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Transcription

1 L ECTURE N OTES 4 XIU ND INIU VLUES PRT lrgest 06 I 3 OTIVTING E XPLES * I He B obtins! g growing Keep s! mximum No lrger 4 mximumo4tv 7 or There Wh Wouldnt Y th c mximize our proit our 5 two pprox 5 x~~ which like There s dierent x~~ plces lues imize n 22 Th plces There 3 ou between s one obtins Ft locll 09 Smllest or would Or he 09 pproximel s xo consumption uel D EFINITIONS Let x be with do D let c be n x in D Then c n mximum i C z x ll in x 2 n i th c 3 locl mximum i 4 locl i c E smllest ccltlx c ll IND X th ll X Close dose lrgest Y c Vlupeossible possible z D u#! } Do ou see dierence 4 ximum inimum Vlues prt

2 RE WE LL ON THE SE PGE Wh st o ceg mximum niml vegetble number point x erl BER x it lws not point output 2 Cn hve me thn ONE mximum mxtnin The hve cn one locl dierent mximums j s x Yes mximum multiple m multiple hve cn most Q C no most one s hhh 3 Cn hve neir mximum n Sure Some 3 include exmples hx gle 4 Looking our erlier pictures wh st o plces do mximums s pper th where round turn smooth ~ turnround shrp where ils x ext o P OWERFUL T OOL # The Extreme Vlue Theem Given x such th 2 x do restricted Is continuous [ on intervl closed b] [ b] won _ pictures nt F C x m Cz b m xc cz ~ x turnround mt o n b end intervl n x gurnteed hve mximum intervl 2 x m m [ b] d b turnround endpoint 4 ximum inimum Vlues prt

3 P RCTICE P ROBLES F ech with designed region sketch grph detere its mximum its i ext I do not ext explin wh Extreme Vlue Theem not violed x x2 on [ ] 2 gx cx thn Y x xo o # on [0 2] Itl 2 x To * g6 [ 02 ] intervl on continuous not 3 hx cos x on 0 4 kx x + 3 on [ 2 3] q * µ t 2 D m I x 2 P OWERFUL T OOL #2 Criticl Points in Deinition criticl number o x n X c such th eir c o b c do k o undeined YtneFI#se&Hmxie4x3 Is Wh do we c bout criticl Criticl numbers Note we must check becuse Just give necessril Exmple c x x NOT s end criticl where o s i re does T µ qz In x s s n not c men Summr tu# djoins mm x 4 s s end criticl cresponding ind m we point D Criticl s x end 3 4 ximum inimum Vlues prt

4 z ORE PRCTICE PROBLES F ech below ind ll criticl b identi ll locl mximums locl s i n where c identi ll mximum s where I no do explicitl sted ssume ou using nurl do o s written You expected provide cler coherent explnions o how ou deduced our nswers 0 He# x grphed below i3 E undeined ± rluesoyoyiomninlw2iexr xq YoItIh0lQEIYiiIhsi ; 2 obs x 7 5 2x + i 0 pple x< 3 gx lnx on 0 2] 2 x 4 2x i pple x pple 3 Y^ criticl Y criticl ^2 X locl s F DbD # µt s locl mx ln2 Fg gµ II Dc s % locion 2 l#tmonns/g2ny * locl VY s L mxs t# nx s ln22 mx 3 x x i 2 x I 4 4 ximum inimum Vlues prt

5 4 t + t3 + t2 5 hx + 4t3t3E+2t t4 i t4t2t3t+2$0 So t using numbers locl x s s Yl Yimiionn Yointmon x X mx mg#cee4i88i xo 2x3 on [0 4] Cx L 6 x x criticl o x 8g 4 µ x4 545 grph in gg#ogx4+j2k4k7xi /8gµ_ sketch locl 0 criticl Grph ct out #i Tt Criticl end i o 42 hlh±sx% x roots no x4/5 x 3 X 7 gx x + x3+x 3 g x on [02 4] DIE Cx C +! Notin do grphs criticl do [ 04 goopsx x0 no±x I Skip xixtnnoetsinpeciied2o tn#hedsendpo;nsgrpksketgh53j**9_f4t~ritiip+ 0 & locl mx 3 locion x h3 5 2 X YgYnYYIlo5;2z/4j25 3 lyl+oinns# ± locl x3 xo CO mx s ximum inimum Vlues prt *

6 Cnt * ok help msel! Grph 4 xtxt on [0 Y tt t Noe d [ 02 4 ] q Noted I time permits Closed Intervl ste lgithm explicitl Wh It does does NOT tell ou