ࡱ> 7 B[bjbjUU X7|7|+l"""8#|#}:$:$"\$\$\$'''^|`|`|`|`|`|`|$; [<|'U'@'''|H\$\$q}HHH', \$\$^|H'^|HHH2zv:{\$.$ P]ό."Gx.:{$}0} 0, f(x) ( f(-b/2a), deci f admite un minim pe R; dac a < 0, f(x) ( f(-b/2a), deci f admite un maxim pe R; Fie funccia f : R(R, f(x) = ax + bx + c, a ( 0. Dac a > 0, minimul funcciei f pe R este  /4a = f(-b/2a) iar punctul de minim este  b/2a. Dac a < 0, maximul funcciei f pe R este  /4a = f(-b/2a) iar punctul de maxim este  b/2a. Sensul de variacie (intervalele de monotonie) Exemplu. Vom studia intervalele de monotonie ale funcciilor g _i h definite pe R, g(x) = (x - 2( + 3 _i h(x) = -(x + 3( + 1. Avem: g(x) = x + 1, x ( 2 h(x) = -x - 2, x ( -3 -x + 5, x < 2 x + 4, x < -3 Funccia g are minimul n punctul x = 2 (g(x) ( g(2), adic (x - 2( + 3 ( 3 sau (x - 2( ( 0, ( x ( R) _i este strict descresctoare pe (-"; 2], strict cresctoare pe [2; + "). Funccia h are maximul n punctul x = -3 (h(-3), ( x ( R) _i este strict cresctoare pe (-"; -3], strict descresctoare pe [-3; + "). Fie funccia f : R(R, f(x) = ax + bx + c, a ( 0. Dac a > 0, atunci f are minim pe R _i vom arta c se comport analog cu funccia g. Dac a < 0, atunci f are un maxim _i vom arta c se comport analog cu funccia h. Fie u, v ( R, u ( v. Raportul de variacie asociat lui f _i numerelor u, v este (f(u)  f(v))/(u-v) = (au + bu - av - bv)/(u - v) = a(u + v) + b S studiem semnul raportului de variacie n cazul a > 0. Dac u, v ( (-"; -b/2a], atunci din u ( -b/2a, v ( -b/2a, rezult u + v ( -b/a sau a*(u + v) + b ( 0. Avem a*(u + v) + b = 0 ! u = v = -b/2a, situacie care nu poate avea loc, deoarece prin ipotez u ( v. Rezult a*(u + v) + b < 0, deci n cazul a > 0, f este strict descresctoare pe (-"; -b/2a]. Dac u, v ( [-b/2a; + "), deducem analog a(u + v) + b > 0, deci n cazul a > o, f este strict cresctoare pe [-b/2a; + "). n mod analog se studiaz cazul a < 0. Fie funccia f : R(R, f(x) = ax + bx + c, a ( 0. Dac a > 0, atunci funccia f atinge minimul n punctul  b/2a _i este: strict descresctoare pe (-"; -b/2a], strict cresctoare pe [-b/2a; + "); Dac a < 0, atunci funccia f atinge maximul n punctul  b/2a _i este: strict cresctoare pe (-"; -b/2a], strict descresctoare pe [-b/2a; + "). Reprezentarea grafic a funcciei ptratice Considerm un reper n plan. Reprezentarea grafic a funcciei f : R(R, f(x) = ax + bx + c, a ( 0, adic mulcimea punctelor M (x, y) ale cror coordonate verific relacia y = ax + bx + c, este o curb numit parabol. Vom nota aceast curb prin (f. Condicia ca un punct din plan s aparcin curbei (f Fie M (p, q) un punct din plan. Punctul M (p, q) aparcine curbei (f dac _i numai dac q = f(p), deci q = ap + bp + c. Dac q ( ap + bp + c, atunci (f nu trece prin M (p, q). Punctul V(-b/2a, -/4a) aparcine curbei (f pentru c -/4a = f(-b/2a) _i se nume_te vrful parabolei. Exemple A (2, -3) ( (f ( -3 = 4a + 2b + c; B (-1, 0) ( (f ( 0 = a - b + c. a + b + c = 0 ( C (1, 0) ( (f ; a - b + c = 2 ( D (-1, 2) ( (f. Axa de simetrie a curbei (f Fie o funccie f : R(R. Dreapta de ecuacie x = h este ax de simetrie pentru curba reprezentativ a funcciei f dac f(h + x) = f(h - x), ( x ( R. Dac are loc relacia f(-x) = f(x), ( x ( R (avem h = 0), atunci curba este simetric n raport cu axa Oy _i f este o funccie par. Funccia ptratic f : R(R, f(x) = ax + bx + c, a ( 0 verific relacia f(-b/2a + x) = f(-b/2a - x), ( x ( R. ceea ce se poate demonstra direct sau utiliznd forma canonic. Curba reprezentativ a funcciei f : R(R, f(x) = ax + bx + c, a ( 0 admite ca ax de simetrie dreapta de ecuacie x = -b/2a. n particular, dac b = 0, f(x) = ax + c este o funccie par. Interseccia curbei (f cu axele de coordonate Se _tie c Ox = {(x, y)(x ( R, y = 0}, iar Oy = {(x, y)( x = 0, y ( R}. Rezult: M (x, y) ( (f ( Ox ( y = ax + bx + c _i y = 0 ( ax + bx + c = 0 _i y = 0. M (x, y) ( (f ( Oy ( y = ax + bx + c _i x = 0 ( x = 0 _i y = c. Dup cum  = b - 4ac este strict pozitiv, nul sau strict negativ, ecuacia ax + bx + c = 0 are dou solucii reale x1 _i x2, o singur solucie real x = -b/2a, respectiv nici o solucie real. n consecinc: dac  > 0, (f ( Ox ={A(x1, 0), B (x2, 0)}; dac  = 0, (f ( Ox ={A (-b/2a, 0)}; dac  < 0, (f ( Ox =. De asemenea, reprezentarea grafic a oricrei funccii ptratice intersecteaz axa Oy, _i anume (f ( Oy = {C(0, c)} Pentru c = 0, curba asociat funcciei f(x) = ax + bx trece prin originea reperului. Trasarea curbei reprezentative a unei funccii ptratice Pentru a reprezenta grafic o funccie ptratic f : R(R, f(x) = ax + bx + c, a ( 0 adic pentru a trasa curba sa reprezentativ (f , numit parabol, se procedeaz dup cum urmeaz. Se determin _i se nscriu ntr-un tabel de variacie coordonatele unui numr finit de puncte ale curbei (f , printre care este bine s se afle: punctele de interseccie ale curbei cu axele reperului; punctul V (-b/2a, -/4a), vrful parabolei. Se reprezint aceste puncte ntr-un reper al planului, ales astfel nct s putem figura toate punctele. Se unesc punctele reprezentate printr-o curb continu, cinnd cont de: Intervalele de monotonie ale funcciei ptratice; Simetria curbei (f n raport cu dreapta de ecuacie x = -b/2a. Cu ajutorul curbei astfel obcinute, putem obcine o bun aproximare a coordonatelor oricrui punct al curbei (f. Semnul funcciei ptratice Cazul  > 0 x-" x1 x2 + "f(x) semn a 0 semn contrar a 0 semn a Cazul  = 0 x-" -b/2a + "f(x) semn a 0 semn a Cazul  < 0 x-" + "f(x) semn a  Partea aplicativ S se construiasc tabelul de variacie _i reprezentarea grafic a urmtoarei funccii f : R(R, f(x) = x - 4x + 3 ( > 0, a > 0) x-( 0 1 2 3 + (F(x) 3 0 -1 0 x - 2x  8 = (x - 1) - 9 f.c. = a[(x - b/2a) - /4a] x - 2x - 8 = [(x - 1) - 36/4] = (x + 1) - 9  = 4 + 32 = 36 f : R(R f(x) = px - (p - 6)x + p - 1 = min x, x = 5/2 p > 0 y (min) = f(5/2) = -/4a f(5/2) = p(5/2) - (p - 6)*5/2 + p - 1 = -3/2p - 25/4p + 14  = p4  12p + 36  4(p - p) = = -12p - 4p + p4 + 4p + 36 = -/4a = (12p + 4p - p4 - 4p - 36)/4p -p4 + 4p + 12p - 4p  36 = 4p(-3/2p + 25/4p + 14) -p4 + 4p + 12p - 4p  36 = -6p + 25p + 56p -p4 + 4p + 12p + 6p - 25p - 60p - 36 = 0 -p4 + 10p - 13p - 60p - 3 = 0 p4 - 10p + 13p + 60p + 36 = 0 P(-2): 16 + 80 + 52 - 120 + 36 = 0 Se descompune polinomul din stnga ecuaciei, n factori de gradul II _i se egaleaz cu factorii cu 0. Ecuacia se scrie (p - 5p - 6) = 0 ( p - 5p - 6 = 0 ( p1 = 6; p2 = -1 f : R(R f(x) = 2x - 3x + 1 f(x) ( [-1/8, + (), (() x ( R a = 2 ( a > 0 ( min minf = -/4a = -(b - 4ac)/4a = -(9 - 8)/8 = -1/8 f : R(R f(x) = x - 8x + 12 ( Ox: y = 0 ( x - 8x + 12 = 0  =64  48 = 16 ( ( = 4 x1 = (-b + ()/2a = (8 + 4)/2 = 6 (A (6, 0) x2 = (-b - ()/2a = (8 - 4)/2 = 2 ( B (2, 0) ( Oy: x = 0 ( y = 12 ( C (0, 12) a = 1, a > 0 ( xmin = 8/2 = 4 ymin = -/4a = -1 ( V (4, -1) x-1 0 2 4 6 7f(x)21 12 0 -1 0 5 NSEMNRI PAGE  PAGE 10 "$:<j$&< P b < Jfh "HνίίίΈwi5B*CJ\mHphsH!56B*CJ\]mHphsH j5CJ\mHsH j6CJ]mHsH56CJ\]mHsH6CJ]mHsH>*CJ CJmHsH5CJ \mHsH5CJ\mHsH56CJ \]mHsHmHsH 5CJ\jUmHnHsHuCJ%   "&(*,.02468:> [@[>@BDFHJLNPRTVXZ\^`bdfhj "$&(*$a$*:<         " $ & ( * , . 0 2 4 6 8 : < > @ $a$@ B D F H J L N P ` b d <  p b JL 3 7`7$a$fhh~0TH$ & F 3^`a$ ^ & F S^S & F Gr^`r7`7 & F w 3^`H^NR*,246<b|~.26BDPRln|~դÓՈՈxnxnanananan j6CJ]mHsH6CJ]mHsH6B*CJH*]mHphsHB*CJmHphsH! j6B*CJ]mHphsH jB*CJmHphsH! j5B*CJ\mHphsH6B*CJ]mHphsH6]B*CJmHphsH!56B*CJ\]mHphsH5B*CJ\mHphsH&RTVZj>@RTHBZ\|r"$*,>@`bõÆxõ j"B*CJmHphsH jB*CJmHphsH6]B* CJmHphsH6B* CJ]mHphsH j jB*CJmHphsHB*CJmHphsH5B*CJ\mHphsH6CJH*]mHsH6CJ]mHsH6B*CJ]mHphsH/HJhRB|vpr 7`7 & F [ 7^`7 3 & F S^`S & F 3 $ 37`7a$ 37`7 & F 3S^`S 3bjlnp~ (*02fh  6 ^ h v !!\!^!!!!!!!""""""""# jB*CJmHphsH jB*CJmHphsH jB*CJmHphsH j"B*CJmHphsH6B*CJ]mHphsH jB*CJmHphsHB*CJmHphsH? !!"J###"$$%%%&F''' 7`7 3B & F ;^`; & F   & F  $ 7`7a$ 7`7 & F [ / 7^`7##2#<#b#d#l#n#p#r############$$:$H$\$^$$$$$$$%%:%<%T%h%j%t%%%%&&&&&&&&''<'>'ͿͱͿͣͿͿͿͿͿͿͿͿͿ͕ j j|5B*CJ\mHphsH jB*CJmHphsH jB*CJmHphsH6B*CJ]mHphsHB*CJmHphsH jB*CJmHphsHB*CJmHphsH6B*CJ]mHphsH7>'F'''''''( ((( ("(0(2(<(>(@(B(J(L(R(T(() ))P)R)X)Z))** *"*(***V*X*****++0+2+B+L+++++++,,P,R,^,`, j j j6] jB*CJmHphsH j"B*CJmHphsH j|B*CJmHphsH jB*CJmHphsH6B*CJ]mHphsHB*CJmHphsH>'''''''''()))`*+N,,F-/000@1`23  & F  $ a$  7`7`,Z-\-------. ...>/@///80:0000111 16181v1z1112222333T4X4^4`44444z555Ⱥ jB*CJmHphsH jB*CJmHphsH6B*CJ]mHphsHB*CJmHphsH5B*CJ\mHphsHB*CJmHphfsH jB*phf jB*phf6B*]phf B*phf6] j j j13335586*77l8|89999: & F `  & F 3^` 3 7`7 & F  / n^`n O7`7 & F O|`| 7`75555264666F6N6666666687:7f7h7l77777788H8l8|88888888888888"9$989:9<9>9B9b9d9z9|9~9ȷ֛֛֩֍֍֛֍֛ jB*CJmHphsH jB*CJmHphsH jB*CJmHphsH! jC6B*CJ]mHphsH6B*CJ]mHphsHB*CJmHphsH6B*CJH*]mHphsH jCB*CJmHphsH7~99999999999*:::::::::: ;;;;(;*;0;2;;;;; <<<<< <L<N<v<x<<<<<<<D=ȷ֛֩֍֛֍֛֩֍ jB*CJmHphsH jB*CJmHphsH j"B*CJmHphsH jB*CJmHphsH! jC6B*CJ]mHphsH6B*CJ]mHphsHB*CJmHphsH6B*CJH*]mHphsH jCB*CJmHphsH1::;v<<B=D=<>>>???T@@XBvBBCHC & F  7`7  & F  ```^7`7  7`7$ 7`7a$D===========r>t>>>>>>??F?H?L?N??????????????漫zl^ jB*CJmHphsH jCB*CJmHphsH jB*CJmHphsH j j|B*CJmHphsH6B*CJH*]mHphsH! jC6B*CJ]mHphsH6B*CJ]mHphsH jB* CJmHphsH jB* CJmHphsH6B* CJ]mHphsHB* CJmHphsH$??@@f@h@j@l@p@r@t@|@~@@@AAAABBBBBBBBBBBBBB0C2C4C6C8CHCDDDD|Dʺʺʺʺy CJmHsH jCJmHsH6CJH*]mHsH jCCJmHsHB*CJH*mHphsH jB*CJmHphsH6B*CJH*]mHphsH jCB*CJmHphsH jB*CJmHphsHB*CJmHphsH jB*CJmHphsH+HCD0DDDLEFGHHHrIJdJJKKKLLL$$ &#$/Ifa$^ & F & F  ` & F S^`S & F7`7$a$ 3|D~DDJEEEEEEEEENFPFTFG"GGGGJJJKKKKK0L2LjLnLLLLLLLL MM.MMMMMMMMMM*N,N4NPNpNNNBOFOLONOTOVOǼǼǼǼ>*CJmHsHB*CJmHphsHCJH*mHsH6CJH*]mHsH jCCJmHsH jCJmHsH jCJmHsH5CJ\mHsH CJmHsH6CJ]mHsH?LLLL MMMMMMweVwPPPP^$ &#$/If$$ &#$/Ifa$t$$Ifl0n& 6 064 la $ &#$/If M.M0M2M6MMMMMZK$&#$/Ift$$Ifl0nF 6064 la$ $&#$/Ifa$$$&#$/Ifa$^MMMMMMMMM(Nn[ $&#$/If$$&#$/Ifa$^t$$Ifl0nF 6064 la (N*N4NfNhNjNlNnNpNNN|xigggge_7`7$&#$/If$$&#$/Ifa$t$$Ifl0n O 6064 la NOOOOOOOP P PPPolom$$IfTl0M 1 064 la$If $$Ifa$ & F w S7^`7 VOOOOOOOQQQQ,Q.QQQQQhRlRRR S$SNSTSSS&T,TTTTTTTnVpVVVVVVVVVVVVVVVWWWW$W&W.W0W6W:WFWHWVWXWnWrWWW j"CJmHsH jCJmHsHCJH*mHsH jCJmHsHCJH*mHsH jCJmHsH6CJ]mHsH jCJmHsH CJmHsHDPPPPPPPPTPPPQQ QQQQR^RRRDSST~T`7`7 & F g`g^ & F Sg`g~TTUZUnVVVVV6WbWWWWXRXXX4YYY*ZnZrZZ$If $If7^7^ & F g`g`WWWWWXX,X.XXXXXXXXX Y"Y*CJ$\mHsHCJH*mHsH jCJmHsH jCJmHsH jCJmHsH6CJ]mHsH jCJmHsH CJmHsH7ZZZZZZZ[[[ [ysi_ 7`7 ^^7`77^7$If $$Ifa$k$$Ifl0n 064 la7 [ [[ ["[:[<[>[@[B[&`#$ 37`7 2[6[8[:[@[B[B*CJH*mHphsH0J j0JU0JmHnHu+ 0&P . 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