Vtyjkyuk

$f$ and $g$ are two continuous functions in $[a,b]$.

Suppose that $g$ has a constant sign (For example $g>0$).

Let $I=int_a^b f(x).g(x).dx$

Does it exist $cin]a,b[$ so that $I=f(c)int_a^b g(x).dx$ ?

Let $a=-1,b=1$, $f(x)=x, g(x)=x+2$. Then $g(x) >0$ and $int_a^b f(x) , dx=0, I=frac 2 3$. So there is no such point $c$.

For the modified question the answer is yes.

Let m and M be the minimum and maximum of $f$ on $[a,b]$ and assume, without loss of generality, that $g(x)gt 0$. Then $mint_a^b g(x),dx=int_a^b m,g(x),dxle int_a^b f(x)g(x),dxleint_a^b M,g(x),dx=Mint_a^b g(x),dx$, so
$$mleint_a^b f(x)g(x),dxover int_a^b g(x),dxle M$$ and because of the continuity of $f$ there is a $cin[a,b]$ for which $f(c)$ equals the quotient. If the quotient equals $m$ or $M$ then $f$ must be constant and any $c$ in $(a,b)$ will do. If $f$ is not constant then $c$ can be found in the open interval between a point where $f$ is minimal and a point where $f$ is maximal.