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Lebl-Basic-Analysis-I
· Problem 4.2.7
The Derivative
Mathematics · derivative-sign-darboux
Problem
Suppose
f
:
(
a
,
b
)
→
R
f : (a,b) \to \mathbb{R}
f
:
(
a
,
b
)
→
R
is a differentiable function such that
f
′
(
x
)
≠
0
f'(x) \neq 0
f
′
(
x
)
=
0
for all
x
∈
(
a
,
b
)
x \in (a,b)
x
∈
(
a
,
b
)
. Suppose there exists a point
c
∈
(
a
,
b
)
c \in (a,b)
c
∈
(
a
,
b
)
such that
f
′
(
c
)
>
0
f'(c) > 0
f
′
(
c
)
>
0
. Prove
f
′
(
x
)
>
0
f'(x) > 0
f
′
(
x
)
>
0
for all
x
∈
(
a
,
b
)
x \in (a,b)
x
∈
(
a
,
b
)
.
Related theory
Monotonicity And The Sign Of The Derivative
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