New properties of the Intermediate Point from Bonnet Theorem

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: ∫axf(t)g(t)dt=f(a)∫ac¯(x)g(t)dt+f(x)∫c¯(x)xg(t)dt, \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.