7/23/2019 Comment montrer que l'quation f(x)=0 admet une unique solution ?

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JlfbnClgltns&Fditliult

Gtbd`nsntAiiacns

bttp?//nxds8gatb.jrnn.jr/

Gditrnr qun cquatldi j(x) 7 5 a`gnt uin uilqun sdcutldi.

Xappnc `n fdurs

Pbdrgn(PZL)

Vdlt juin jdiftldi `fiiln nt fditliun sur ui litnrvaccn L`n Xnt ant h`nux rncs `n L.

Udur tdut rnc m ]j(a)= j(h)W, lc nxlstn au gdlis ui rnc ]a= hWtnc qun j() 7m.

Xngarqun ? Cn tbdrgn `ns vacnurs litnrg`lalrns (PZL) nst ui tbdrgn `nxlstnifn nt idi `uilflt ! Di utlclsnra`dif cn fdrdccalrn fl-`nssdus pdur `gditrnr cuilflt `ais cn fas partlfuclnrj() 7 5.

Fdrdccalrn(Pbdrgn `n ca hlonftldi)

Vdlt juin jdiftldi `fiiln, fditliun nt strlftngnit gdidtdin sur ui litnrvaccn ]a= hW`n X.Vl j(a)j(h)3 5, acdrs cquatldi j() 7 5a`gnt uin uilqun sdcutldi `ais Wa= h].

Xngarqun ? ant hpnuvnit trn kaux . @ais fn fas di tu`lnra ca clgltn.

Gtbd`n

Udur appclqunr cn fdrdccalrn prf`nit lc jaut vrlfinr `ais cdr`rn cns pdlits sulvaits ? Ca jdiftldi nst fditliun sur clitnrvaccn L7 ]a= hW. Ca jdiftldi nst strlftngnit gdidtdin sur fnt litnrvaccn. Nifii, lc jaut hlni sr qun 5 ]j(a)= j(h)W.

Nxnrflfn 1 (Asln 8516)

Di fdisl`rn ca jdiftldi `fiiln sur Xpar ? (x) 7 8(x 1)nx + 1.

1. Facfucnr cns clgltns `n ca jdiftldi ni nt +.8. tu`lnr cns varlatldis `n ca jdiftldi sur X. Urflsnr ca vacnur `n (5).6. @gditrnr qun cquatldi (x) 7 5 a`gnt nxaftngnit `nux sdcutldis `ais X.0. Di idtn ca sdcutldi ikatlvn `n cquatldi (x) 7 5nt ca sdcutldi pdsltlvn `n fnttn quatldi.

cal`n `uin facfucatrlfn, `diinr cns vacnurs `n nt arrdi`lns au fnitlgn.

Nxnrflfn 1

1. clgx

8(x 1)nx 7 clgx

xnx 7 5, `aprs cns

Frdlssaifns Fdgparns.@dif par sdggn `n clgltn ? clg

x

(x) 7 1.

clgx+

8(x 1)nx 7 clgx+

xnx 7 +.

@dif par sdggn `n clgltns ? clgx+

(x) 7 +.

8. nst `rlvahcn sur X fdggn sdggn `n jdiftldis`rlvahcns sur X? (x) 7 8nx + 8(x 1)nx 7 8xnx.

Fdggn di a pdur tdut x X

? n

x

95, cn slkin `n(x)nst fncul `n x. @dif surW= 5] , (x)3 5 ?ca jdiftldi nst `frdlssaitn sur fnt litnrvaccn nt surW5=+] , (x)9 5 ? ca jdiftldinst frdlssaitn surfnt litnrvaccn. @d cn tahcnau `n varlatldis ?

x

(x)

5 +

5 +

11

11

++

5

5

6. Vur W= 5W nst fditliun.

Vur W= 5W nst strlftngnit `frdlssaitn.

5 ]()= (5)W 7 ]1=1W.@aprs cn fdrdccalrn `u tbdrgn `ns vacnurs litnr-g`lalrns (du tbdrgn `n ca hlonftldi), lc nxlstn uirnc uilqun W= 5]tnc qun () 7 5.Cn ggn ralsdiingnit sur clitnrvaccn ]5=+]gditrn qulc nxlstn ui rnc uilqun W5=+] sd-cutldi `n cquatldi() 7 5.

Fdifcusldi ? cquatldi(x) 7 5 a`gnt nxaftngnit`nux sdcutldis `ais X.

0. Ca facfucatrlfn `diin suffnsslvngnit ?

(8) 5, 12nt (1) 5, 0;,`dif 83 3 1 =

(1, ;) 5, 516nt (1, :) 5, 5>,

`dif 1, ;3 3 1, : =

(1, :2) 5, 551nt (1, :;) 5, 55>,

`dif 1, :23 3 1, :; =

(1, :;