo);(a , + ao); (-00, a]; (-00, a); (-00, +co)

3 . G i i hn hm slim f(x ) = b o Vs > 0, 35: 0 < I x-a < => f(x) - b t < sx-> alim f(x) = bVs>0, 3 A :|x |> A ^ I f(x) - b I < 8x - 00

4. T nh l i n tc c a hm s"y = f(x) lin tc t i Xo lim f(x) = f (x j

x->x0

5. T nh kh v i (c o hm) ca hm s'

y = f(x) kh v i t i Xt, o 3 f '(Xo) = )x->x0 x - x 0

6. C c i v cc t iu ca hm sGi s y = f(x) lin tc-trn. gian D.im Xo eD gi l im cc i ca y = (x) o 3s > 0 sao cho

ffe ) = M ax f ( x ) . K h i f(xo) = fC = fa**xe(x0-6, x0+s)

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9/463

Chng* I: L thuyt tng quan v hm r>

im x0 e D gi l im cc tiu ca y = f(x) o 3 > 0 sao chofOO = Min f ( x ) . K h i fix,,) = fCT = 4 in

xe(x0-e, x0+s)

7. Hm s chn

Cho hm s y = f(x) xc nh trn tp hp X . Km y = f(x) gi l hmchn nu n tha m n h.a iu kin sau:* Nu X e X th -X e X (Tp X nhn X = 0 lm tm i xng)* V X 6 X ta c f(-x) = f(x)

Cc hm s chn n gin ; y = cosx, y = x2,y = xa (n e Z )Nhn xt: th ca hm s' chn nhn trc tung lm trc xng.8.H m s l

Cho hm s y = f(x) xc nh trn tp hp X. Hm y = f(x) gi l hml nu n tha m n ha i iu kin sau:

* Nu X X th - X e X (Tp X nhn X = D lm tm i xng)

* V X X ta c f('X) = - f(x)* Cc hm s l n g in : y = sinx, y = tgx, y = cotgx, y = sinax,y = tgax, y = cotgax, y = arcsinx, y = arctgx, y = arccotgx, y = x211*1(n Z)* Nhn xt : th ca hm s" l nhn gc ta lm tm i xng.9. Hm s tun hon

Cho hm s' y = f(x) xc nh trn tp hp X. Hm y .= f(x) gi l hmtun hon nu 3 T > 0 sao cho V X 6 X ta c:

* X - T e X v x + T e X -* f(x + T ) = f(x)

S' T > 0 b nh t (nu c) th mn cc tnh cht trn gi l chu k c

s ca hm s tun hon y = f(x)* Cc hm s 'tun hon n giny = sin x, y = cosx c chu k c s2n;y =tgx, y=cotgx c chu k c s K.

2tt"y = s in (ax + b), y = COS (ax+b) c chu k c s l v i a * 0

a

y = tg (ax + b), y = cotg (ax+b) c chu k c s l vi a * 0a ]

10. Hm s ngc ca m t hm s kh n gh chCho hm so'f: X R -Tp xc nh : X

X -> y = f(x) V1 Tp gi t r : Y = f (x )

Nu V-r e Y , phng trn h n x : f(x)-= y c nghim duy nht X, thbng cch cho vi mi y e Y tng ng vi nghim duy nht , ta xcnh c hm s' g: Y R

y - g(y) = X (sao cho f(x) = y)Hm s X =g(y) xc nh nh vy gi l hm s ngc ca y= f(x)Mt hm s c hm s ngc c gi l hm skh nghch

*Nhn xt: th ca hm skh nghch y=f(x) v 'th ca hm s ngcy = g(x) lun xng vi nhau qua ng phn gic gc phnt th nh t (dng y = x).

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10/463

Chng-1: L thuy t tngL quan v hm s

c hm sngc thng dng

o y = sin x v i X c hm s ngc l y = arc sinx-

y cosx vi x e [0, 7t] c hm s" ngc l y = arc cosx

y = tgx vi X - . ,j c hm s ngc l y = arc tgx

9y = cotgx v i X e (0 ,7) c hm s' ngc l y - arc cotgx9y = a* Vi 0 < a #1 c hm s ngc l y = logaxy = e* c hm s" ngc l y = lnx

1. H m s hpCho hm s' y = f(x) c tp xc nh X, tp gi tr T v hm s' y = g(u)

tp xc nh Y cha tp T . Kh i vi mi gi tr X e X ta c mt gi

m s' f v g theo th t ny v c k hiu h = g o f .2. H m s s cp

u) Mt hm s c gi l hm s s cp 1 bin s nu n dc xcnh bi mt cng thc y = f(x) trong f(x) c to nn bng cch thcin theo th t trn bin s X v cc hng s' mt s hu hn cc php on sau y:

Php ton i s^Cng, Tr, Nhn, Chia, Ly tha s' m nguynhai cn.

e Php ton siu vit:xa (a v t); ax; logax, sinx, cosx, tgx, cotgx,

rc sin x, arc cosx, ar c tgx, arc cotgx.b) Mt s'v d v hm s s cp

y = anxn + a^ x"-1* ... + a2x2 + X +a0

y______ 2x2 -3 x + 7_____. y _ 2 sinx- y = earctgx

J x 4 - X3 + 3x2 -1 +

1 ( I 9. , 2sinx-3cosx L o'terctgxy = In X + V X +1 + --------------;-------- +12 + x z f

V ) 3c o sx-4 sin x + 8

c) Cc v d v hm s khng s cp[1 n u X h u t

* Hm Drchlet: D(x) =[o nu X v t

D(x) khng l hm s cp vi n c xc nh bi 2 cng thc.* H m f : N - > N

n >f(n) = n!Hm ?(n) khng l hm s cp v s php tnh nhn tng ln khi r

ng tc l s php tnh nhn khng l hu hn.inh l:Mi hm s" s cp u lin tc trn min xc nh ca n.



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11/463

Chng I: L thuyt tng qnan v hm s

C h u y n 1

T P X C N H C A HM s

. PHNG PH P

Tm tp xc nh ca hm s y = f(x) l tm tp hp tt c cc gi tca bin sX sao cho f(x) c ngha.2 . K NNG CH

y = -7- ) c ngha o v(x) # 0

y = 2 f(x) c ngha o f(x) > 0

fo < a 1y = loga f(x ) c ngha

f(x)>0

o y = tg u(x) c ngha o u(x) * 4- k~ (keZ)2y = cotg u(x) c ngha o u(x) *k (keZ)e y = arc sin u(x) c ngha o -1 < u(x) < 1y = arc COS u(x) c ngha -1 u(x) < 1

3. CC B I T P M MINH HATm tp xc nh ca cc hm s sau (B i 1 n bi 5)

B i 1: y = VX2 -2-\/x2 -1 + Vx -3 + 2a/x - 4 :

Hm s c ngha kh i v ch khi' 2 - l> 0

X2 - 2V/x2 - = -\/x2 - >0

x - 4 > 0

Lx - 3 + 2-Vx - 4 = [Vx - 4 - l ] 2 > 0

B i 2: ^logx(x 3+ l) .log x -2 [ 8 I I - B TTS]

Gii :Hm s' c ngha kh i v ch kh i

0 < X ? 1

0 < x +1*10 < X3 +1

2 < logx (x 3+1) -log X = logx+1(x 3 +1)> ? x+1

o 0 < x

[x 3 +1 > (x + l ) 2 X >

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12/463

Chnsr : L thu tng quan v hm s

Bi 3: y = . f3 - 2 x - x 2>ilogi2 x + 2

Gi i :Hm s' c ngha kh i v ch kh i

3 - 2 x - x

x + 2f

log2

>03 - 2x - X

x + 2>0

3 - 2x - X2x + 2 >0

X +2x- 3x + 2 >0

3 - 2 X - X2 s 1 x 2 + 3x - 1 0

_ -3 - -2 _ 1 +------y/A'//V/i = ----- y ///////zfa-

-3 -3+NS

Bi 4: [i hc Dc H Ni 1999} y = lg

x + 2

- 3 - V 32

-3 + -n/3

< X < -3

< X < 1

21~x - 2x +12X-1

Gii :

t f(x) = 21_x - 2x + 1 = 2 . j- j + (-2x) + 1^>f(x) nghch bin

m f (I) = 0 n n B ? T f ( x )> 0 o f (x ) > f (l) 0 g(x) > g(0) o x > 0 , suy ra g(x) cng du vi X

c ngha Oo ->0 o 0 0Hm s xc nh ' o - '( )(x + l * 0 x + 1 *0

3x + l)> 0Nu m = 0 th X +1 >0'X > -1 .

[x +1 *0.

Nu m * 0 th (mx + 3) (x+1) c 2 nghim l (-1) v



B

I

D

N

GT

O

N

-

L

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0

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N

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N

H

N




13/463

Chng- L thuy t tng- quan v hm s

Xt hiu u(m) = (-1) - I = "_ m -> Bng xt du u(m)m

m - 00 ) ' 3 + co

. u(m) - + 03 _

Nu m > 0 th u(m ) < 0 -1< . K h i m

I x < H () o j m o

X + 1 * 0

-1 < x m

Nu 0 < m < 3 th - < -1, kh i (*) 0 vte R (tha mn) Nu m = -1 th f(t) = (t-1)2 = 0 ti t = 1 > 0 (Lo) Nu m = 0 th f(t) = t2 = 0 t i t = 0 (tha mn) Nu (m < >1) u (m > 0) th A'f > 0 nn f(t) - 0 c 2 nghim phn bit

tj < lc f(t) > 0 t e (-C0, t^U (t-2, + co) = ITa c (0, + co) c l o tx < ts 0.

16



B

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O

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-

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N




14/463

Chng Z \Jj u tng'gg&a rMaaLSt

m < -1m > 0

m < -1_m >

|m < 0 ^ V H gta 111t jt 2 = -m > 0t x +12 = -2m 0

Kt in : -1 < m 0Cch 2:Phng php hm s

t2 + 2mt - m > 0 v t > 0 m (2t - 1) > -t2 v t > 0 (*)

Nu t =>0 th B P T (*) lun ng. Nu m = 0 th (*) lun ng.

Xt hm s' g(t) = => g (t) = => B B T2 t - l (2t - 1)

t -OD 0 1/2 1 +00

g(t) - 0 + + -

g(t)

4-00 +00

CT . / \-00 -00

Vi m * 0 ta c

g(t) < m v t >

K t l un : -1


15/463

hoxi&I^L .u g- quan, v hm s

t. i - ;;in 2x [-1, 1] ta c

hm sxc nu V x o I ' y - - mt > 0 V t e ['1, ]

cr. g(t) ^ tz V 2m i - 2 < 0 V t s [-1, 1]fg ( - l )= -2 n : - l< 0 -1 1

' < /V _ c? -- < m < Lg(l) = 2 m -I< 0 2 2

4. B I r,..;- NH CHO BN CT GII'.rin: :;j.xc ni ca cc hm s sau

ir.x2 -3x-j-?.) -Jx+ 4}(x + 1)y ' '-77 = ^ ; y - - n r r =

Vi>T-iX X a/' ^B a i 2 .;/ - n [j - ig (x2 - x 4- 16)]

B i s. H Cnh St 1939] y = Vx2 + x-2 .1og 3(s - x 2)

B i 4. H Y H Ni 1997] y = 2^x_3H8_xt + l~ 0g3~ ~

V Vx 2 -2 x - 8Bs . y = VX I+V ~ .Z+ ;

y - ^25^"x2+1 -34.152*'*2 +92x_x2+

B l 6. 31, -- g23x + 32;; - 7X -i- 2 - 14x) ; y = n --- -~--X^ j

B: 7. y - Jx^ - sxj + 2in(2x-1)

B-. fc. y f e_m_ {x2+ 2 )- l +

3 U ^ 'l i l - BTTS3 Cho y ^ - ^ S I - v i < a * Ioga n x -531+ 2]

) i? xc nh khi m = 2

b) Tm ;r.i hm s xc nh vi mi X > 1Bk ;.0. Cio y - n [m . 4X+ (m-l)2s+2 + (m-1)]

Tm m hm s xc nh vi mi X > 1Bi 1. Tm tp xc nh, ca cc lii s sau:

a) y = arc sin - - -] b) y - arc COS (2sinx){1 + x }

c)y = gcos (Zgx] ) y = arc sin g10,

e) y = g[l - 2g(x2- 5x)] f) y = lgsn(x-3)+ V l6 - X 2

18



B

I

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N

GT

O

N

-

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16/463

Chng L JL^Lthuyet tng- quan vk$7?. s j '

B i 12. Tm tp xc nh ca cc m s:

V a) y = VsinV x ; b) y = siTtx

B i 13. Tm tp xc nh ca y = V sin 2 x- sin x

B i 14. Tm tp xc nh ca y = log3 -

B i 15. Tm m hm s

1 + log + loga x I

y = V- sin2X - 2m cosx + 3m -1 xc nh. V X R.

B i 16. Tm m hm s

y = log2 [ m-4* + (m - l ) 2" 2 + - ! ) ] xc nh Vx e R

B i 17. Tm tp xc nh ca cc hm s

'2x + (2/5_la) y = Inlogj

5(1-X)

b) y = log23

logi

B i 18. Tm m hm s sau xc nh. Vx [-1,1]

_____________ 1____________Vx + 3m -2 W m + 2 -x

B i 19. Tm tp xc inh ca hm s"

y - V8.3^ ^ * + 91+ - 9 ^

B i 20. TDpL m hm s

y = g[


17/463

Chtdgr L' L tmvt tnnfr quan v hm s

C h u y n 2

TP GI TR| CA HM S

1. PHNG PH PCh hrp sy f(x) c tp xc inh D tin tp gi t r ca hm s l tp

hp T = {y 3 X i D : = f(x)

2 . CC B I T P MU MINH HA

B i X: Tm tp gi tr i ca hm s y = x + -2x -1

Gi

Tp xc nh:D = R \ |

Tp gi t ri : Vi X e D xt P T : y = y(2x-l) = 3x +52 x - i

o 2xy - 3x = y + 5 o (2y - 3) X = y + 5 (*)

Nu y * - th (*) o X = * -2 2y~ 3 2

V y tp gi t r ca hm s l T = R \

2 2_X +1B i 2: Tim tp gi tr ca hm s' y = --x z + s + l

Oi.*Tp xc nh : D = R

Tp gi t r : Ta c y = x ~x +1 o te 2+ X + 1) = X2- X + 1X* + X + 1

o (y -l)*2 + (y+ )x + (y-1) = 0 (*)Nu y = 1 th X 0. X t y * 1 v thuc tp gi tr cua hm s' P T

(*) n X c nghim .

y * 1 v A = (3y-l)(3-y) 5 :0 y *1 < 3.3

_ _ T i K t lu n : Tp gi tr ca hm s l T , 3

B i 3 : 'Xin! tp gi tr ca Hm so y = X4 -6x2+2 v i xe [-2,1]Gi

t t = X2. Vi X [-2 ,1] th t e {Min21X I , M ax21X ] = [0,4]K h i d y = g(t) = t2 -6t +2 vi t [0,4]Ta c hoh. nh ca Parabol y = g(t) l t = 3 nn

20.



B

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GT

O

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-

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18/463


19/463

Nu 3 -1 tx + i

>: - 10 V X 51: loi

N>. ?. * -1 th y = 4 r " ~ y(x 2 + a)= x + 1x ' -h&

>yx2 - X + ay - 1 = 0Xt y - 0 ta c ( * ) x + l = 0 o z = -lXt V 5* 0. Ta c y e Tp gi t r ca hm s(*) c nghim

A = I - 4y(ay - 1) > 0 g(y) = 4ay-2 - 4y - 1 < 0 tp gi tr ca hm s" cha [0,1] th g(y) 0 ng Vye [0,1]N: u a 0 th g(y) = -4y ' X < 0

(*)

^ S y ^ O ,! ]

4a-5 0 < a <

- l O 4

* 1Tu. R < 0 th s(y) < 0 Vy e [0,1] lun ng4ag( l ) 0 th g(y) < 0 C> ) V |4ag{o)


20/463

bZ


21/463

. B i 11. [ 67-B TTS]H- cosx + 2 sin x + 3 _ , .Tm M ax, M in ca y - ------------- v i X e (-71,7t)

2C0S X-S1X + 4Giu

Do xe (-71, Tnn 6 1 I => t = tg e (- co,+co)2 \ 2 2 ) 2

_ - _ 2t _ 1 - t 2 ts +2t + 2:T a c s in x = --r- ,cosx = - =3-y = - 1 + t 2 1 + t 2 t - t + 3

- y(t2-t+3) - t2+2 t + 2 (y-1) t2 - (y+ 2)t + (3y-2) = 0 (*)

Nu y = 1 th (*) 1 - 3t = 0 t =

Nu y * 1 th (* ) c nghim =>A = (y+2)2 * 4(y-l) (3y-2) > 0y * 1 v 11 y2 - 24y + 4 < 0 o - < y 51 < 2

Chng-i> h thni tng qzmn ve hm s

Tng hp cc kh nng xy ra => y e

Max y = 2 o t= tg = 4 =>X = 2arctg2 3

2 _ . X - 8 . 8M in y = tg = -> X = -2arctg

11 2 3 3B 12. [HSP Qui nhn 1999].

Tm M ax, M in ca y = " Sln ^ vi X e [0, 2 + cosx

Gii

Ta c y sino y (2 + cosx) = sinx 2y = s iix - ycosx2 + cosx

=> (2y)2 = (sinx - ycosx) < [l2+ (-y)2] [s in^ + cos2x] = 1 + 3^ 1 1 ^

=> 3 y2 1 j=< y < - j= .Do X e [0, x]nn => 0 < y < v3 V3 V3

1 1 yM axy = - p r O = - > 0 o 1 - J L ---------

V 3 sinx cosx cotgx = ~ 3

cosx < 0 _ 2t

1 => x = -

Min y = 0 o s in x = 0=>x = Qhoc X = n

3 . B I TP DNH CHO BN C T G I I2x + 3B i 1. Tm M ax, M in ca =

[H SPTPH CM 2000]

Tm M ax, M in ca y =

3x2 + 4x + 5S i 2. [HSPTPH CM 2000]

3x 2 +10X + 20X2 +2x + 3

24



B

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N

GT

O

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-

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0

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22/463

Chnsr : L thuvt tnr quan v hm s

i 3. Tm a, b y =* k c tp gi tr i T = [-2,5]X +1

i 4.Tm a y = --- x+-- c tp gi tri R

2x~a/? X -1 i 5.T m a y = T---- c tp gi t r cha on [-1,0]

r- a i 6. Tm tp gi tr ca

- 2 s in 2 X + 3s in xc sx + 4 COS2 X - 3 sin2 X - 4 sinxcos X +COS2 X + 2

i 7. [ 3 4 III - B TTS ]

T m k y c M in y < - 12 + co sx

i * rr* n*-*. __ L 2 sin s + co sx-3 i 8. Tm M ax, M in ca y = 3c o sx-4 sinx + 7

i 9. [ 13SII - B TT S]

cos3x + asin 3x + l2 + cos3x

^ l + Vx + 3a2 w


23/463

. PH NG PH P

f(x )l h m c h n /D P / j x^ quax = 0f(- x) = f(x) V xe D

- '- -i m ~ I D i xng qua X = 0f(x) l hm le/D[ f ( -x )=- f ( x ) VxeD

2. CC B I T P MU MINH K AB i 1. Xt tnh chn , l ca cc hm s" sau:

a) f(x ) = X10 - X8 + Xs - X4 + X2 - Ix + X - ~ + ~X X

b ) g ( x ) = X 9 X 7+ x X 3+ x + ~ ~X X3 x

Gii :a) Tp xc nh :D = R\{0 } i xng qua X = 0Vx e ta c:

c-x) = (- x)10 - (- x)8 + (- x)6 - (- x )4 + (- x f - - x | + l - + - ( - x f ( - x )

f( - x ) = x 10 -X 8 + x4 + x2 -|x | + l ---V + ~xr X4=> (-x) = f(x) V x e ==>f(x) l hm chnb) Tp xc nh :D = R \{0 } xng qua X = 0

Vx e D ta c f(-x) = (-x)9-(-x)-7+(-x)5 - (-x)3+ (-x) - + X ( x )3 (

f ( - x) = -X 9 + X7 - X5 + X3 - X + - ~X X X5

Suy ra f(-x) = -f(x) V xe D = > f(x) l hm l V / 1. X 12 *.1A * * _ _

-3 x 2 -7

B i 2 : Xt tnh chn, l ca cc hm s sau

a) f{x) = w g(s) = ,-V4-X \X 2

a) Tp xc n h : 4-x2 > 0 o x s (-2, 2) = D i xng qua X = 0

Vx D = (-2,2) ta c f(-x) = (~x)5 + ( - x)3 - ( - x )

26



B

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O

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24/463

Gh& ng l L thuyt tng- quan. vMm S

(-x) = - ^--+-X- = - f (x ) => f(x ) l hm lV 4 - x 2

b) Tp xc nh : Xs - 2 > => D = cOj-Viiju 2,-i-wi i xng qua X =

w _ r\ * - ( - x )4 - 3 ( - x )2 -7 _ X4 -3 x 2 -7 __ ^ {\ 1"Vx D ta c g (-x ) = -i r : - ' =----= .7 = g(x) => g(x) lV ( - z J * -2

hm chnB i 3: Xt tnh chn, l ca cc hm s

a) f(x) = a/3 +x + -y/3-x b) f(x) = 2*

Gi i :a) Tp xc nh'.

Xt f(x)

l hm chn0; f(x) = 2* c tp xc nh D = R xng qua X = 0

Ta c f( l) = 2 v f(-1) = - nn f(- l) *f( l) v f(-l) * -f(l)A

Vy f(x) khng l hm chn v cng khng l hm lB i 4. Xt tnh chn l ca cc hm s

ex - exa) Hm sinliypecbolic: shx =

2X , -x6 + 6

b) Hm cosinhypecbolic: chx = 2Gii :

Tp xc nh:D = R = (-00, +00) i xng qua X = 0

a) Vx eR ta c sh (-x}= --------- = ------- = -sh x => shx l hm lv 2 2t \ 6~x + ~b) Vx e R ta c ch.(- x ) = ----- -------chx => chx l hm chn

B i 5 . Xt tnh chn l ca hm s" vi 0 < a *1

a) f(x ) = l ~ b) g ( x ) = x .^ a x - l a x - lGiu

a) Tp xc nh : = R \{0 } i xng qua X = 0

1V xe D tac f ( - x ) = a~- ---- = - ^ r

v J a"x - l l - a x

27



B

I

D

N

GT

O

N

-

L

-

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0

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N

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25/463

Chng I : L . thut tng quan 'v hm 8

f {- x) - - s . = -f(x)=> f(x) l hm l

b) Tp xc nh :D = R \{0 } xng qua X - 0

i _ 1Vx e D ta c f(-x)= (-x )-a~ ^ --- = ---- = (-x )a ~ l A r - 1

l + a}l - a J

ax +1= f (x) => f (x) l hm cTin

a x 1B i 6 : Xt tnh, chn , l ca cc hm s

> .f - i * ( x +V ^ ) b) g(x )= g^ |2 z | J

Gi

a) Tpx c i : Ta c X + V l+ X 2 > x + Vx " = x + |xj>0=> D = R

Vx e R ta c f(-x) = Zn3 - X +J l +(- x )2 = n 3 V l + X2 -XVx e R ta c f(-x) = Zn3

V * 2) - - 2

~x + -\jl +( -x )2

= Zn3 ^ = d L -- = Zn3^ x W l+ x 2 j = -Zn3^x+-\ft + x 2 j

= ( - l)3.ln x +V l+X 2) = - In x +Vl+X 2'j = -f(x) => f(x) l hm l

; T/? xc c?rc: Xt . > 0 X (-2000 ,2000) = D

Vx s D ta c g(- x ) = lg5

/2 0 0 0 -X N_1

2000+ x~2 0 0 0 -(-x )2000+ (- x )

= lg52000+ x2000- X

lg [2000+ x

= (-i)5-id

_ J 2 0 0 0 - X\2 000 + x

2000-X " = _ . / 2000- X 2000+ x j ^ 20 00 + x

V y g(x) l hm l

B i 7. X t tnh chn l ca cc hm s

= -g (x)

a) f(x ) = X3+ X2- 1 b) g(xj = V x2 -3 x -1 0

V W N _ . 3 ' ,v , V s in ^ f l- x 2 + COSc) h(x) = s ii rx + cosx d) p(x) = H xl

Gi i :a) f(x) = X3+ X2 - 1. Tp xc nh: D = R

f(2) = 11, f(-2) = -5 => f(-2) * f(2) nn f(x) khng l hm chn vcng khng l hm l.

28



B

I

D

N

GT

O

N

-

L

-

H

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0

B

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26/463

Chng I: L thn tng-quan TO h.m s

b) g(x) =V x2- 3 x - 1 0 .X tx 2 - 3x -10 > 0 oX < -2

X > 5

=> D = (-00, -2] u [[5, + o) khng i xng qua X = 0V y g(x) khng l hm chn v cng khng l hm l.c)h(x) = sin3x + cosx c tp xc nh D = R

J W | f - J z f L = h .V 4 J 2 J + 2 4 ; U J [ 2 J + 2 4

Do h^ 5* h j nn h(x) khng l hm chn v khng l hm l

d) Sa i lm thng gp: p(-x) = p(x) => p(x) l hm chn

guyn nhn 1 -x 2 >0

[x = 1Tp xc nh :Hm s sc nh

1 -x 2 >0

X2 - 1 > 0

1-x^O :X * 1

Vy p(x) khng phi-l hm s' i 8. X t tnh chn, l ca cc hm s sau:

a ) f ( x )= e ' X k h X 0 b)|x + e kh iX O |x + | | x

i i :a) Tp xc nh: D - R i'xng qua X = 0

N u X < 0 th (- x )> 0 =>f(-x) = -x + e = e- x = f(x)N u X > 0 t h (-x) < 0 => f(-x) = - e - (-x) = e + X = f (x ) .Vy f(x) l hm cn

b) Tp xc nh:X eD Ix+2 # x*2 > x * x O ' X * 0[X + 2& 2 -X

Vy D = R \{0 } "i xng qua X = 0^ -X + 2 + - X - 2I |x + 2 + |- x - 2 , \

Vx ta c f( - x )= ----- - H -----7= - -------- 4 = -f (x)' ' IV 4- 9l IY 9l X + 2 - - X - 2 xJ

Vy f(x) l im l i 9. G i s hm s" f(x) c tp xc nh D i xng qua X = 0

CM R: f(x ) c th biu din l tng ca mt hm s" chn vi hm s li

t ft (x)= fM + fH , f2(x )= W z M

Ta c f2(x) v f2(x) cng c tp xc nh D . M t khc

Vx e D th f1(-x)= ^ ,x)+f(x) _

29



B

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GT

O

N

-

L

-

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0

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27/463

Chng-1: L thuy t tn g quan v hm s

f2( - x ) = t t ! W = _f2(x)

Vy f(x) = fi(x ) + f2(x) trong fx(x) l hm chn, f2(x) l hm l

B i 10. Cho f(x) = ax3 + bx2 + cx + ;

Gii :

g(x) - ax4+ bx3+ cx2 + dx + e

Tm a, b, c, , e f(x) l hm l; g(x) l hm chn. i:

f (x ) = ax3 + bx2 + cx + d |g (x )= ax4 + bx3 + cx2 + dx + e

f (- x) = -a x 3 + bx2 - cx + d [g(- x) = a x 4 - bx3 + cx2 -d x + e

f(x) l hm l f(-x) = -f(x) Vx - bx2 + d = -bx2 - d Vxo 2(bx2+d) = 0 V X O bx2 + d = 0 V x o b = d = 0

g(x) l hm chn f(-x) = f(x) V x o 'bx3+ dx = -bx3 - dxo 2[bx3 + x] = 0 V X o bx3+ dx = 0 V x o b = d = 0

Kt lun tng qu i :f(x ) l hm l Cc h sbc chn bng 0g(x) l hm chn Cc h s bc l bng 0

3. BI T P NH CHO BN C T G

B i 1. X t tn h chn, l ca h m s y = f ^ f ^ + tg x + s in *V X4+ xa+ 1-C0SX

B i 2 . Xt tnh chn, l ca hm s' sau

a) y = x b )y = c) y = --- d) y = s in (3x-2 - X * V x2 - ! s in s + tg x

B i 3: Chng minh, cc mnh sau y:

a) Tng ca hai hm chn l hm chn, tng ca ha i hm l l hm lb) Tch ca ha i hm s chn l hm s chnc) Tch ca ha i hm s l l hm s chnd) Tch ca mt hm s' chn vi mt hm s l l hm s l.

B i 4. Cho phng trn h;a2002x2002 + a2Ix2000 +...+ a4x4 + a x2 +1 = 0

Phng tr nh n y c th c ng 1001 nghim c hay khngB i 5. Xt tnh chn l ca f(x) = h[g(x)] nu h(x) l v g(x) chn.B i 6: G i s f(x) l hm v dn iu tng.

CM R: Nu a, b, c tha mn a + b + c = 0 th

f(a) f(b) + f(b) f(c) + f(c) f(a) < 0B i 7 . X t tnh chn l ca cc hm s

a) y = ----- b) y = V l + x + x 2 - V l - X + X24X

B i 8. Cho f(x) = 2002* + 20022'x. Tm a y f(x+a) l hm chn.

30



B

I

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N

GT

O

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-

L

-

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0

B

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N

H

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N




28/463

Chng : L thuyt tng- quan vhm SQ

Chuyn 4

X T T NH T U N HQM C A HM s

1. NH NGHAf(x) tun hon / D o 3 T > 0 sao cho V X e D ta c.

X- T D, X + T'e D v f(x + T) = f(x)S" T > 0 nh nht (nu c) gi l chu k c s ca f(x)

*Cc hm y-sin(a x+ b); y = cos(ax-Hb) vi a * 0 c chu k cd s T = --la 1

* Cc hm y = tg(ax+b); y =cotg(ax+b) vi a 5*0 c chu k c s T =la l

2.CC B I T P MU MINH HAB i 1. X t tnh tun hon v tm chu k c s ca cc hm s sau:

a) y - sin2 X b) y = sin6x + cos6x c) y = Xcosx d) y = cos3x(I+cosx)

Gii \ ;_ 2 l- cos2x , , wa) y = sirr X= -----c chu k c so T = z2

b )y~ s in6x + cos6x = (sirx^ T co s^)3 - Ss in ^ co s^ (s in ^ + cos2x) =3 5 fl t 3 l- co s4 x . 5 + 3cos4x , T_v ^ i m 2 X=1-sin2 2x = l - . - = ----- c ch k c sT = = 4 4 2 8 4 2

c) Xt phng trnh (x + T) cos(x + T ) = X cosx Vx R vi T > 0

Cho X = - =>[ +tco s 4 + t = 0 o cosf? + T ! = 02 ^2 ) ^2 J 1 2

-sin T = 0 o T = kx . Do T > 0 nn chn T = %K h i (x + 7) cos(x+7t) = xcosx V x e R o -(x +7) cosx = Xccsx Vx Ro -7UCOSX= 2x cosx V xs R -7 = 2x Vxe R => V lVy hm y = X cosx l hm khng tun hon

d) y = cos3x(l+cosx) = cos3x+cos3x.cosx = cos3x+ [cos4x + cos2x]

Do cc hm y = cos3x, y = cosx, y = cos2x l cc hm tun hon vi

chu k c s , v BSCNN ,-,51 = ' L J

nn hm y = cos3x(l+cosx) tun hon vi chu k c s T = 2k

B i 2. a) Phn nguyn ca s' thc X l s nguyn ln nht khng vtqu X v c k hiu l : [. CM R: y =-[x] khng l hm tun hon,

b) Phn l ca s thc X l {x} = X - [x]CM R: Hm y = {x} l hm tun hbn

Gii:a) G i s y = [x] tun hon vi chu k T > 0,k h i iun tm c s" n

nguyn dng nT> l= 5x + n T > x + l

31



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29/463

Chng I: L thuvt tng quan vhm s

=>[x + nT] > x + 1] > [x] =>[x + nT] [x] =>V l =>Gi s sai.Vy y = [x] khng l hm tun hon.b)t [xj = kk 0. kh i

sin 1 1 = s in oX + T X

- =+2kiz Vk zX + T X

- = 7T-+2k7 Vk z Lx + T X

+ = (2k + l)jr Vk zLx + T

1

Chn k 0 v X . ln ta c lim

lim--- ( +) = 0 nn cc phng trnh . (1) v (2) u khng cx-m-co x + T x j

nghim T > 0.

Vy hm s y = sin khng tun honX

c) Gi s hm sy = sinx2) tun hon vi chu k T > 0

IV-*'


30/463

~ ~ 11 n u X hu tGii :Xt hm ichlet B(x) -

[0 nuX v t

Gi q > 0 l mt s hu t . K h i , hu tnu xh u t(x + q) l s < nn

[v t n u X v t

x ln uxh ut _ , N vD(x+q) = - - D (x + q) =^^x) Vx

[0 nu X v t

Vy D (x) l hm tun hcnV trong tp hp Q+khng c s nh nht nn D(x) khng c chu k c sCh : f(x) = c hng s' cng l hm tun hon nhng khng c chu k

sd,i 5. Gi s f(x), g(x) l hai hm s tun hon trn I v c cc chu k

n lt l a, b vi l s' hu t. CM R: Cc hm s F(x) - f(x) + g(x) vb

(x) = f( x ). g(x) cng l hm tun hon trn I .i i

T gi thit suy ra 3m. n G N+vi (ffi, n) = 1 sao cho = => na =

b = T . K h i x T e V x e IF(x+T) = f(x+ T) + g(x+T) = f(x+na) + g(x+mb) = f(x) + g(x) - F(x)

G(x+T) = f(x+T).g(x+T) = f(x+na).g(x+mb) = f(x) . g(x) = F(x)Vy F (x ), G (x) l cc hm s" tun hon trn .

i 8. Cho hm s f(x) xc nh trn D v mt s at0

G i s f(x + a) = (x )^ Vx e D . CM R: f(x) l hm tun hon.f ( x )+ l

i i :

fix + 2al = f(x + al+ al = (x + a)~J- _ f( x )+ l _ ~ _[ 2aj |*x a) j f (x + aj +1 f(x) - f(x)

f ( x )+ l

^ f{x+ 4 a )= 7 R s r ^ l = f(x)

Vy f(x ) tun hon vi chu k T = 4 a h n x t:Tp hp cc hm f(x) tha mn yu cu b i ton l khc

tp rng

VD: f(x) = cotgx v a = 4



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31/463

Chng : L thu tng' quan v hm s'

B i 7. Cho g(x) xc nh trn D v mt s a > 0

Gi s f(x+a) = + / f( x )- f2(x) Vx e R+. CMR: f(x) l hm tun hon

Gii : T gi thit >f(x+a) > - Vx s E +-> f(x) > Vx e R +2 2

Ta cf(x+2a) = + + a } - f 2(x + a) = +-J f(x + a |l - f(x + a)J2 2

= r | r 4+Jf ( x ) - v ; 2 = + f ( x ) - | = f(x )Vy f(x) tun hon vi chu k T = 2a

B i 8. G i s f(x) tun hon tha mn iu kin af(x) = f(ax) V x e R v vi a l 1 hng s khc 1 v 0. CM R; f(x) khng c chu k c sd.Gi i ;

Gis T > 0 l chu k c s ca f(x) v a I >1Ta c f(ax + T ) = f(ax) = af(x); m t khc

tYI I T )

Nu a < i th f(x + aT) = faT )= f a + T y = a f + T j

= at~ j = ( a .* ) = f(x) => |a . T l chu k ca f(xM |a | , T < 'T V I a < 1 => T khng l chu k c sVy f(x) khng c chu k c s '

g(x) vi chu k T sao cho f(x) = ax . g(x)

Chn a = k1 bay. k = aT, kh i ta c f(x+T)= aT.(x) Vxe Df.

t g(x) = => f(x ) = ax.g (x ). X t Vx Df ta c

..............Vy g(x) l hm tun hon vi chu k T v f(x) = ax.g(x)

B i 10. Cho hm s y = f(x) xc nh trn R v tha mnf(x + 3) f(x )+ 2 V x e R .

34



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32/463

Ghng.L-L thn v t tone? quan v hm s

Ta c: g(x + 6 ) f(x + 6 )- (x + 6) = f[(x + 3 )+ 3 ]-(x + s)< f(x + 3)+ 3 - (x + 6)< f(x)+ 3 - (x + 3) = f(x)X = g(x)

M t khc g(x -1-6)= f(x +6) - (x+ 6) = f[(x +4)+2] - (x + 6) > f(x + 4 )+ 2 -(x + 6) = f[(x + 2 )+ 2 j-(x + 4 )> f(x + 2 )+ 2 -(x+ 4 )

> f(x) + 2 - (x + 2) = f(x) - X = gx)fg(x ^ g(x)

) ( ) ( nn g(x+6) = g(x) => g(x) tun hon

g(x + 6) >

a. CC BI TP NH CHO BN C T GIIB i 1. Xt tnh tun hon v tm chu k c s ca cc hm s" sau

a) f(x ) = 2tg+ 3tg + 4tg b) f(x) = sin x + sin (x . V2 )2 3 4

Bi 2. Tn ti hay khng tn ti cc hm s f, g: t R, trong g l hm s tun hon v tha mnf([x]) + g([x]) - X3 V X s ,

B i 3. Cho f(x)+32V xe R w w %

B i 4. H m f(x) xc nh, trn D gi l phn tun hon vi chu k T > 0

nu f(x) tha mn ha i iu kin sau* Nu Xs D th X T D f(x+T) = -f(x) Vx s D

CMR: Nu f(x) l xm phn tun hon th (x) cng l hm tunhon. iu ngc l i c ng khng.B i 5. CM R: f(x) l hm phn tun hon chu k T trn D k h i v ch kh if(x) = g(x+T) - g(x) vi g(x) l hm tun hon chu k 2 T trn D .Bi 6. Gi s f(x) xc nh trn R nhn 2 ng thng X = a, X = b lm trc xng (a < b). CM R: f(x ) l hm s' tun, honB i 7. G i s f(x) xc nh, trn R v nhn I(xo, yo) lm tm ixng,

nhn ng thng X = a (a # Xo) lm trc i xng. GMR: f(x) l hmtun hon.

B i 8. Cho f(x) =2 + tg^x

-' Vx = + k (k z)0 -' Vx = + k (k z ) 2

CM R: Hm g(x) = f(x) + f(ax ) tun hon a e Q

35



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33/463


34/463

Ch n s-: L th tng- quan v hm s

i 4: Cho f(x) = ^r-r v g(x) = xc inh g 0f1 + x l _ x|

Gt' g[f(x)1=gf e ) =r j L =xl+ !x

i 5: Cho f(x) =V l + X5

a) T m f2 = fo fv f3 = f o f o fb) Xc nh fn = fofo...of

n lni i :

V l + X2

V T + ?

1+/ \2

X

vVl+X2 )

, t / i + 2 x 2

b) Ta s chng minh fofo...of(x) =

_______X_________X

V1 + X

_______ Xl+3x2 Vl+3x2l + 2 x 2

nln 4 (*)

+ nx

Vi n = 2 ta c cng thc (*) ng theo cu (a)i

y =fofo...of(x) = f fo ...of(x ) = f . x I =Qc+l)lln L rT J l + kx2 )

X

V l + k x 2 __________ X X

177 X i j i + ( k + i )x2r u / w j 1+kx2Nh vy cng thc (*) ung vi n = k+1 nn theo nguyn l qui np

uy ra cig thc (*) ng V n e N.

> " M-J r l F l l r i Fg(x) = X4 - 4x2 + 2

37



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35/463

Chng minh rng: f [f(x)j " g[f(x)] V X > 2G:

Do X> 2 nn c th t X= t + - vi t > 1. Ta ct

r 2 I 2 2g(x)=[x2 - 2 f - 2 = + 0 - 2 - 2 = | V +-i.J - 2 = t4 + i

= J M 1 - 1 = V - i . j Suy ra

to |g(*)f I ^ = t 4 v gM t e l l , ! . - L2 V 4 2 V4 . t4

Do : f[g(x)]=t4/3-f^

M t khc: ^ - 1 + = f

Nn: g[f(x)j = gVt + i j = t 3 + - I j-

Vy [g(x)] = g[f(x)]B i 7: Cho f(x) = X2+ 12x + 30.

Gi fk(x) = f[fk.j(x ) vi 2 < k e N.Tm giao im ca th y = f200i (x) vi trc honh

Gi i :

Ta c f(x) = (x + 6)2 - 6 suy ra f(x ) + 6 = (x + 6)2 vfs(x) = f[f(x)] = [f(x) + 6]2 - 6 = (x + e -6

T ta c fk(x) = (x + 6)2 - 6X t phng trnh

f200i(x) = O o (x +6 f 2001 - 6 = 0/ o\20Dl

o X - 6 6 (1/2)

B i 8: Ch.0f(x) = 2x + 1. Tm hm g(x) bit g[f(x)J = 7x +5Gii :

38



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36/463


37/463

Ohrtng : t thuyt tnsr quan vhm, s

c T/ , _2 * / .'-2 r / : 2 500') / 2503js ,= 2 sin2~ k f sin2- - - ++ s i r r - + f s i ir , \ 2002J , 20022002 ^ 2002

+ fs in 2 = 2 i f fsin2 + filCOS2 +... +

< 2 j LV I 2002j i, 2002) )( :_2 O7O / 250071: 1+ sin - + COS -[X 2002) , 2002JJ

+ - = 1000+ - = 1000-6 3 3

+ (i) = 2 l + 1 + ... + l

B i 11. Cho f(x) = X2 - 2. t fn(x) = f[fn.i(x )] Vn > 2

Gii :CMS,: Phng trr i fn(x) = 0 c ng 2n nghim phn bit

f2(x) = f[f(x)] = f (x ) - 2 = [x2 - 2]2 - 2 = X"1- 4X2 r2Bng qui np suy ra fn(x) l da thc bc 2a nn c nhiu nht l 2n

nghim phn bit.. Xt cc nghim X e [*2,2]t X = 2cosa vi a e [0 ,7t] => f(x ) = (2cosa)2 - 2 = 2cos2a

=> f2(x) = f(f(x)) = (2cos2a)2 - 2 - 2{2cos22a - 1) = 2 cos22aBng qui np d -dng chng m inh fn(x) = 2cos2na

, V ~ .... n , 71T d suy ra fn(x)= 02cos2na = O o 2na = +kt 2

f k (k e Z )2n 2n

Do a s [0- Tnn 0 < 71r + k ,< 72n 2n

Suy ra

Suy ra k ly cc gi tr 0, 1, 2, 2n -1Vy phng trn h 4 (x ) = 0 c 2 nghim phn bitB i 12: Cho f(x) = X2 + bx + c. G i s f(x) = X c 2 nghim phn bit

v (b+1)2 > 4(b+c+l). CM R: f[f(x)] = X c 4 nghimGi

, Gi s f(x) = X c2 nghim phn bit Xj, x2. Khi X i )] - X i = f ix ^ - x r = 0[ f[ f(x2) ]- x 2 = f(x2)- x 2 =0

f [f(x)3 - x = ( x * x 1} ( x - x 2)[ x 2 + (b+l)x + b + c + 1]" ( x - X i X x - x g ^ O

g(x) = x2 + (b + l) s + (b + c + ) = 0

Ta c Ag = (b+1)2 * 4(b+c+l) > 0. Suy ra f[f(x )] = X c 4 nghim.B i 13. Cho f(x ) = ax2 + bx + c (a 9* 0)

Gi s f(x) = X v nghim. CM K: f[f(x)] = X v nghimGi

Gi s f(x) > X V x =>f[f(x)] > f(x) > XNu f(x) < X Vx=> f[f(x)] < f(x) < XVy phng trnh f[f(x)] = X v nghim.

Do d f(f(x)] = X



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38/463


39/463

Uing-1: JLVthuyt tng- quan v hm s

C h u y n , 6

HMS NGC

1. HM S NGC C A CC HM LN G G IC

a) sin : R [-1, 1]X h- sinx

arc sin; [1 ,1 ] >-71 711 ~ 2

arc sin XQuan h:sin (arc sinx) = Xc) tg: R -> R

X tgx

a rc tg :R _ _ J i ,|

b) cos: R >[-1, 1]X 1- cosx

arc cos: [-1, 1] [0 ,7t]

X arc cos XQuan h:COS (arc cosx) = Xd) cotg: R R

X !-*cotgx

arc cotg: R -> (0, TC)

X 1- arc cotgxQuan h: cotg(arccotgx) = X

X arc tgxQuan h:tg(arctgx) = X

2. CC B I TP MU MINE HAB i 1. Xc nh cc hm s" ngc ca cc hm s sau:

a) y = Vx b) y = V 2x- 3 c) y = 4x3 -7 d) y = 3x2 +2Gi i :

r~ (y - 0a) y = VX - => hm ngc X = y2 vi y > 0

[y = X

i li k hiu ta c y = Vx c hm ngc l y = X2 vi X> 0

r - f y 0 [y - b) y = V 2 x - 3 o o i v 2 + 3

y = x * = ^

+ i li k hiu ta c y = V2x - 3 c hm ngc l y = ------- vi X > 0

c) y = \/4x3 -7 o y 3 = 4x3 - 7 y3 + 7 = 4x3o x = ^y+^

il i k h iu ta c y~ ^4x3 -7 c hm ngc l y =

r_ ___ v >2 [y >^2 y> /2d) y = /3x2 +2 ' { 4 _ 9 o - | (

y4 =3x2 +2 X = .

42



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40/463

Chng : L thuyt tng- quan v'hm s

Nu X < 0 th y = ^3x2 +2 c hm ngc l X = hay i li

L 4 _ 2k hiu ta c hm ngc y =J------- .

Nu X < 0 th y =ydx2+ 2 c hm ngc l X = kay i l i

lx 4~ 2k hiu ta c hm ngc: y = -J - .

B i 2 . Xc nh cc hm s ngc ca cc hm s' sau

a) y = - x + b) y = X2 - 4x + 7 c) y = a/i - x^ x - 2

G i i ;

a) y ss y(x-2) = 3x + 4 o x(y-3) = 2y + 4x - 2

X= , i li k hiu ta c hm ngc l y = * 4y - 3 x - 3

b) y = X2 - 4x + 7 = (x-2)2 + 3 (x-2)2 = y - 3J y -3 > 0 y > 3

x - 2 = Vy-3 x = 2 -y/y-3

Nu X 2 th hm ngc l X= 2 + ^ - 3 hay i li y = 2 + V x-3

Nu X< 2 th hm ngc l X= 2 - -Jy-S hy i li y = 2 - V x-3

/ T T F ^ y - ^ y - ^ - y - 1c) y = V l - X 2 r 2 1 2 , 2 i ' r ~ z 2y2 = 1 - X [x = 1 - y [x = Vl-y

Nu -1 < X < 0 t h hm ngcl X = - ~ y 2 h a y i l i y = - V I-X 2

Nu 0 < X < 1 th hm ngc l X = - y2 hay i li y = V l-X 2

B i 3 . Xc nh hm s ngc ca cc hm s sau

a) y = 10x+3 b) y = e2* 1 c) y = 1 + lg(x+4)

Gii

a ) y = 1 0 - o ( ? ' 0 { y > .lgy = x + 3 x = g y -3Hm ngc l X = lg y - 3 hay i l i k h iu ta c y = lgx - 3

ry>0

1 + lnyb) y = e2*'1 iy > " o[Iny = 2x - 1 X =



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41/463


42/463

Chng-I: L thnv tng' quan v hm S

Do tga = nn tg2a = - =>tg(2a - 5 1-tgza 12

=> s = sin^2arctg- arctg j = sn(2a ->) = 04 2 2i 7.C M R: arcsin + arccos-y=r = arccotg5 V5 11

t cotgy = v 0< YC-T-11 11 2

m ' _ cotgyTa c cosy =2/11

y/l + cotg2y . R 5V5T suy ra cos(a T P) = cosy. D o 0 < a + 3,y arctg+ 2arctg

a = arctg |e o,f)

t p = arctge

Y = a rc tg e (o ,|]

. _ 43

tg3 = v -4

+ - 1t g y - i

aE(-f)P +2Ys o , | j

tg(5 + 2tgyTa c tg(j3 + 2y) = ft --------- L J _ =

l-tgj3tg 2y t tgP- 2tgy 13l - t g 2y

= Y



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43/463

Chng-I: L thu i tng- quan v hm s'

3 . B I T P DNH CHO BN C T G I IB i 1: Tm hm ngc ca cc hm s sau

xv ix


44/463

Chng-1: L thuyt tng-quan v hm s

C h u y n 7

PHNG TRNH HM

Cc b i ton v phng trnh hm thng xut hin trong cc k th i

Olympic Ton quc gia v quc t vi nhiu cc dng ton khc nhau.Phhg trnh tuyn tnh , phng trnh phi tuyn, phng trnh hm

trong cc tp ri rc, phng trnh mt n hm, phng trnh nhiu nhm. Do tnh cht khng chuyn su ca sch nn trong chuyn nychng ti ch gii thiu mt dng ton c bn ca phng trn h hm v ik thut oi bin s".

B i 1. Tm hm s f(x) bit rng f ( x - l) = - ? * Vx * 4x - 4

Gi

_ __

_

, V__ 3(t +1) + 2 3t + 5t t = x-1 => X= t+1 => f(t) = - (t +1) - 4 t - 3

Vy f(x) = V X 4, X * 3X 3

B i 2. Tm hm sf(x) bit rng { - 1 = Vx 5 2V .X-2J X + 5

Gii

t t = =>t(x- 2) = X+ 3 =>X= => x - 2 v ' t - 1

o (2t + 3)f ( t ) = A L - = j t e I z = 9 j^ d L

2t + 3 j 5 (2t + 3)2 + 5 (t - l f 9 t2 + 2t +1 4

-/ \ 10x 2 - 5 x - 5Vy f(x ) = - --- ------

9x + 2x +14

Bi 3. Tm f(x) bit rng f^x+ j = X2 +~ + 2 Vx ^ 0

Gii :3 9 f 3 0 9

t t - X+ => t = X+- =x +-^- + 6X { x j , X* _______

=s* f(t) = t2 - 4. Mt khc ta c X2 2J x 2. ~ =6X V X

=> t2 > 12 => t > 2J3. Vy f(x) = X2 - 5 vi IXI > 2V3

47



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45/463

Chng-1: L thuyt tig- quan v hm s '

Bi 4. 'Rai f(x) bit rng f f X + = x3 + -4-Vx * 0V. X) x a :

Gi i :

t t = x + =>t3 = f x + = x 3 + -^ --3x.fx + 1X ^ x ) X 3 XV. x j

=> X3 + = t3 -3 t => f(t )= ts -3 tX3

Vy f(x) = Xs - 3x V x 0

B i 5 . Tm hm f(x) bit x + V x2 + l j = X - V x2 +1 Vx e R

Gi

b t t = X+ Vx2 +1 > X+ =x + x| > 0=>t >0

Ta c x - V x 2 +1 ------ = - => f(t) - - vi t > 0X H - V ^ T t t

Vy f (x) = vi X> 0X

B i 6.11111 hm f(x ) bit 4f(x)+ (x - l)f^ = lV x ri 1

Gii:

t t - x =>x = =>x - l = v t * lX ~1 t - 1 t - 1

=> 4 f[- -j + f(t) = 15 . i l i k hiu t bi X ta c h

4f(x)+ (x-l)f^-^ -j = 15

r f(x)+4- f e ) =15

G ii h phng trnh ta c

f(x) =

15 (x-l)

15 41

x-1

6 0 - 1 5 ( x - l ) -16-1

= 5 - x V x 1

B i 7. Tm hm f(x) bit 2f(x) + 5 x . f(-x) = 4x + 3 VxG i i :

Thay X bi (-x) vo phng trn h cho ta c2f(-x) - 5 x . f(x) = -4x + 3. Do ta c h phng trn h2f (x) + 5x .f(- x ) = 4x + 3

- 5x.f(x) + 2 .f (- x ) = -4 x + 3

G ii h phng trnh ta c



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46/463

Chng I : L thuyt tng quan v hm s'

_ 2(4x +3)- 5x(3 - 4 x) _ 20x2 - 7x +64 + 25x2 ~~ 25x2 + 4

4x + 3 5xS - 4x + 3 2 2 4x +3) - 5x(3 - 4x) 20x2

f(*)= 2 5x ' ------------------------ 2

-5x 2 i 8. T m hm s f(x) bit rng f(x)+xf^ x j = 2

i i :X t 1t t = ----- =>X = vi i 5 -- th vo ph

2 x - l 2 t - l 2 vi i 5 th vo phng trnh cho ta c

2 t - l 2

Vi X = 1 => 2 f(l) = 2 o f( i) = 1. Vy f (x) =4 x - 2x - 1

1

i 9. Tm cc hm s f(x), g(x) bit rng[f(x + l)+ 2g (3x+2 )= 2x+7

[f (2x - 1) - g(6x - 4) = -2 x + 3i i :t t = X + 1 => 3t * 1 = 3x + 2 v 2x + 7 = 2t + 5K h i (1) o f(t) + 2g (3t-l) = 2t + 5L i t t = 2x - 1 th vo (2) => f(t) - g(3t - 1) = -t + 2

f(t) = 3

v i X = 1

2

| f (t) - g(3t -1) = -t + 2 g(t)=t + 4

Vy f(x ) = 3 v g(x) =x + 4

i 10. Tm cc hm s' f(x), g(x) bit rngf(x + l)+xg(x + l) = 2x Vx * 0

f| I +g(2E = x . 1 Vx* 1

()

-1 1-1 (2)x - 1) l^x-1i i : t t = X + 1 th vo (1) => f(t) + (t-1) g(t) = 2t - 2 (3)

t t = -+th vo (2) => f (t)+ g(t) = (4)X - 1 t - 1

49



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47/463

Chtng- : L thuyt tng-quan v hm s

2tGii h: (3), (4) => f(t) = -2 v g(t) = nu t * 1, 2 hay g(x) = 2x

t - 1 x - 1f(x)= -2 vd i X * 1, 2. Mt kh c trong (1) tha y X = 1 v trong (2) th ay X ==>f(2) + g(2) = 2

Vy | ff x " a|g(2) = 2 - a

2. CC BI TP NH CHO BN C T G l iB i l . Tm f(x) nu bit 2f(x3) + f(-x) = X

Bi 2. Tm f(x) nu bit 2xf(x)+ f j = 2x Vx * 1

B i 8. Tm f(x) nu bit f (x - 1)+3f( -x-~1 = 1 - 2x V x i v ' U - 2 x j 2

B i 4 . Tm f(x) nu bit (3x + 1) f(x) - (2-5x) f(x) = X2 + 2Bi 5. Tm f(x) v g(x) bit rng-

xf(x + 1)+ g(x + 1) = 2x(x + 1)+ 11A . _ 2 - 2 x - 1 0 x 2 w - 1

A * J s l x j V x * B 6. Tm f(x) v g(x) bit rng

(2 x - l) + g (l- x )= x + l ' .

1.x + i ; 2g["2x + 2

Bi 7. Tm f(x) v g(x) bit rng

= 3 Vx 9* -1

f(x + 6)+ 2g(2x +15) =x + 2

f - j + g(x + 5) = x + 4

Bi 8. Tm f(x) v g(x) bit rng[ f ( 3x - l )+g (6x - l )=3x

[f (x + 1) H- xg(2x + 3) = 2x2 + XB i 9 . a) Tm hm f(x) = ax3 + bx2 + cx + (a s 0) sao cho

f(0) = 0 v f(x ) - f(x -l) = x2 Vx

b) Tn h tng S 1= l 2 + 22 + 33 + ... + n2; S2 - l 2 + 32 + 52 + ... +Bi 10. Cho f(x) c D = R v f(x + y) + f(x - y) = 2f(x).f(y) V x, y e Rng thi gi s f(x) khng l h m hng.

a) CM R: f(0) = 1 b) CM R: f(x) l hm chnGi s thm rng 3 mt s a sao cho f(a) = 0

c) CMR: f(2a) = -i v f(a+x) = -f(a-x) Vx e Rd) CMR: f(x) l mt hm tun hon

. e) CM R: f(4ka) = i ; f[(4k+2)a] = - l; fr(2k+ l)a] = 0 Vk 2

50



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48/463

Chng I I: Gii hn hm s

L CC NH NGH A V G I I HN1. G i i hn hm s

limf(x) = A o V s> 0 ,36 > 0 : vx-a |f(^)-A lim f(x n) = Ax-a x-*2. G i i hn bn tr i

lim f(x) = A o Ve > 0,35 > 0: Vx e (a - 5,a) =3- f(x) - A| < x-a-

3. G ii hn bn ph ilim f (x ) = A o V s> 0 ,36>0: V x e(a ,a + ) =>f( x )- A j < e

x-a+

4. Gii hn v ccm f(x) = A o V e > 0,3M > 0: vjxl > M=> f(x) - A| < x00

lim f(x) = V s> 0 ,3 M > 0 :Vx> M =>f(x)-A< sx->+co

lim f( x ) = A Vs > 0,3M > 0 : V x < -M :=> f(x) - A| < sX>-

5.G i i hn l v c c (khng tn tigii hn )lim f(x ) = c O .VM > 0,35 > 0 : V x - a .f(x) > Mx->a

lim f(x ) = co o VM > 0,3N > 0 : V X > N =>f(x) > MX-HO

6. Quan h gii hn ph i, gii hn tr i vi gii hn hm s

llm f (x) = A lim f(x ) = lim f(x ) = Ax-*a x-a" x-*a+

n . CC NH L V G I I HNGi s lim f (x) = A v lm g (x)'=B , kh i

x->a x->a

1. lim [f (x ) + g(x)] = lim f (x) + lim g(x) = A + Bx-a x-a x-a



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49/463

lim |f(x) - g(x)] = lim f (x) Hin g(x) = A - Bx-a x->a x->a.

lim [f(x) - g(x)] = lim f ( x ) . lim g(x) = A . B.

GhnngJJ; G ii xnJinLSQ

limX>3

f(x)]_ Xhi f(x) _ A = -^2 = . vi B * 0limg(x) Bx->a

2. Nu 38 > 0 sao cho f(x) g(x) Vx (a - 5, a + S) v tn t ilim f ( x ), lim g(x) th lim f(x ) < im g (x).x->a x->a- x-a x--a

3. Ng uyn l gii hn kD giaNu 35 > 0 sao cho f(x) < h(x) < g(x) V x e (a - 5, a + ) Vlim f (x ) = ln g(x) = A th lim h.(x) =

: x-a x-a x->a4. Cc dng g ii hn c bit

lim sm x = 1; lim ( + x)x = ex-0 X x->0

V A * - r eX 1 _ 1 I n ( l + x ) ,lm 1 + = e; lm------- = 1; lim - = 1x-0 X x->0 X

I I L CC DNG B I TP V G I I HN

Chuyn 8

GII HN DNG XC NHS dng:

Nu c l hng s' th lim c = cX-Mt0

Nu f(x).l hm s cp v x0 e TX : D th lim f(x ) - f( x 0). x-0

B i 1 . Cho f(x) = X4- 5x3 + 8X2 - 6x + 3- Tm l n f(x )x->2

Gii: limf(x) = f(2) = 24 -5 .2 3 + 8.22 -6 .2 + 3 = -1

B i2 . Cho f (x )= 2x3~4x2+9x~3 .T m Hm f(x )X - 3 X-1

Gtk KmfM = f(l) = - 4;12; 91- 3- : =-2x-*l 1 - 3 - 2

; B i 3. Cho f(x ) =2dJ z 1 + 3x + 8 . Tm lim f(x )V x - 2 X 1

Gii : lim f(x ) = f ( - l) = 200Cp- 1) l . 4 + 3 .(-l) + 8= 1x^-1 V - 1 - 2

52



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50/463

Chng" II : G ii hn hm s'

2 7CX

i 4. Cho f(x ) = (2x2 X +1) 6 . Tm lim f(x )X~>-1

- - _

i i : lim f (x ) = f ( - l) = 4 6 = 4 2 = Vi' ==2x->-i i 5 . Cho f(x ) = 24*2 -X.V2+tg(arcsinx) T m l i l? f ( x )

x->l/v2

: lim f(x ) = f(l/V2= 2 g4 = 22 = 4X-+1/V2 v

.2, 1. J x +x-lOg2

i 6 . Cho f(x ) = 2000 v 2~x J .Tm Hm f(x)X>1

i i : lim f(x ) = f ( l) = 2000log21 = 2000 = 1.X>1

Chuyn 9

Glr HN DNG V NH s 0

D N G

___ 1 . p(x) , . fP (x ): a thc , p (x0) = 0I = lim vi {

x-x0 Q(x ) [Q (x ): th c, Q( x0) = 0

hng php :T _ 1*_ P(x) _ (x - x 0)P1(x) p ,(x ) P ix ) nI= lim =lim ( _ C c \ ( \ =lim 7 T T T = 7 T T T vi Q ita) * -x-0 Q (x ) X*XQ (x - x 0 )Q 1 (x ) X>x0 Q 1(x ) (x )

Nu P 1(x) = Qi (Xq) 0 th phn tch tip 1^ _ .' [Q 1(x ) = ( x - x 0 ) Q 2 (x )

Qu trnh kh dng v nh l qu trnh kh cc nhn t chung

- Xo)k s dng li khi nhn c gii hn xc nh tc l Qk(Xo) 5*0.

t 1= lim ^ - = ] m = .x->x0 Q(x ) x^x0 Q k (x) Q k (x)

C B I TP MU MNH HA

Y^_4.v 4- 4y _Q i 1.11111gii hn A-L = lim -------------------

x~>3 X - 3x

A, - lixn = U m ^ 2 i = 1x->3 ( x - 3 ) x x->3 X 3



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51/463

g x 3 _ 2Bi 2. Tm gii hn Ao = lim ----------

x-* /2 6x - 5 x + 1

(2 x- l ) (4 x2 +2x + l ) _ . . 4x 2 +2x + 1 _ -Gii: A o = l im - ------- = l i m -------- ----- = 6

X>1/2 (2x -1 ) . (3x -1 ) x->i/2 3x -1

B i 3. Tm gii han Ag = lim 2x4 ~ 5x3 * 3x2-* x " 13 X->1 3x - 8 x +6x -1A ( x - 1 ) ( 2 x 3 - 3 x 2 + 1 ) . . 2 x 3 - 3 x 2 + 1

Gii: Ao = li m --------------_ _ :-= l i m 5------------------ =(x - X)(3x - 5x + X +1) X-*13x3- 5x + X +1

_ ( x - l ) ( 2 x 2 - x - 1 ) .. (x -l)( 2 x + l) 2x + l 3= lim---- - ----------- lim ------ = lim -

(x - l)(3x - 2x -1 ) X->1 (x -l )( 3 x + l) xU 3x + l 4

T,iS _ A 2x3 - (4V2+ l)x2 + (4+ 2V 2 V - 2Bi 4. Tm gii hn A4 = lim. = f~T=- -Af p f ------

*k /s X (2V2 + 1 Jx + (2 + 2V2Jx 2

| x - V 2 f ( 2 x - l ) =I v - J9 r /S- - 1 'I X-W2 X 1 V2 - 1

G i ; A 4 = l i mX 2 (x - V 2 f . (x - 1 )

CC B I T P DNH CHO BN C T G I I

A 1- x 2 - 5 x + 6 * 1. x 3 - 3 x + 2A s = l i ra ----- A fi = lim ; -----x-3 X - 8x +15 x- X - 4x + 3

A x 3 - 2 x 2 - 4 x + 8 . x 3 - 2 x - 1A 7 = lim --- --------------- A q= lim ----------X-V2 X ~8x +16 *H*x5-2 x - l

A10= i im ^ 4 :4 t ^9 X-+0 X 10 x->0 x 2 + x s

X100 - 2x +1

X-+1 X3 - 2 x + l

t 2-x-2FAn ~ lim -------- - ; A-2 = lim

^X-+2 (x3 - 12x +16) - 50

. . . x + x 2 + . . . + X11- n . . x m - lA-,o=lim --------- -------- ; A 1 i =l im ----

13 x4i X - X 14 X->1 x n _

A xn+1- (n + l )x + n . (xn -an ) -nan-1 ( x -A ; s S 5

A Um ( i2 < l % < i 5 k ; Av_vA V V ''vH 2x->0 X x-0 X

f (x) if(x 0) = g(x0) = 0lim vi i *-** g(x) I f(x ),g (x ) cha cn thc ng bc

II. DNG 2:

54



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52/463

Phng php:S dng cc hng ng thc nhn lin hp t v mu nhm trc

cc nhn t (x - Xo) ra khi cn thc.rr a _ iva +1 ='1 . V - - 1 - ;

Va +1

. V - V b - i ^ - ;Va + Vb

3 . ^ - 1 = a _ 1

V - 1r /r_ a-bVa + Vb = 7= J=

Va - vb

^ 2 +3V I +i

4. fy-$lh =

5. - l =

a +

a - b . 3/ ~ , 3/T_ a + b

$ ? + & * + $ ? - , ^ a -1

\/ 4-1

?,%!&+%!b =

\/an 1 + an"+... + + l

2n+ + i=

6. l - b -

a + 1

2n+2^&r _ 2n+ a2n-i + _ _ 2n+^ + xa -b

2n+']^ 2+l^ ___ ________________ a + b______________ .__________2n+v ? - 2n+l /a 2n- l_b + _ _ 2n+ l/ab 2n- l + 2n+l / ^ r

CC B I TP MU MINH HA

B i JL. Tm gii hn B i = limx^ o X

Gi i : = lim

+ x* -1

a ^ V j i m x = - = 0V l + X 2 + 1 x- * V l + x 2 + l 2

B i 2. Tm gi hn B 2 = lim - 2 x^2 x 2 - 4 (x + 7 ) - 9 1 _ 1Gii: Bo = limr------ / 7= * = lim -------- / 7 = \ = 777

(x2 - 4)(Vx + 7 + 3 j XH>2 (x + 2)(a/x -h 7 4- 3j 24

B i 3. Tm gii hn Bo = lim 3 ^ l y I E - 2

Gi i : B 3 = lim + = lim ^ E +2 = 2x--i [(x + 5) - 4][vx+2 +1 x--i Vx + 2 +1

B i 4. Tm gii hii B 4 = lim... /l + x2 -1



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53/463


54/463

Chnp II: Gii hn hm SQ

ng php :

Xt 1= u m M =um - = um 'g(x) X-X0 g(x)

in i:I = lim - limx-x0 g(x) x-+x0 g(x) x^ o g(x)

n y cc gii hn c tnh theo dng 2.

C BI TP MU MINH HA

i 1 .Tm c x =1 X.1X->1 X2 - 3x + 2

i i:C 1 = l im E z 2 ] = i^ z 2 ) = i in?^ z i _ i im ^ E z i

X-1 X -3 x + 2 x-*x -3 x + 2 x h > x 2 - 3 x + 2(x + 7) - 8_______________ (x + 3) - 4

/(x + 7)2 + 2%fx + 7 +4 *-S(x- l)(x-2)fVx + 3+2]

= l i mx-l

L( x - l ) ( x 2) /(x + 7)2 + - 2 ^ + 7 + 4 ^ ( x - l X x - :

1 ______________ _ _ r 1____

(x - 2)^(x + 7)2 + 2^x + 7 + 4 j 1(x - 2)[Vx +3 +2]

12 I 4 J 4 12 6

i 2. [HQG 1997 Tm gii han Co=x->0 X t

i: C2 = lim fe ^ L z ! E H ) = i J ^ ^ +^ ^ 'x k) X x-*0 X X

~ limx-v

= limx->0

2x X

x ( v i + x + l j x ^4 + 2 ^ f s - - x + ^/(8 - x)2 j

2_

___________1V+x+l 4 + 2 ^ x +(8-x)2

= 1 J L = H+12 12

i 3. [H Th y Li 2001].

qv *; 1. n _ V V l + 2x - ^ l + 3xTm gii hn c 3 = lim

i : Co = limx-0

x-*0 X2

VT+ 2x - ( l + x) ( l + x)-\/r+ 3 x2 + T 2X X

57



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55/463

Chng II: Gii hn hm s'

= lim 7 = --------+ lim ----------------- ------------ =1*-> V l + 2x + (l + x) ^ 0 (1 + K)2 + (1+ x)^ /I+3x+%l(l+3x)2 2

CC B I TP DNH CHO BN C T G I I

= C . = lim t/ + - ^ x +43X O X x-2 2x2 + 3x - 2

c 6 = K m ^ ^ ; c 7 = im ^ E p E Il*->1 X 1 x->2 X - 3x + 2

C9 = ]im L t - ^ +-2x-0 X + X X->1 X ~ 1

[ H S P II 1999AJ

; c XH>1 x - 1 11 x->0 r X

.. V4x 4 + 5 - ^ 8 x 6 +2 1. ^ s/l + ax -^ tfubC12=lim --------------------------- ; Cl3 = lim-

X~*1 V x -1 x-0

+ a x , l + bx -1C14 = lim - ;*-io X

0 - V l + 4x . l + Qx . y r+ 8x .^/l + 10x -115 X -S X

[ HSPI 2000B]

r- _ T _ \/x + 7 - V - X 2C16 = lim ------- 1 --------X->1 x -

[HQG 2000D]

r _ r -s/2x + l -/x2 +1c 17= li m---------------------x--0 sin x

x-? Vx + 9 - 2

- V l + 4x - \/ +6xC19 = limX-X) X2

0 , - mx->0 Vx -1

58



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56/463

Chng'I I: G ii hn hm s

Chuyn 10

GII HN DNG V NH:

_ p/x )Ph ng p hp : Xt I = lim ' . vi P (x )f Q(x) l cc a thc hoc cc

x-*Q(x)

hm i s.Gi bc p(x) = p; Bc Q(x) = q v m = Min(p, q), khi chiac t v mu cho xmta c k t lun sau:

Nu p < q th ton t i gii hn Nu p > q th khng tn t i gii hn

CC B I TP M U MINH HA

____

. , _ _ 2x

3 - 3x

2 +4x

- 1Bi 1. Tm gii hn: Da = lim ---- ---------x-* X - 5x + 2x - X + 3Gi :

Dx = limX50

X4 , 5 2 1 31 + 2 ~ ~ 3 + 4. X X X X4 J

= lim-X-KC

2 - 1 +4 - iX X2 X3

, 5 2 1 3

. X X2 X3 X4 J

=

. . - 6 x 5 + 7 x 3 - 4 x + 3B i 2. Tm gii hn.: Do = lim 57-----~- 8x - 5x + 2x -1

Gi

Do = lim

J __ _ 4_ _3_X2 X4 ' X s J -35 2 1 ^ x-xo rt 5 _ 2 1'4

x | 8 X X 3 X58 - - + 3 ------ 5

X X3 x

____ 3x2 (2x- l)(3x2 +X + 2)B i 3. Tm gii hn D3= lim ---- --------5 :-------

3 X-** 2x + 1 4x2

Gii :

, , 12x4 -( 4 x 2 - l)(3 x 2 +X+2) - 4x3 - 5x2 +X + 2D s = l i m ------------ - -------------- --- l i m ---------- 7- ---------------x->


57/463

Chng I I : Gii hn hm s'

B i 4. Tm gii hn_ . (X + 1)100 + (x + 2)100 +... + (X +99)100+ (x + lOO)100

limx-xc X100+10xLU+10010 ao

Gii D4= limX-WO

x 1 0 0

H)*H ...............(1

10Qj100h-00 , 1 0 O O 101 + +X 10 x a o o J = 100B i 5. Tm gii hn Ds = lim

G i i : D = limX>00

Dg = limX-+00

D 5 = lim

W-16 + - X . g +

x .m + - x .sI+

. Xt 2 gii hn 1 pha

3 - 0 32 -0 2

- X . * 7 * * 7

Do D5 = D5= nn D5= lim f(x ) = 2 x-x 2

3 + 0 3+

B i 6: Tm gii hn

Dg = lira

lnGi i : D g = lim

ln(x44- 3x34- 2x2+ X +1)x-+w ln(x5 + 5x4+ 4x3+ 3x2+ 2x +1)

, 3 2 1 1

5 [, 5 4 3 2 1X | l + - + - W - ^ + - y + - r

X X X X X

' 41nx + ln| 1 + + ~2 +~^ + - l X Y2 V3 V4= lim ----- - ---------------------r = lim

x->+co 7_ , 5 4 3 2 1 1 x +0051nx + ln i + _ + - + -i- + _L +{ X X2 X3 X4 X 5 )

4Inx 451nx ~ 5

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58/463

Chng IL . Gii hn hm s

C B I TP DNH CHO BN C T G l l

D7 = Im (2x 3)!.4x + 7)3 . Dg = lim J

X-W (3x3 + l) (1 0x + 9) *-*>

(2 x - 3 ) 2 0(3x + 2)30

50

Dg = lim(x - l ) ( x -2 ) (x-3 ) (x - 4 ) (x - 5)

D10 = lim

D12 = lim

(5x -1 )

(x + l) (x 2 + l). . . (xn -i-l)

V4x2 + 3x -7

Dn = lim

(2x + 1)

Vx2 + 2x +3x

x^ V 4 x 2 + 1 - x + 2

^27x3 +5x + 4 +xD13 = lim

n _ X + V ? + xD14 lim ----------7= X w3x--vx2 + l

D15 = lim

5x + 3v -X T x

ln( l + Vx +\/x)X+c o 1jjh + ^/x+3/x

^ ^x5 +1 - ^x2 +2D is = s r r = r r = = ;

x- ^ x 4 + l - V x 3 +2D17 = lim Vx2 + 4x + 5 + 2x + l-im . = ----

x~* V3x2 - 2 x - 7 + x

Chuyn 11

GII HM DNG V NH: 00 - 00 ng php:

Bin i a v dng gii hn * ' c o

C B I TP M U MINH HA

i 1. Tm gii hn E-L = limx->+L

i: Ej = lim + = lim _ _ = _ = !

4 O + 1 | 2u VX J

2. Tm gii hn E 2 = lim L/(x +a)(x +b) -xjX->+oo

Vx + Vx - V x j

i: E = Um (x + a)(x + b )~x2 - lim .X->-KO^(x + a)(X + b) + X x-y+co

(a + b) + X a + b

1 + 1 1 + 1 +1

61



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59/463

Chng XL xii hn hm s'

B i 3, Tm gii hn E , = lim ^/(x + a1)...( x + an) - x l

______________ (x + ) -(x + a ) - x n

Gi i :

E 3 = lim

E$~ lim

r I11-.1 r ,n 2

[(x+ a1).. .(x + an)j n + l(x + a1)... (x + an) f7 T .x + ... + x n llim (ai +a2 +-4-an)xn-1 + A n. 2x n"2 +.. .+A 1x+ A 0

[(x+a^-.ix-f-an)]""^" +[(x+a1).(x+an)jr,x+...--x

xn_1 (a1+a2+...+an+^^+...+-^L+-^-(12 n-1

Eo =n

B i 4. Tm gii hn:

E 4 = lim [\/(x + 5) (x + 6)(x + 7) - \/(x + ! ) (x + 2)(x +3)( x + 4) ]x-*+co *

Gii; E4= l imR/(x+5)(x+6)(x+7)-x] . j jmr^(x+l) (x+2)(x+3)(x+4)-x->*e J X-+oo*" 5 + 6 + 7 1+ 2+ 3+4 _ 10 7

3 4 4 2

B i 5.Tm gii hn: E 5 = limx~*lm n

Gi i : Ec = limX>1

mLl-Xnl 1-Xn.

n 1

(m,n e Z +)

E s = limX-+1

E 5 = limX>1

1 x m 1 - x j u - x n 1 X,m-l>m - ( l- f x + x z + ...-t-xm~x) n -C l + x-hx^ +... + x n~ )

l - x m i - x n

( l - x ) + ( - x 2) + . . . + ( l - x m~ ) (1 - x) + (1 - X2 ) + ... + (1 - x n-1( l - x ) ( l + x + x 2 +... + xm_1) ( l - x ) ( l + x + x 2 + ... + x n~1)

1 + (1 + x) +... -h(14-X + + xm"2) 1 + (1 +x) + ... + (1 +X 4-... + (xn~2

E e

Ee

17 i : _ l + ( l + x) + . .. + ( l+ X + . .. + Xm''2) l + ( l + x)- g iim --------------- ~ --------- ------------ ---- -----x-*1 l + x+x +... +x m 1+2_ 1 + 2 +. .. + (m -1 ) 1 + 2 4-... 4- (n - 1)

m n_ m (m - l)/2 n ( n - l)/ 2 _ m - l n - 1 _ m -

m n 2 2 2

1 + x + x +... + x :n.~l

62



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60/463

Chng I I: G i i hn hm sn

CC B I T P DNH CHO BN C T G II

Eg = limX-+c

Eg = limX>"Kc

; E 7 = lim fx + Vx + Vx - Vx

-Jx + Vx + Vx -- ^ x- V x- V x ;

V x - * \ / x - V x - V x

E = lim (2 x- 5 )-V 4 x2 - 4 X ; E 10 = lim ^x3 +7x2 -\/ x3 +8xx->-Ko|_ J X00L

Chuyn 12

GII HN DANG V NH: OO.0

Ph ng p h p: a v dng v nh 0

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