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Finding Critical Values and Global Extrema in Functions

School
Fairfield University**We aren't endorsed by this school
Course
BIOL 234
Subject
Mathematics
Date
Dec 10, 2024
Pages
73
Uploaded by MateValor4972
ExerciseI441.get)=t:e-t.ttEli2)TofindtheglobalextremaIneedtotestyC-11,got&¥4functions←whereccanbemorethanistheitvaluesonevahej①wheneithery'lt)is0orundefinedCinotherwordsfindthecriticalvaluesofthejunction)
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I.g.C-1):3174galiEe-thg.c-D=-Fe'gal=ÉTeyell=lifegtD=legt-usingoakg.C-11=2.718ga:of(2)=cafe-4of(2)=if¥hint¥215glad--Iyank
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1.Findingcriticalvalues:""Igot)=t?e-tg'Ct=É•[étTte-t.lt:]•g.A)=#•le-tl.EDY-e-t.at#Yg'ltI--(t:.e-t.-)te-t.oge-÷)gits:(e-t.-t:)-1¢-t.E.e-¥1a.bIa•ca(Btc)Jrg'lt)=éttt÷+}t"-
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5124I.Needtosolvewheny'It:O&wheng'It)isnotdefined.g'Ctl:Oe-tft:+3T¥)=0A-INeedtosolvea-0&b=0A--O••€-5--o%Afollowstheformofet.Therangeoféis⊥,a)thustherewillbenovalueoftthatwillmakeA--0true.
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61741.B-t?+Est=O✗3✗3→t?t2T¥=0T¥(-31-+2)=0ITNeedtofindwhenf-0&D=of=0,Cfollowstheformfix5-X?wherenisnegativewhichhasarangeoftoo,0)0COt).Thusthereis'noanswerforG-0
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1.D=o-Ky-31--12=0-2-2-3t=-2=3=3t=3☐Nowneedtocheckwhentythereareany)valuesoftthatmakef'lt)undefined.Givenmyexplanationofwhyc--o,isfalse,Ñltisundehnwwhent=0T
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I.80thMy2.71800.392¥2f-••••y-I02-32AtcriticalvaluesThelargestvalueofyct)withinEliz]happenswhent=-1Thustheglobalmaximumiseiht"ihepointinwhichtheglobalmaximumisobtainedisc-ligtl))whichisthe)
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1474:÷¥÷÷÷.hlo)=0-+24hCO)=}hH:=has=Y¥÷,h(4)=18¥hC4)=Y
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2-Needtofindcritical""numbers:fh'is/=¢""KsYL%¥¥4Gs+☐•)_n'a)="""Y?¥Y¥4laD_nisi="¥¥¥Y
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"h'(g)=(452+25)-(25+4)=1%44it4s+Imsn.EE?.:-;nisr-::::-*WiUfiydmhenh'Hisundefined:__4it4s+I=012¥12=o
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2.A=024¥±gnmaÉ2-2=-1-2isnotwithins←aWillnowfindwhenhisequals07¥:¥⾨=o
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2.Needtopluginall1%4relevantcriticalnumberHintohIs).TherelevantcriticalnumberG)are5=1.th(1)•nT¥¥,h111=h(it=}h
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1.fa)=(✗4×-2162%4f'CH=6(✗4×-2)5•[✗4×-5jail--61×4×-2112×+1)y'1×1=64×+211×-11%2✗+1)f'CH:OVr0=6/1×+211×-11/{2×+11Needtosolve✗+2=0→×I¥=0✗-I:O→I2×+1--0→2✗=-I→x=
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ABCD24>y----p-1++1--1,0-----apt-1-1++1--1-2-£IA:willusef'C-31ft-31=611-3+211-3-11%21-31+11f'C-3)=611-11-1151-1y'C-31=414-1y'C-31=411-1f.'C-3)=-B:willuseflitof't)=611-1+211-1-1114241-111fat=611+11-115f)fat=c-Pt)→j'tis-1-11-1→f'c-11=-1
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4-•221>4C:willusef'MJ'61=640+216-1114261+1)f'(01--611+11-1154)y'101=1-1%+1f'co)=(-11-1)jcoi=-D:willusefulf'(2)=642+2112-111%2121-111f'G)=611+11+1%+1jlzk(-44-1)f'121=1-111+11421=-1
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2.Thusthedomainif""hot)isC-oooo)hCtI=Ct+1)e-¥,h'CtHt+111e-¥11(e-¥/(t+1)•h'A-1--Ct+11k¥File¥1111ah'A)=¢+4ice¥I:[¥1:'-(e-¥1a•(b•c+1)h'(t)=(e¥)(It+it1¥I•+1Ih'little¥Nt+it1¥,I+it¥]I"""%;Y;tY-
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2.÷[¥]=t"Y!-K%,¥[¥]=-9¥¥[¥]=¥tplugintohit)n'A)=le¥Ñt-1111--11-1+11Needtofindwhenh'Ct)=oo-te-ENt-iyt-i.tl-111Needtosolve:e-¥=o&t-1111%1-1+1=0
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2.e¥=o,since¥4therangeofe"isG.•)thereisnorealsolution.(tee)ttzt)-11=0-l-ld-+111%1-1=-1-ItEat=-1•••2Bo-2t•+t=2-2-2thtt-2=0It-12kt-11←t=-2,I
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2%42.tilt)=0when11=1,-2ABC----pet-1+-1++1--1-1g,-------sayingrange-2✗isloadyhllttltflt-111%1-1+11A:willusef4-3)f'C-31=(-1%-3+111%-3)-111y'1-31=(-11/1-2113-2)-11)y'1-31=(-111-5-+1)f'C-31=(-111-3-11)f'C-31=411-1F'1-3)=-
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2.B:willuseylco)%,JYOKG)(10+111-1.01+1)fico)=H1(0+1)ft):(TIG)f'G)=-1c.Willusef'121f'(2)=HK2-411-1-2.21+1)y'(2)It)((3)(-11+1)2421-1-1/(-3-11)1921-1-1/f)L'(2)i-
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3%43.a+baa'cr1=2r-1121nA)-1)Gr)a(b-11)a'G)=2r(2)ncrl-I+ya'Cr)=2r(2Incr))a'G)=4rInG)Needtofindwhena'G)=0+0=4rlnlrlNeedtosolve0=4r&InIrk0
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3%,3.E-¥In(r)=or=e°a'G)=4rIn(r),r>OTeb-asincerhastobepositive,awillalwaysbe+bsinethisfollowstheformInCH,bcanoutputallrealnumbers
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3.Thusa'lrl=G)ln(r)331>4A.B✗g-----•,-1+-1+-4+-4A:WiduseaY㱺a'(E)=G)lnltla'(E)=G)f)a'(E)=-Biwitlusea'121a'(2)=G)Ink)a'(2)=H(t)a'(2)It
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Exercise#Myfcx)=ln1×2-4×+5)xintfduginoforaio-lncih-4x.is)e°=X'-4×+5-4=5-4×+1,Am1-"-ya-14=00
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B=0361,4✗-2=0-12+2✗=2×mtisatlyint[plugin0fora]•.¥Éy=In(o-o+s)y=In(s)yintisat4n
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37/74willfindwhenfisincreasing&decreasingff'(x)=¥51s•E-4×+59f'G)=×¥+s(2×-4)jail=¥÷Escriticalpointslocatedatavaluesofwhen2×-4=0&F-4×+5--0
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2×-4=0criticalpoint"-14+4✗(2)Jk))24=4(2,1%12^-41211-5)=:/12,1^14-8+511✗'-4×+5=0-Thediscriminantis(-45-41116)=16-20=-4←sinceitislessthan0,thereisnorealanswerfornAB-------.gg-1+-1+-1-1←fad←✗
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A.mil/usej-#Fgjan=::.-I+sjc-n--+sjai=jH=÷y'tis=-B:willusef'131jÑ?
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j'on=s¥4f'l3)=¥f'(3)=-1ThusfisincreasingatGin)&decreasingatC-•,2)Willfindwherefconvenesupanddowntojail=f"in=*""k¥÷¥;;÷"*㱺_
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j"H="""¥¥¥%¥k×I-¥j"ex)=k×%¥¥;:%×jail=_•÷¥÷willfindwhenLifitoccurs/j"G)isundefined.(x2-4×+55=0✗2-4×+5=0Thediscriminantis(-41^-41116)=16-20=-4←sinceitislessthan0,thereisnorealthismeanthatanswerfora•%)isdefinedforallrevalues.
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Willfindwhenf"(a)=042¥jail=_•÷¥÷o=-•÷¥¥÷0=-2×2+8×-60=-242-4×+3)0I-2(x-1)(x-3)X=l&x=3✓ABC←six------1+-1+-1-141-+1+1--1-1+-1+-1I3←✗Infectionpoints:
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A:WiAusej"¥4J''(D)=-201+8101-6✗of-4Costs12j"co"f"co)=F-=-B:willusej"14:f"(z)=-2(21t8C2#¢2T-Y(2)-15121"N=-¥÷÷
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J"(2)=¥=-1C:willuseof"(4)¥4J"(4)=-2(4)+8CH-6④f-texts12J"""=!%¥→y"Cy=E-=-1ThusfisconcaveupwardsatG,3)uGN)andconcavedownwardsatC-•,1)
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Findingiyvaluesforinflection4¥,points:✗=I&✗=3☒Ifly=In11^-4(1)1-s)jl1)=In11-4+s)JC1)=In(2)¥3)=In(3^-413)+s)f(3)=In(9-12+5)f.(3)=In(2)Thusthereisaglobalminatthecriticalpointtho)
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GraphofJCX)4¥㱺÷÷÷
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EYY.ie#..iiE+red•÷•greena••-•I;•-&Thismeansfatisformedinthismayfcctbwherebisanyrealnumber.Thismeansfluwillbeahorizontalstraightlinewhichdoesnothaveanyextremism.
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Ify2.JK)=of'(c)=of"G)=oaibiccanbeanyvaluesaslongasa<beCistrue.&Thismeansy''(c)=0forsomeCECaib],willbetrue,butfatdoesnothaveanyinflectionpoints.
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3.fix)=-✗•f4×k-×?-✗a¥f4×1=✗4•:convenesup✗,¥;umaresdown4.Jake✗Leta=D,b=S&G-2Let'stestifjcb)-JCa)=L'G)J(b)-fCa)-1J'cc)
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4.es-E-ja¥4of'cake✗flake"muybackines-e0=?etes-1Ieaes
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¥4s::#PJ1×1isalways70ontheinterval[a,b)✓sinceJaiisahraysincreasingontheinterval✓Cab]y'1×1isalnay>0ButifItconceivesdownnotup✗t
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1.Iimfix)¥,✗→atpluginalimp1×1=3Ink-1/-281-17✗→a1*81×1=3Ink)-2calsome13ln(&)<4-2(•⊥_|¥1,4K)=-2cal=-a
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2.Needtofindcritical""numbersf4)=3In1×-11-2×+7J'G)=3•¥•[x-D.-2y'all=÷,-2winfindwhenf'1×1=00=¥,-22=3×-7
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¥2.21×-4=32✗-2=3+2-12¥✗=I←cnhéae@Willfindwhenf'CHisundefinedj'CH=÷÷-2
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2•d=o%,✗-I=0.✗=/←ignoreasiisoutsidetheinternal4,a)ABO+Égg←j'CH•§f-XA:Willtesty'Caljcxs=÷#J'(2)=£,-2
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2.f'14=3--2%1,f'121=3-2f'121=-1B:willtesty'DjcH=÷,-2J'131=3-1--2J'(3)=3-2f'(3)=-
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"Thusfisincreasing¥4ontheinterval:(1,E)anddecreasingontheinterval:(£,a)3.Needtofindwhenf'H'"o=¥,-22=3×-721×-1)=32✗-2=31-2-12
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4.63-4testinys.8-kl-B.SIUsingcalcJ(5.87)=0.01JIs.88)=!%%dMYtesigns0Thusthereisarootintheinternal:C.8>is.88)
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6¥y5.part3:Thereisaglobalmaximumat✗=sz.INcontentthismeansprofitisthehighestwhen2soomotonylesaresoldTheCordinateoftheglobalmohimumis⾨,3tenth+2)*(£,3.227.Incontentthemanamountofprofitis3.22millioneuros,whichonlyoccurswhen2,soomotonyksaresold•
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5.⾨Part3:ThereisnoglobalmanasI:pan=-o.Incontentthismeansthatthereisnominimumamountyoucanlose,asintheoryrompingproductionCentremly)willleadtogreater&greaterlosers,thegreatertheamountofmotoighsproduced.
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EmerineVI#%y1.Revenue=price.quantityR→amountof$5pricereductions.price=400-s×numberofsubsimkas=1000-12ont(a)-1400-s×)(1000-120×1f(X)I1400.I000)+(20×400)-(s×.1000)-(5×-20×1f1×1=40Oooo+8000×-SO00✗-I00×2p(x)I-I00×2+3000✗c-400000
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I•f'1×1=-200×-13000%4Needtosolvewhenp'Cx)=0-200✗+3000=0-3000-3000÷÷÷÷a=¥=I5r"(X)=-2Oo
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1.sinerealisahnaÉ"negative,thereisaglobalandlocalmanimumata-IS.ThustheGoldenMembershipshouldbepricedat$325pass=¥-s&pUS)=325
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3.¥4Revenue=price.quanityR→amountof$5pricereductionsprice=400-S✗numberofsubscribers=10001-lonr(X)=¢100-SX)(1000+10×1p(y)=400.1000)+(400.10x)-(s✗•1000)-(s✗•10×1ff4)=400,00Ot4,000✗-Soo0✗-so✗2p(X)=-SO✗2-Iooo✗t4oo,ooop'(X)=-I00✗-l000
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3.Needtosolvewhen?⾨r'4)=o-I00✗-I000=0+IoootI000-100✗=I00OFoTooR=-109notacriticalnumberasxisnegative.Thusrevenueisminimizedwhen✗=0•
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