Recall Intermed!ateValuetheorem Let fAbe cont!nuous on lab and I dlet d be between fla and fb thenthereex!sts a c c a b suchthat f D f

Hava az adafcaı a b

Recall Mean Valuetheorem Let f!x be cont!nuous on a b f

and d!fferent!able on a b then there ex!sts ceca.by II jjmsat!sfy!ng f'a f

Reca! Rolle's theorem Let f!x be cont!nuous on a bd!fferent!able on a b andf then there ex!sts cEca b ah!re ısat!sfy!ng fk O a b

a Let f Kt 3 7 FAI !s polynom!al palynam!are cont!nuouseverywhere saf !s cont!nuouseverywhere

µf 7 fala 3 az

µ !s cont!nuous on 0 23 and ford O µf 2 Cd Lf so by TUT there ex!stsEl 2.01 such that f D

has atleastone solut!on

Assume that Ax has Mevetken 1 roots call two ofthan!n order Ç 2 c ç oppos!te order same calculat!ons

wh!ch means f 4 o f D polynom!als are

d!fferent!able everywherebe!ng polynom!al fıs d"f"blealso So we can apply MUT or Roles on cınca

fKI !s cont on Ccı az d!ffbkoncce.czand ftp.flcz o then there !s a CECG.czsuan that f flcI.IE a0 jofforRollesfka o

But f M 3x42 3 for 32 2 3 0 D 1 4 3.3 3220

f W has no root contrad!ct!on our assumpt!onAssume that hasmarethanonerat !s wrong sofcan not have more than one root

Comb!n!ng FW has exactly one root

b fW x4txtx 10 ı 2! fCok 10 flaş 12 fl!t 8

FM !s a polynom!al so !t!ş cont!nuous and d!ff.tkeverywhere 2 1 worksalso µ !n Trtf!x !s cont on 2,03 fC27 8 Ö sf

by TNT 7 GEC 2.01 such that fca D

S!m!larly fu !s cont on 0.23 f 10 LOCKby EVI 7 az C 0.21 such that f D

GEC 2,0 Cz C Caz 2,01 MOR D saç Czwh!ch means fal has atleast 2 roots

!! Assume that FM has more than 2 salnscall three ofthem !n order Ceşczuz ftp.fkd fkD o

fA !s cont on k!nc! d!ff'ble on 4 cz so byMUT zdfccr.cz such that f'dı Halefk !s cont on cz.cz d!ff'ble on cz.cz byRolles thmzdzECcz.cz s.tn f d 0

But f 4 3 2 1 call th!sgx for s!mpl!c!ty ! e 911 4 3 2 1g 12 2 2 70 forany so 9W !s !ncreas!ng !t !s 1 1

because If Xzxncxzsofcxucfworxpxzsoflx.lt ysog fk cannot have more than 1 root Contrad!ct!on

all because of our assumpt!on Hence our assumpt!on !swrongTh!s means f" 0 can not have more than

2sdns!ff!ncrea.FIIfpa lfx oe0s o 1 1 !nequal!ty !s sat!sf!edIf O let fav e f H!s contdd"ffbkeveywhbecause"t"sexp.fm

apply MUT on 0

of contoncax d!ffbkonco.nl then 7 c C CAN

s!t fk f o occcxys!!sYII.uso 1 Da

d ö e ı et! x

µNİYE wn.f!exsxh""lXcosfWcont n x o

d"ffbleonlx.olthenzcEH.dstfka.frH ff fxccao secceo 1

mü he tw l c1 xco

X xe 1 e x.cl

sf """ ı also works lexerc!sesb for no equal!ty holds Ö 0tl 1 1

Let f et_ l!near comb!nat!onaltımacontdd"ffbkt.de2 funct!onseverywhere so cant Ed!ftkleCont d!fbleeverywhere

everywhere

for o apply MUT on G

f !s cont on G d!ffble on G x then byMUT 3 c E CAN such that

f FW FA f et_ et XX o

t.ecc E 1 from parta ekaxel

o

Eyı ek Iz 1

2nd way öz x 1exerc!se

o 1 C 3 4dgTÜFÜu

f ex!sts on G 9 so f !s cont!nuousor"ff bu cat

of"scont 133 d!ffsla a 3font f!sd!H.dk

SobyMUTthere!scEC1.3 such that

f FM tl 5 1 z3 ş 2 Iz

f !s cont on 4,83 d!fble on 4,8 then by MUTthere ex!sts a d suchthat

f'a fg

w4oz

apply Mu Roles than us!ng !nterval a d f gwg f'W and f IN ex!sts on G 9 gw!sd!ffbk.es9 g !s cont!nuous

gandg"ant and ak"le on Cad byRollez

theorem there !s a number k C c d such that g D

ağ f f O

üOR lfxrtxzthe.fm flxz

u

2 7 2T cos XI 2T

7 2 7 7 2T cos 7712K µXp X2

5 4 ftp.flxzv

f 7 Zıtaos x O f !s !ncreas!ng FWIWfw !s one to one

f !s !nvert!ble f k! ex!stsDa ray

b flu !s def!ned for every xEIR fa.ro f Asdün f R range R Ö b f B A

f ax 7X 2s!dxl 2.4jflrx.IEzs"nHI zj!.mx'Tefexera.se I"mjzsTx o

a

s!m!larlyf takes everyraluefren ta no

Bugünrange f Hara doma!n of f IN

c f 6 1 flakalıta 2s!ntakz AHf

6 fYHI.HN 1

f 6 af"afbfa 7a zs!nra.fr HmY f

azluf 1

ı

""" "ley" µ

2 !mpl!c!tlyd"ff 5 Üye121 et 2x fa"l Öt f Ce 3 Ö

2 et 2X.FMf'w fH.eEtfCxI.eX 3ex

Ist way put 1 and 2nd way solve kytl4solvethen put 1

2 fHHFH.ehtY.ee3ftp.xe 4e4 3eX2etKfweftp.Tze el OFTI O µ 3 ex ze.tt fWeXyk

e

fkDy flD 0

1 5µ y 5T

d!ff boths!des of yj.frm

fkgegtthesec2lXyl1 yY cosCxy 1y xyl DIDXsecYxtyltsedx ylykkoscxy.ly cos x yy j put CE A y t.EE

g m A KI

2ndway

slope of tangent Iş so cons!der g fanfan xtfMI s"ncx.fmseczlXtfcxN.C1tfkxII cosCx.fcxN.CfCxIX.fkxDputCx.yIF Ful fArt 7

sec!c! F HAYAL F trutlıtrf"ftfkrA1 H f F 1 C EFKAN

f A 1 A Tt 1

f F 1FITI