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f(x)=e−1/x2x f(0) = 0
n∈NPn
∀x∈R∗, f(n)(x) = x−3nPn(x)f(x)
Pn
fC∞
Pn
f: [a;b]→R
f0
f0(a)f(b)−f(a)
b−a
f0f0(a)f0(b)
f: [a;b]→RC1
α∈[a;b]f(α) = f0(α)=0
f: [a;b]→RCna1< a2< . . . < an
x0∈[a;b]c∈]a;b[
f(x0) = (x0−a1)(x0−a2). . . (x0−an)
n!f(n)(c)
K
f(x0) = (x0−a1). . . (x0−an)
n!K
n x 7→ f(x)−(x−a1)...(x−an)
n!K
f: [a;b]→RC2f(a) = f(b) = 0
∀x0∈[a;b],∃ξ∈]a;b[, f(x0) = (x0−a)(x0−b)
2f00(ξ)
sup
[a;b]
|f| ≤ (b−a)2
8sup
[a;b]
|f00|
f:R+→R
f0R+fR+
|f0(x)| → +∞x→+∞f
R+

P0(x) = 1 n∈N
Pn+1 = (2 −3nX2)Pn+X3P0
n
n > 0 deg Pn= 2(n−1)
f n ∈N∗f(n)(x)−−−→
x→00
P1= 2
f0(0) = limx→+∞f0(x) = limx→−∞ f0(x)=0
f00 ]0 ; +∞[ ]−∞ ; 0[
P2P2
f00 ±∞
]0 ; +∞[ ]−∞ ; 0[
P3
y f0(a)f(b)−f(a)
b−a
ϕ: [a;b]→Rϕ(x) = f(x)−y(x−a)
ϕ
ϕ(a) = f(a), ϕ0(a) = f0(a)−y < 0, ϕ(b) = f(b)−y(b−a)> f(a)
ϕ0(a)<0ϕ f(a)
α∈]a;b]ϕ(α)< f(a)
ϕ α b
x∈]α;b[⊂]a;b[ϕ(x) = f(a)
a x c ∈]a;x[
ϕ0(c)=0 f0(c) = y
f0
f0(b)f(b)−f(a)
b−af0(a)f0(b)
(an)f
(an)
(aϕ(n))α f(α) = 0
aϕ(n)aϕ(n+1) bn
f0(bn)=0 n→+∞bn→α
f0f0(α) = 0
f(α) = f0(α) = 0
x0∈ {a1, . . . , an}c
x0/∈ {a1, . . . , an}K
f(x0) = (x0−a1). . . (x0−an)
n!K
x7→ f(x)−(x−a1)...(x−an)
n!KCna1, . . . , an
x0n+ 1
n c ∈]a;b[
dn
dxnf(x)−(x−a1). . . (x−an) n!K=f(n)(x)−K
K=f(n)(c)
x0=a x0=b K
g:x7→ f(x)−(x−a)(x−b)
2K
x0
gC2a < t < b ξ ∈]a;b[
g00(ξ)=0
sup
[a;b]
∀x∈[a;b],|f(x)| ≤ (x−a)(b−x)
2sup
[a;b]
|f00|
x7→ (x−a)(b−x)
2x=a+b 2
|f(x)| ≤ (b−a)2
8sup
[a;b]
|f00|

sup
[a;b]
|f| ≤ (b−a)2
8sup
[a;b]
|f00|
f0R+f
f ε = 1 >0
α > 0∀x, y ∈R|y−x| ≤ α=⇒ |f(y)−f(x)| ≤ 1
x∈R|f(x+α)−f(x)| ≤ 1
ξx∈]x;x+α[
|f(x+α)−f(x)|=α|f0(ξx)| |f0(ξx)| ≤ 1/α
|f0(x)| → +∞