31 increasingandDecreasing
Functions
Ext
afindtheintervalsof increasinganddecreasing for function
fansans in son tooo o nooo positivethenthefunctionmusthavemo or asymptote
tetra
fansonsonso
Oonasonso
i
o anim o un o t t t
o on out no o
n to no sign t o o
no
Increasing
c o is coin ner
Decreasing
cn einer
Exa
Qproduce a graph with thefollowing properties
fans o c2.2 is o is
nano o 2 ins n o
fans cz.si
fan s a a s
Ex3
a produce the properties that definethe followinggraph
fin o
IX
s fi sis
a
fiasco c3as aas
É a
fearone
himfino
nigrato
r
3.2maximaandminima
localextrema akaextremevaluesakaturningpoints
Alocalmaximumoccursat a point onfansifthe ycoordinatesofallotherpointsinthevicinityarelessthantheycoordinateofthepointten
famchanges frompositive to zerotonegative ietinalso to piano to finaco
or
teaislocatedattheendpointofthegraph
, r e at
oalocalminimumoccursatapointon ifthe ycoordinatesofallotherpointsinthevicinityaregreaterthanthe y coordinate ofthepoint on
fin tens
finchangesfromnegativetozerotopositiveie tineasoto pianoto finaso
or
teaislocatedattheendpointofthegraph
thecollectionoflocalmaximaandlocalminimaarecalledlocalextrema
onfunctionhasanabsolutemaximumatapointma iftheycoordinateatthatpointisgreaterthanallotherycoordinates inthedomain
foundatlocalextrema or attheendpointsofthe
interval
onfunctionhasanabsoluteminimumat apointna ifthe ycoordinateatthatpointislessthananotherycoordinates inthedomain
Foundatlocalextremaorattheendpoints oftheinterval
oa criticalnumberof afunctionoccurswhen icano orteadoes
notexist na isacriticalnumber
Ifreanothencancanis a criticalpoint
s absolute
s
absolute
minimum
É local
s
f
ExI
aFindtheabsolutemaximumandminimumoffans n asu e on theinterval c s s
s
so wemustfindfanso fisimi
L
fanssus as
osusas
fiariza
fearizz
o antainas maximumoccursat
thereforetheabsolute
n n e andis isa
i thecriticalnumbersare theabsoluteminimumoccursat meandis is
naand n e
n sandus
Ex2
aFindandclassifythecriticalpointsoffinsn'tsusout
sowemustfindthecriticalnumberssincethereisnointervalforourdomainthenwedon'thave tobe concernedabouttheendpoints
fanssuitonas
onionas
Functions
Ext
afindtheintervalsof increasinganddecreasing for function
fansans in son tooo o nooo positivethenthefunctionmusthavemo or asymptote
tetra
fansonsonso
Oonasonso
i
o anim o un o t t t
o on out no o
n to no sign t o o
no
Increasing
c o is coin ner
Decreasing
cn einer
Exa
Qproduce a graph with thefollowing properties
fans o c2.2 is o is
nano o 2 ins n o
fans cz.si
fan s a a s
Ex3
a produce the properties that definethe followinggraph
fin o
IX
s fi sis
a
fiasco c3as aas
É a
fearone
himfino
nigrato
r
3.2maximaandminima
localextrema akaextremevaluesakaturningpoints
Alocalmaximumoccursat a point onfansifthe ycoordinatesofallotherpointsinthevicinityarelessthantheycoordinateofthepointten
famchanges frompositive to zerotonegative ietinalso to piano to finaco
or
teaislocatedattheendpointofthegraph
, r e at
oalocalminimumoccursatapointon ifthe ycoordinatesofallotherpointsinthevicinityaregreaterthanthe y coordinate ofthepoint on
fin tens
finchangesfromnegativetozerotopositiveie tineasoto pianoto finaso
or
teaislocatedattheendpointofthegraph
thecollectionoflocalmaximaandlocalminimaarecalledlocalextrema
onfunctionhasanabsolutemaximumatapointma iftheycoordinateatthatpointisgreaterthanallotherycoordinates inthedomain
foundatlocalextrema or attheendpointsofthe
interval
onfunctionhasanabsoluteminimumat apointna ifthe ycoordinateatthatpointislessthananotherycoordinates inthedomain
Foundatlocalextremaorattheendpoints oftheinterval
oa criticalnumberof afunctionoccurswhen icano orteadoes
notexist na isacriticalnumber
Ifreanothencancanis a criticalpoint
s absolute
s
absolute
minimum
É local
s
f
ExI
aFindtheabsolutemaximumandminimumoffans n asu e on theinterval c s s
s
so wemustfindfanso fisimi
L
fanssus as
osusas
fiariza
fearizz
o antainas maximumoccursat
thereforetheabsolute
n n e andis isa
i thecriticalnumbersare theabsoluteminimumoccursat meandis is
naand n e
n sandus
Ex2
aFindandclassifythecriticalpointsoffinsn'tsusout
sowemustfindthecriticalnumberssincethereisnointervalforourdomainthenwedon'thave tobe concernedabouttheendpoints
fanssuitonas
onionas
