énoncé

\
f
∀x∈R×
+, f(x) = x2−xln(x)−1f(0) = −1.
ϕ
∀x∈R×
+, ϕ(x) = 2
x+ ln(x)
f
x= 0,5 1 1,5 2 2,533,5 4
f(x)' −0,400,6 1,6 3 4,7 6,9 9,5
fR+
f
fR∗
+
f(x)x
fR∗
+J
f−1f−1(x)x
k xkf(xk) = k
x0
f x1x2
xkf−1(xk)
k
(un)u0=3
2∀n∈N, un+1 =ϕ(un)
ϕR∗
+
ϕ(3
2)'1,73 ϕ(2) '1,69 ϕ3 2; 2⊂3 2; 2
ϕ0∀x∈3
2; 2,|ϕ0(x)|62 9
x=ϕ(x)f(x)=1 x1
x=ϕ(x)
n
3
26un62 ; |un+1 −x1|62 9|un−x1|;|un−x1|62 9n
.
(un)
u10000

\
(fn)n∈N∗]−1,+∞[ (Un)n∈N∗
fn(x) = xnln(1 + x)Un=
1
Z
0
fn(x)dx.
U1
a b c
∀x∈[0,1],x2
x+ 1 =ax +b+c x+ 1.
1
Z
0
x2
x+ 1 dx.
U1=1
4
(Un)n∈N∗
(Un)n∈N∗
(Un)n∈N∗
∀n∈N∗,06Un6ln 2
n+ 1.
(Un)n∈N∗
unn>0
+∞
P
n=2
un
u0= 2 u1= 1 n>0
un+2 =3
4un+1 −1 8un.
u0= 1 u1= 3 n>0
un+2 = 4un+1 −4un.
u0=−2n>0
un+1 = 2un−3.
A
A=3−1
−1 3 
fM2(R)
f(M) = A2M−AM.
f
M∈ M2(R)f(M)∈ M2(R)f
f
Vect[...]
f
Vect[...]

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P=
1 1 1
−111
0−1 2
, A =
111
111
113
, D =
000
010
004
.
A
A
λ A −λI
AX = 0 AX =X AX = 4X X =
x
y
z
∈ M3,1(R)
S= Vect[...]
P P −1
A=P DP −1
AnP D n n >0
DnP A n
E F
E={M∈ M3(R)AM =MA}
F= M=
3a+ 2b+c−3a+ 2b+c−2b+ 2c
−3a+ 2b+c3a+ 2b+c−2b+ 2c
−2b+ 2c−2b+ 2c2b+ 2c
(a, b, c)∈R3
E F M3(R)
F⊂E
E=F
MM3(R)N=P−1MP
M∈E⇐⇒ DN =ND.
DN =ND N M3(R)
E
f x f(x) = (1− |x|x∈[−1; 1]
0
R1
0f(x)dx R0 −1f(x)dx
f
X(Ω,A,P)
f

\
X
X FX
FX(x) =
0x < −1
1
2+x+x2 2−16x60
1
2+x−x2 20< x 61
1x > 1
Y=|X|Y
(Ω,A,P) FY
FY(x)x
x
(Y6x) (−x6X6x)
FY(x)FX
Y g
g(x) = 
2(1 −x)x∈[0; 1]
0
Y
U V (Ω,A,P)
[0; 1]
I= inf(U, V )ωΩI(ω) = min( U(ω), V (ω) )
a b
min(a, b)a b
a b
I(Ω,A,P)
xP(I > x) = P( [U > x]∩[V > x] )
FII
FI(x)x
I Y
10000 Y
R2x
x
1
√t2+1 x
f
∀x∈R, f(x) = Z2x
x
1
√t2+ 1 dt
f

\
f C1R
x
f0(x) = 2
√4x2+ 1 −1 √x2+ 1.
fR
t26t2+ 1 6t2+ 2t+ 1 t
∀x∈R∗
+,ln(2x+ 1) −ln(x+ 1) 6f(x)6ln 2
f(x)x+∞
f
f(x)=0