Stories About Maxima and Minima

200 Pages·1991·2.04 MB·English

by  V. M. Tikhomirov

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V.M . Tikhomirov
AmericaMna thematiScoalc iety
Mathematical AssociatioofnA merica

MathemaWtoircl• adVl o lu1m e
Storiaebso ut
Maxima
andM inima
V.M . Tikhomirov
Transflraottmeh de Rubsy sian
Abe Shenitzer
AmericMaant hematSioccaile ty
MathematAiscsaolc iaotfiA omne rica
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Translaftreodm tRhues sibayn Shenitzer
1991 OOA07.
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OOA300.0 A350,1 -014.6 -01. 49-01.4 94199-90 3.
LibraorfCy o ngrCeas!>t alo-giinn-gPublDiactaat ion
TikhomirVolv,,t diMm i(rV ladiMmiirk hailovich), 1934-
Storidebso umta ximadn dm inima/MV Tikhomirov
p em Md(t hemdtiwcoarll dI.S SNI 0 55-942I6 ,)
ISBN0 -8:!18-0165-1
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QA306T 55 1990 90-21246
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such permis�hsoiuolbend addrestsoet dh eA ssisttaont th eP ublisAhmeerr,i caMna them.ttSioccdile ty,
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TabloefC ontents
Iondturction
. . . . . . . . . . . . . .i x . . . . . . . . . . . . . . . . . . . . .
ParOtn e. AncieMnatx imuma ndM inimuPmr oblems
Tefih rsstto ry
WhyD oW eS olMvaex iamnuMdmi nimum
Problems?
. . . . . . . . . . . . . . . . . . . . . 3
Teshc eond story
The OPlrdoebslte m-Dido's Problem
.. . 9.
Tehti hsrtdo ry
MaxiamnMadi niiNnma aet ( uOrptics)
. . 1.9
Tefhu osrttho ry
MaxiamnMadi niiGnme ao metry
.. . .2 7. .
Teh fifth story
MaxainmMdai niiAnml ag aenbidrAn an alysis
37
Teshx istyth o r
KeplPerro'b.sl em
. 47
Teh seventh story
ThBer achistochrone
55
Teehg tihsht ory
NewtAoeonrd'ysn aPmriocb.al..le. .m.
. 65
ParTtw o. MethodsS oolfu toifoE nx tremal Problems
Tehn isnttohr v
WhaiatsF unc.t.i.o.n..? . .
81
Teht esnttohr y
Whaiatsn E xtrPermoab.ll. e m?
93
Teeh elnsettrho rr
ExtreFmuan cootffi oVOnansre i able
99
viii CONTENTS
Teht wseytl oftrh
Extroefm aF uonfMc atnViyao rneissa .b l
ThLea grParnignec iple
. . . . .1 0.9 . . . . .
Tehti hernsttteho ry
More PSroolbvlienmg
. . . . . 1.1 9 . . . . . . . .
Tefhu orstteoernyt h
WhaHta ppLeanitetendhr T e h eooEfrx yt remal
Problems?
. . . . . . . 1.4 3. . . . . . . . . . . . .
Tehl satsotr y
MorAec curaDa itseclu.ys.,.s ..i..o.n
. 179
Biiobg.lr aphy
187
Introduction
In ldeiai fitcl oyin ss nteacnettscloshy ao roybs eeps otts h(seoi pbtlei mal)
soluAtt ireomne.nn udmoobusfesu rp c rho balreiimenssc e o naonmidin c s
technIosnlu cocaghisy te.fs ru ieesqn utsletyfro ue lst omora tt hematics.
Inma thetmhasett iuocdfmsy a, x iamnumdmi nimum bpegraonb lems
av elroytn igam geio nf,ca tt,w entcye-nfitavugeroF i.oae rl s o tnigm e
thweernreoe u noirfwma oyfts a cpkrloibnflgrofie nmdsei xntgrT ehmea .
firgsetn meertaholiof nd vse isoatnnsi dgo alotueftx itorpner bmolaewlme sr e
create3d0y 0e aaabrgoasoutt ,th t ei omfte h fero matofim oant hematical
analysis.
Theibnte ccalmteeha caret r stpaeiocnpi tailm pirzoabtlieaom ns play
crurcoiiltanehl n ea tsucriaeSlnp ceecsii.wfit ac fsauo lntldhy ma,at n y laws
of enc aabtnedu errfirmvos eod- c"avlalreipdar tiinocAnicapcllo ersd.i"n g
ttoh persien gciivapenlcnyeo sl,lo efac dtmiiosns ibwlheda itsm totiino­ns,
guitshhaeec stm uoatloi fom nea c hasnyisoctrael lmio ,fge hltec,at ricity,
fluagi adas,n,s d o iotsnh i,at t maxmiimniiizzcmeeessr q tuoaarin nt ities.
Somceo ncerxettrpeerb molaewlmh soc,so en dteernfitrv otemhsn e a tural
sci(etnbhcreea sc hipsrtoobNclehewrmtop,onr neo' basln oedtm h,we errse) ,
posetdhe enao dtft hsee tveeencnetnyht.T u hre tnose oeltdvh eea sse ,
welmla noayts ph reorb olgfee mosmm eetcrhyaa,nn pdih cysls,eit dco s ,t he
creoaftn ieaobw nr anch ofa nmaatlthychasaetimtm seoa k tnbioecaw asnl
thcea locvfua lruisa tions.
Thien tednesvievleoo ftp hmceea nltco ufvl aiurastc ioonuntesfid ron
abotuwctoe ntMuarnioyef s fi.tn hesesc ti eonft eitishgtehs at nede nth
nienteceenntthtu oroikie nsp patrrhotiac sne dsts,h, be be yg ionftn hiinsg
cenitsuter eyam,ise tf dh eye xhhaatduh set etdo pic.
Buittt nu erodut th tahtwi asns o tc atsThehen.e e eodfps r acleti.if cal
espeicenic aolnlaoynmt die ccsh ngoarlvioetsg noeye p ,wr obtlhceaomtus l d
nobtes olbvyte hodel m de thOondhesa t.dao d vaInwtca ens.e cessary
tcor eana etfiwee olfmd a theamnaatlikycnsaoaiwl"ssn c n, ov aenxa lysis,"
invotlhsveti uondfgy c fonuncvteaixnoc dno sn vexp reboxltermesm.a l
ix