IV Dérivabilité sur un intervalle

f0f:I→Rf
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f0(x0)
f(x)x x0
f(x0)f(x1)x1
x0
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a < b f : [a, b]→R
f[a, b] ]a, b[f(a) = f(b)
c∈]a, b[f0(c)=0
[a, b]f[a, b]f
[a, b]x0x1∈[a, b]f(x0)≤f(x)≤f(x1)
x∈[a, b]
x0x1]a, b[c=x0x1f
c]a, b[c f0(c)=0
x0x1[a, b]f(a) = f(b)
f(x0) = f(x1)f(x0)≤f(x)≤f(x1)f[a, b]
f[a, b]f0(c) = 0
c∈]a, b[
a < b f : [a, b]→R
f[a, b]
]a, b[c∈]a, b[f(b)−f(a) = f0(c)(b−a)
θ=f(b)−f(a)
b−ac∈]a, b[f0(c) = θ
g: [a, b]→Rx7→ f(x)−θ×(x−a)θ×(x−a)
[a, b]g
g(b) = f(b)−f(b)−f(a)
b−a×b−a=f(a) = g(a).
c∈]a, b[g0(c) = 0 g0(x) = f0(x)−θ x ∈]a, b[
f0(c) = θ
IRf:I→Rf0(x)≥0x∈I
f0I
a b ∈I a ≤b f(a)≤f(b)a=b a < b
f a b c ∈]a, b[f(b)−f(a) = f0(c)(b−a)f0(c)≥0
b−a > 0f(b)−f(a)≥0

f(a) = f(b)f0(c)=0
f a b a b
a b (T)f
(C)a b c f0(c)
θ=f(b)−f(a)
b−a
T
C
b
a
f(a)
f(b)
a b |cos c| ≤ 1c
|sin b−sin a|=|cos(c)(b−a)|=|cos(c)|×|b−a|≤|b−a|.
|sin b−sin a| ≤ |b−a|a b b =x a = 0 |sin x| ≤ |x|
b
a b ≥1|ln b−ln a| ≤ |b−a|
1x0
f x0
f(x) = f(x0) + f0(x0)(x−x0)+(x−x0)(x−x0),
lim0= 0 n
n
f:I→RI⊂Rn
f(n)n∈Nf(1) =f0f(n)= (f(n−1))0f(n−1) f
Cnn f C∞n f (n)
n
C∞
Cnexp(n)= exp n
cos(2n)= (−1)ncos,cos(2n−1) = (−1)nsin,
sin(2n)= (−1)nsin,sin(2n−1) = (−1)n+1 cos .
n(1/x)(n)= (−1)nn!/xn+1 x∈R∗n∈N∗
f:I→R
I⊂Rn f(n)(x0)x0∈I  :I→R
lim0= 0 x∈I
f(x) = f(x0) + f0(x0)(x−x0) + f00 (x0)
2(x−x0)2+··· +f(n)(x0) n!(x−x0)n+ (x−x0)n(x−x0).
f n x0
n x0

n= 1
n= 1 n= 2
f x0
ϕ(x) = f(x)−f(x0)−f0(x0)(x−x0)−f00 (x0)
2(x−x0)2,
ϕ(x)/(x−x0)20x0
 > 0α > 0|ϕ(x)| ≤ |x−x0|2x∈I|x−x0| ≤ α
α ϕ
ϕ0(x) = f0(x)−f0(x0)−f00 (x0)(x−x0).
f0x0n= 1 f0
ϕ0(x)/(x−x0) 0 x0α > 0|ϕ0(x)| ≤
|x−x0|x∈I|x−x0| ≤ α
ϕ x0x y x0x|ϕ(x)−ϕ(x0)|=
|ϕ0(y)| × |x−x0|y x0x|y−x0|≤|x−x0| |x−x0| ≤ α
|y−x0| ≤ α|ϕ0(y)| ≤ |y−x0| ≤ |x−x0|ϕ(x0)=0
|ϕ(x)|=|ϕ(x)−ϕ(x0)| ≤ |x−x0|×|x−x0|=|x−x0|2,
x|x−x0| ≤ α
0 exp C∞exp(n)(0) = 1 :R→R
lim0= 0 x∈R
exp(x) = 1 + x+x2
2+x3 6+··· +xn n!+xn(x).
0
n
n x0= 1 f:x7→ 1/x
n
0f
f