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maximum nor a minimum value]

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Limit &amp; maxima

  • 1. it L-1 €6."^nrr & Dc=v fi^A Gr- ^*DU rtl x+E 551d1 cnl 6-ct+f[rro x+5 = E'- QnD+3 oE ao rr3:-*a g A*s: '{^n tc! , ou,; ur2->'ft dr) oo Lralh ryf+ { 5r*) i.q : ef* 4Um x+2 ?(- -b E f t4 0t^ c It frr,d Urn a)2 x-3 A* 44*-4 -* = _41 I / t $ </--*--Z!.^- i i j I
  • 2. t eu.t:h01 '" ,lctrfi ot'. fi"d ,n,..-l , 5c'tucf I cn 1 unn {"4 2_ ->< - 6x+? *-*b 2'/- +8 )t+1< - 2- - -r(^- lO'Y t25 d-* 1- -lr( - I C}, 1- dr ) r) I €- n ,(^- zx - D D t.. U rr) x. -4 Lt rn 'X-t b t Lrm {+6 rm *4) t "rC- I -..IR- 5 s) (12 - - t-T q Lir,^ x-> L f,tm G-, g9GI"-, x- 1 IUrn . {"{ J_ C=+ a oL i I (L r) ( @ )
  • 3. qwcI *nr L-, Jg L/ C*oq a- L* 2 6 -21xL3 zt+i% 'N>3 {h!- ^pf(eabta pa"t + the.t ,m fr.A C9 L,tnq u+-Q ca^A a.o d' ro-oor ed-jr Z- Tr.un, JeU"n "(+-f fc G)_g Jto px"o*eFur6 ns- U''"')L ' ' v(+3 +hq fI f"ad-/ (r6'n 4v Joo= )("t n dr. t th'e "i*ruqo-rl - 7 elDprucaehe6 - ? J** p;'.* rt +he f*-*^ J"e E ; Lir', T "C :t,t 9_ t : Lrrn -i- ai-2' @b) o D_G 5 t-l rn {,r-l tl*> 2' _'i-r.d* 61"i.):' L ;ftu"s {-'i ' utq j"D dr.t l x+*g_ i I i
  • 4. $*frcrL"o fr^A 551 Urn U"n ,roj t"o 3x +6 bx+b _B -71-4 +& Crt - irn 3+ Yrt "(-rtrO 6-, I /x z - Urn ; 214 t oO C=* utD Lir.t '*-11 'O 3+ )* )t m 6 u"o G Urn G1, al-e t,O -''{41"O q +n- im CJ I e A-)1a t, t) 3to In tL e-o &( a- .9 (.- )
  • 5. ) frnA6; $,"n 1*-+ - "O 4*-xB 2e{ al^ Q a_ Urq D at x' 2s+ >L+ eD 1--3x a_ 1 x -L* rL (, -d,rli06'0. Urn :(-+ 1,o -t eo Cu> Gd 1-)d = jnn at+ + "O CL yL - 2:>( + L /rx-? =-t'O !r />( +5 -o e+5 + ') >( +5 +'z( 6 Q*hcrn'. fir.A 4f*h +.6 +6 n -x- ,L'^_ +E Unr) X--+'t'O cfn i [-r tq d-+ t rD Urn {-) + ,0 l I = -l "q )(-, + a.txa rG a,t 1E ,,
  • 6. .= Urn -ll.-l Jr r<l L,t"n Dt-" + oO KAt {ffi+-,3 tr/x3 1+-L +1-p1a e) rtc tl q*efr,'cn: ffi",J ! r ln0 :{,*, +,O 3t)?C+6 *t 551*htcft: 5 .i"rt q(*+"O 3 irn ' d- +"D 6yc #+rf +;d._ 3 --)L +X 6 d_ + 35x) *Jr tjr,t *4 +.,o ,2r E>a 'r 7* a + t-rro zL-q +CI q 5{ 6f , a-t_ +x7-rr R T I z ,tjm "yLjU c--- D + -<' x5 +.4 'to {r I O -6c-
  • 7. q.ft1wa St-aqi, 8bh nsqq qftfr-t (-1,-l), f(x) rrfr.nE< Efr&nTd qtrq r {fi x = /3,".t^ fv3)=-)-2=-4<o "u, {/3)= /r- /r- /r, = %, ,. )t. 'T+t{rqe'qfr Sg{{srs x qqr<rwsfi-+<q oRm ln x.l dx '. 1; , rii r':,i.' . " q"rq qfr6{rcffi ** dY = o <Ftrst dx l+lnx=0,St{t y +x qrqF {.rlx=y,"*#=o+(/"f.,,0 {9{i( x = X f**y c< eifrffi qrcq q<(qfr$.qlr WY Eqlqnq ))s qflq fi rq, *=% R*" y=J5sin;r+3cosx €K qftqa qtce r [Plove that .r,=.6sin-r +3cos_l.has a maximum raltre fbr. ^=%l 1i l y'r q3 qfr'ffi ftfg s( r the minimum value of y = xx ] t ,{ ti .: 3 i d .i
  • 8. e Coo ftqfrqen* TiHTo,q Tfl{r{s vft, f1x ) = r6sin x + 3cos x ' ol{lqtm r'(x)=.'.6.orx-3sinx ,!.rq {frthn-.r< wi f '(x)= 0, slcqt .6.osx -3silrx =0=)tallx = l/-*x=T/. /.!3 /o wt<R f "(x )= -.6rin x - 3cos x {fi x =y6 ,E{F f,, w) ":tr I iu 3cos < 2a <-mftco .1, = <i3, 1 u -r, =.'r +sin 2.r for 0 < x <Ztt ) yN]{r.rs vfr f(x) = x+sin2x vrqr' qErc, f '(^ ) = I * 2cos 2x ,{?rdr {kfu"r <l s&eTrr{R q{I w<Qu r .t' ;i J ) ?ir.' r i.-i:i f) --l2) t -4sln 7r/ -/-1 -2J1<i0.',, : ,;,::*tlr. t l:::::r ir::i:: ::.: =-r:,:jj-.-_:=.t ,ii r VJ sln {s-<l( x =r/oAnro 6sinx+3cosx e<{frtTr{qEqr .q< cqt, ntT*fr Sp, $q r< $,qy€ 0<x,<2n ffik6o : )
  • 9. Fr{ftffi E"r"ttI,.j .'. '-i = 4f; A,nw f(x) .r< "r ftqrn qTfq q<( otfu${1afi Co) Efisfi qfr6fi{ qte q<( Frfr& t P%) = r% *,i,.,(+fi) = r% - J|/ ' ", , i{:.''1, , ,L ).:r i lrrcm [" (*) = 2(.ostr x + cosx) rrrt{ x-0. s"ta f ,'(rt)= 2(.u.1.,0+cos0)= 2(t + {E(1( x=0 f+.nro qq*E {n?.rr.r< -fusn" Tfe €<( l)=4>tl :' ,( l I i i : ! i ri I ii I J E i{ 5 :l T EI Eq'R3"t )e3 _f (*)= et + Zcosx + e-^ rfls.lrd1-1 Efr$:flr [Examine the function f (*)=e. +2cosx+ e-., for val ues] sm !r{ r
  • 10. Frflfw E:1'nn " Ce"e )= 2xr -9xr + 12 -3 qti-RF< .{fr8qtq e<( FTFi6TFT fi.ftr oa"r [Find the maximum and minimum values of the function ofthefunction f(x)=xi-5xt+5x3-ll,,,'r ffi5pve x)=5xt' +18x5 +l5x{ -1g q{irllFcqs qfrrTn e<( qfurh{rd rtER iF-( r I Investigate for what values " of x, +18x5 +l5xa *l0has the maxi s.I r(*): 2x3 - 9xr +tz-31 ufr )tqs,us , Nfrfu rrt(,H8< 'fR6flq, .qr( qfrt$q wi-Elutl ,, iEiro { ft)= x't - 5x{ + 5x j - I [Discuss the maximum arrci nrinimum ,' .. r ] fl<.RE< e< r I Discuss the maximurn and mini function f(x)= x-' -3x' + 3x + I I ,i,;i J: '.:,!:d..:ri: +5 ,{l o) cRXls.6r",.x qnq r I Plpve that .l'(.r)=sin.u(l+cos,r.)' haS, a maximunt, r 'rrltrc lor. ^ =%l i )blb ffirya'i'*u, cr f (x)= x3 -6xr il2x -3 e< ,ffr&{1q TfFrqfle '/*, t Show that f(x)=*l-o*'i ii-t maximum nor a minimuml ,/7 >>/m,t;rs c< f (x )= x3 - 3x 3 + 6x ,/ ' nR [Show rhar f(x)= x: maximum nor a minimum value]