DOI: 10.2478/gm-2024-0002 ISSN: 1584-3289
New properties of the Intermediate Point from Bonnet Theorem
Emilia-Loredana Pop, Dorel I. Duca, Augusta Raţiu Abstract
In this paper, the following result is presented: if f, g : [a, b] → ℝ are two continuous functions and f is monotone, then there exists a function ̄c : [a, b] → [a, b] continuous in a with the property:
∫
a
x
f
(
t
)
g
(
t
)
d
t
=
f
(
a
)
∫
a
c
¯
(
x
)
g
(
t
)
d
t
+
f
(
x
)
∫
c
¯
(
x
)
x
g
(
t
)
d
t
,
\int_a^x {f\left( t \right)g\left( t \right){\rm{d}}t = f\left( a \right)\int_a^{\bar c\left( x \right)} {g\left( t \right){\rm{d}}t + f\left( x \right)\int_{\bar c\left( x \right)}^x {g\left( t \right){\rm{d}}t,} } }
for all
x ∈ [
a, b]. Then, under specified conditions, the function ̄
c that is differentiable at point
a is presented, and
c
¯
′
(
a
)
\bar c'\left( a \right)
is also calculated.