Section 5.1 The FirstDerivative

CriticalValues offee N indomainoffsuch that f cul O or f cu PNEEnarpk

Fcal

a.atCritical ValuesF cul o when n acs c dFEmDIVE hen no such sevalues

21 F

mta b d

CriticalvaluesF em 0 when k b dFcul PNEwhen n a c

31 FindCriticalvalues ofa feel 3 4 75 8

f cu 12 3114u

fkn o

12m3414n se 12 2 14 0 se o or Izu 414 0

But kn't 19 0 s Izu 14 no solutions

Thus only n o is such that f cm o

f ca DNEf always existsj i se o

9b fcmften t18 4

118,2

zakatfeel8 2

f em o

3 1 1,2242 0 s 3k see241 0

zuko or zee24 0

k o or N 24

f em DIVEft 181 DIVE But fl 181 DIVE y 18notondomain not a Cv

Cv's 8 0 24

Properties of First peroratovef cm o ou ca b f increasing on ca bf cu Lo on casi s f decreasing on Cab

FIFTH'tcu fadwhere f is increasing and decreasongD Findwhere fCoe _o and f en DNE if Cri o

46 4 n 2 o se zo a

flexi DN E if l DIVE

2 Numberline L l I 3Cwin 2 Can 4 4.2

3 Findsignof fins on eachintervalno 21C94 Try u o ffo I o

f cm o on Cwid s f Pon C w 21

4C 2241 Try se 3 f'G i Tco

flculco on 2,4 f on 2,4

4 tool Try re si f FELT 12 o

fkn o on 14 twi s f on 4 w

4Thus fcm is increasing on C w 2 Uchewldecreasing on Cheol

trample 4Findwhere fCui r 11 is increasinganddecreasing1 f'cm In

fcm o 4ns o se of emDIVE f cu always exists

2 I 1l fkn 4 C113 4 coC w o o Costco f i 4 c a o

fcm Tf cut to 7

Thus fineeasing on C w odecreasing a co eco

ffm Cnet Gcn 6 se fondwhere f increasinganddecreasing

1 V's f cul o when N l n 6 Note e to l

Z Z o 7n L l l S s z co1 6 flea C2 11612 61e

f'cu t 32768e 2

fcm D T Los o cof co 6116Co 61 e

Thus 7776eincreasing 6 tou o

7decreasing C wi 1 UC1,01 fCol 7 I bc76 se O

262144eo

Relataeextrema

µweed th's f 714IT n In

FirstDerivativeTestSuppose fCc o Then

Icf cu t fat relative maximumfcm P

fat relative minimumf Cui tfcm X T

fat neitherf Cui tfcm Por

ProblemiFind Relative extrema of1 feel 2

I f'cul ZUf cul o when N o

f lens always exists2 I tiL l s

C w o 0 Co toof cul f's 11 2 Cof cut f Cis 2 SO

Thus f has local non of foe o at re o

2 fon zze.KZ

I f cu 2rem zuse2e

f'cm oZuck 2n en en zu zu

week2C q2o

2x o or ed o I 2 0

I _o notpossible U I I

flee o when n o 1 11

fKui DNE f'cns always exists

2 2 I E zC l I iCco 1 i ciol ocoin i ci.tw

fcu t t flc 21 12e4sof m x 7 ft z I e't Lo

f't t Ie't of cu 12 LO

N 1,0 i indomainof fRelative min fCo 0Relative max fell e and fci e

3 feel Tea

l f cu Isfcu o Is ofcm DNE fEosDNE Kote fCol DNEl

2 I ie i sCaero o co.tw

floe 1 f c it 2 Of cu P f Cis 2 CO

Note fCol is not rel max since fcol notdefined