Planche 1. Planche 2. Exercice 2. Soit P ∈ R[X] Planche 3.

∼ ∼ ∼ ∼
f g I a
fg a
n t 7−→ cos(t)et
f: [a, b]→R2f(a) = f(b)=0
c∈]a, b[γ∈]a, b[f(c) = (c−a)(c−b)
2f00(γ)
f: [−1,1] →RC1−1,0,1g: [−1,1] →R
g(x)=2x4+x+f(x)
c∈]−1,1[ g0(c)=0
P∈R[X]deg(P)≥2
P P 0
PRP0
f I f I f0≥0
f: [−1,1] →RC1[−1,1] ] −1,1[
f(−1) = −1f(0) = 0 f(1) = 1 c∈]−1,1[ f00(c)=0
x, y ∈R0< x < y ≤π/2
x/y < sin(x)/sin(y)<π
2x/y

cos cos(t)et=
Re(e(1+i)t)n
(cos(t)et)(n)=Re((1 + i)ne(1+i)t
(1 + i)n= 2n/2einπ/4
(cos(t)et)(n)= 2n/2etcos(t+nπ/4)
g(x) = f(x)−A
2(x−a)(x−b)A g(c) = 0
g(a) = g(b) = g(c) = 0 a0∈]a, c[g0(a0) = 0 g0(b0)=0 b0∈]c, b[
γ∈]a, b[g00(γ) = 0
g00(x) = f00(x)−A f00(γ) = A

g C1
g g(−1) = 1 g(0) = 0
g(1) = 3 g0(a) = 0 g
c∈]0,1[ g(c) = 1
a∈]−1, c[g0(a)=0 g(−1) = g(c)
c∈]−1,1[ g0(c) = 0
PCn
C
P
P(X) = λ
n
Y
k=1
(X−ak)
akλ
a1< . . . < an
P[ak, ak+1]k∈[|1, n −1|] ]ak, ak+1[P
C1RP(ak) = P(ak+1) = 0 yk∈]ak, ak+1[
P0(yk) = 0
n−1yka1< y1< a2< . . . < yn−1< anyk
P0n−1P0
yk
PRλ a1, . . . , aN
a1< . . . < aN
P=λ
N
Y
k=1
(X−ak)αk
αk
y1,...yN−1
yk∈]ak, ak+1[P0(yk) = 0 akP0αk−1
P0ykN−1xk
αk−1N−1 + PN
k=1 αk−1 = N−1−N+PN k=1 αk=n−1
P0n−1
P0P0R

f C1a]−1,0[ f0(a) = f(0) −
f(−1)/(0 −(−1)) = 1 b]0,1[ f0(b) = f(1) −f(0)/(1 −0) = 1
f0c∈]a, b[f00(c) = 0
f0(a) = f0(b)
0
f(t) = sin(t)/t ]0, π/2[ x, y ∈R
0< x < y < π/2
2
πf(x)< f(y)< f(x)
f
f
f0(t) = tcos(t)−sin(t)
t2
A(t) = tcos(t)−sin(t)
[0, π/2[ A0(t) =
−tsin(t)≤0A(t)<0t6= 0 A A(0) = 0
A(t)<0f0(t)<0f
f(t)→1t→0f(t)→2/π t →π/2
0< x < y < π/2
2/π < f(y)< f(x)<1
f(y)< f(x)f(y)
f(x)> f(y) 0 < f(x)<1
f(y)
f(x)>2 πx/y < sin(x)/sin(y)<π 2x/y