5.15. Théorème Dérivée et monotonie.

IRf I
•f I f0
I
”∀x∈I, ∀y∈I, [x≤y⇒f(x)≤f(y)]” ⇔”∀x∈I, f0(x)≥0”
•f0I
f I
”∀x∈I, f0(x)>0” ⇒”∀x∈I, ∀y∈I, [x<y⇒f(x)< f(y)]”
•f I f 0
I
•f0I
f I
•f I f0
I
ln exp
x > 0 ln0(x) = 1
x>0 ln ]0,+∞[
xexp0(x) = exp(x) = ex>0 exp R
a > 0x7→ xa=ealn(x)= exp(aln(x))
]0,+∞[x > 0
exp0(aln(x)) ×(aln(x))0= exp(aln(x)) ×a
x=xa×a x=a xa−1
]0,+∞[
f:x7→ 1
xR∗R∗

]−∞,0[ ]0,+∞[
x
f0(x) = −1
x2
R∗f
]− ∞,0[ ]0,+∞[R−1<1
f(−1) = −1<1 = f(1) f f(−1) > f(1)
f g f
I
g f(I)
•f I g ◦f I
•f I g ◦f I
g f(I)
•f I g ◦f I
•f I g ◦f I
g f
g◦f x y
I x ≤y f f(x)≥f(y)g
g(f(x)) ≥g(f(y)) g◦f(x)≥g◦f(y)x≤y
g◦f
g◦f I x
(g◦f)0(x) = g0(f(x))f0(x)
g g0f f0
g◦f g ◦f
P T
ln(P)Texp
exp ◦ln(P) = eln(P)=Pexp ln T
f f
f f0
f0
f

•f0+−
•f
% &
•f f0
• −∞ +∞f
f x R\{−1}f(x) = ex
1 + xf0(x) = xex (1 + x)2
−∞ −1 0 +∞
f0− || − 0 +
0|| +∞+∞
f& || & %
−∞ || 1x
y
IRc I f
f I c f(x)≥f(c)x∈I
f I f(c)f I c f(x)≤f(c)
x∈I f I f(c)
I f f
I
f f
f c ]a, b[
a < c < b f c f0(c) = 0
f(c, f(c))
f0(c)=0
c c
c c
c
c

f]−1,+∞[
x= 0 f(0) = 1
]− ∞,−1[ x= 0 f
R
R
f
]a, b[f0(x)=0
f
f0(x)=0
sin [0,2π] sin0(x) = cos(x)
π
2
3π
2
0π
2
3π
22π
sin0= cos + 0 −0 +
1 0
sin % & %
0−1
sin [0,2π]3π
2
π
2
f x0f(x0, f(x0))
y=f0(x0)(x−x0) + f(x0)
f(x0, f(x0))
f
f(x0, f(x0))
f I f
I I
f I I
−f(−f)0=−f0f0
−f0f I
−f
f I
•f I f
I
•f I f
I

f f
I x0I
(x0, f(x0)) y=f0(x0)(x−x0) + f(x0)
g(x) = f(x)−[f0(x0)(x−x0) + f(x0)]
I g g0(x) = f0(x)−f0(x0)f
f0g
x0
g0(x) = f0(x)−f(x0)−0 +
& %
g0
g I f
(x0, f(x0)) x0I
f I
exp : x7→ exR
exp0= exp Rexp
R
ln : x7→ ln(x) ]0,+∞[x7→ 1
x
]0,+∞[ ln
x
yy=ex
x
y
y= ln(x)
f I
f0f0
f I f0
I f I f00 f0
f I