8/6/2019 Comment montrer qu'une fonction est drivable en un
point ?
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LhaboMrhvethfk
ItbfmosotEkkecos
bttp7//oxfs0ie
tb.lroo.lr/
Ifktror quuko lfkathfk ost mrhve`co ok uk pfhkt.
Peppoc mo afurs
Mstkhthfk
Fk mht quo l ost mrhve`co ok e sh, ot soucoiokt sh, l(e+b)
l(e)
b
emiot uko chihto stkho quekm b tokm vors>.
Co kfi`ro ost ecfrs eppoc kfi`ro mrhv mo l ok e, ot kft
l(e).
Poierquo 7 Aotto mstkhthfk ost uthchso pfur rotrfuvor cos
lfriucos mo mrhvethfks usuoccos. ^fur ifktror quuko
lfkathfk ost mrhve`co ok pfhkt fk uthchsore pcutt ce mouxhio
mstkhthfk.
Mstkhthfk
Fk mht quo lost mrhve`co ok e sh, ot soucoiokt sh, l(x) l(e)
x e emiot uko chihto stkho quekmx tokm vors e 4 ecfrs 7
chixe
l(x) l(e)x e 9l
(e)
ItbfmoHc oxhsto moux aes euxquocs kfus sorfkt afklrfkts pfur
mtorihkor ce mrhve`hcht. Meks tfus cos aes, hc leut afiiokaor
per aecaucor co reppfrt mo ce mouxhio mstkhthfk, ot co iottro
sfus uko lfrio mfkt ce chihto sore leahco mtorihkor.
Oksuhto fk mhsthkjuo cos moux aes 7
Ce chihto mu reppfrt tokm vors, ot meks ao aes ce lfkathfk kost
pes mrhve`co ok e. Qh chi
xe
l(x) l(e)x e 9 chixe+
l(x) l(e)x e 9 chixe
l(x) l(e)x e , ecfrsl ost mrhve`co ok e.
Oxorahao 5
Qfht lce lfkathfk x |x0 5|. l ost-occo mrhve`co ok 5 6 Gusthstor
ce rpfkso jrepbhquoiokt, puhs per aecauc.
Oxorahao 0
Qfht lce lfkathfk x x shk x.5. Gusthstor quol ost mrhve`co sur
Y>4 +T ot aecaucor l(x) pfur tfut x mo aot hktorvecco.0.
Mifktror quol ost mrhve`co ok >, ot prahsor l(>).
Oxorahao 5
Alemiot ok5moux moih-tekjoktos mo pokto 0 jeuaboot +0 mrfhto. Qh
l ost teht mrhve`co ok 5, Al emiot-treht uko souco tekjokto. Fk
pout ecfrs afkacuro quo l
kost pes mrhve`co ok 5.^fur tfut x T5 4 +T 7 x0 5 >, mfka
l(x) 9x0 5.Fr ce lfkathfk x x0 5 ost mrhve`co sur P, mo mrh-vox
0x. lafkahmo surT54 +
T evoa x
x0
5,
mfka 7 chix5+
l(x) l(5)x 5 9l
(5+) 9 0.
Mo lefk ekecfjuo, pfur tfut x T5 4 5Y, x0 5 >,mfka l(x) 9(x0
5) 9 5 x0. lafkahmo sur T5 4 5Yevoa x 5 x0, quh e pfur kfi`ro mrhv0
ok 5,mfka 7 chi
x5
l(x) l(5)x 5 9l
(5) 9 0.
Fk e mfka 7 chix5
l(x) l(5)x
5 9 chi
x5+
l(x) l(5)x
5 , ao
quh prfuvo quo lkost pes mrhve`co ok 5.
Oxorahao 0
5. x x ost mstkho sur P+ ot mrhve`co sur P+4meutro pert, ce
lfkathfk shkus ost mstkho ot mrhve`co sur
P. lmfka ost mrhve`co sur P+ afiio prfmuht mo lfka-
thfks x x ot x shk x mrhve`cos sur P+.^fur tfut x P+, l(x) 9
shk x
0
x+
x afs x.
0. ^fur tfut x ;> 7
l(x)
l(>)
x > 9
x shk x
x 9 x shk x
x .
Fr fk seht quo chix>
x9 > ot chi
x>
shk x
x 9 5. Ehksh,
chix>
l(x) l(>)x > 9 >. Fr l
(x) 9
x
shk x
0x + afs x
sfht chix>
l(x) 9 chix>
x
5
0+ >
9 >.
Fk e ecfrs l(>) 9 chix>
l(x) l(>)x > , ao quh prfuvo per
mstkhthfk quo l ost mrhve`co ok > ot l(>) 9 >.
Ettokthfk 7 Hah fk ktumho pes ce chihto jeuabo aer
occo koxhsto pes ! l ost mrhve`co ukhquoiokt pfur cos
rocs strhatoiokt pfshthls, mfka fk ko seht rhok mo ao quh
so pesso jeuabo mo >.
