I want to prove this:
- If $f : [a, b]\to \mathbb{R}$ is continuous and $g$ is an integrable function that does not change sign on [a, b], then there exists $c\in (a, b)$ (open, please) such that $$\int_a^b f(x)g(x)dx = f(c)\int_a^b g(x)dx. $$
I have seen some demonstrations that find $c\in[a,b]$. How can I obtain such a $c$ in the open interval $(a,b)$?
In one of my textbooks, the author put as an exercise this implication:
- If $g>0$ (the previous conditions are in game) and $$ \int_a^b f(x)g(x)dx = f(a)\int_a^b g(x)dx,$$ then there is $c \in(a,b)$ such that $f(a) = f(c)$.
But, we assume $g>0$ instead (for example) $g\geq 0$. Can we prove this implication assuming only $g\geq 0$? If not, can we directly prove the mean value theorem described above assuming $g\geq 0$? How can we do it?