Suppose $f(x)$ is continuous on a closed interval $[a,b]$ and differentiable on the open interval $(a,b)$. I would like to prove the existence of $c$ such that
$$(c-a)\cdot(b-c)\cdot\ f'(c) = (2c-a-b)\cdot\ f(c)$$ and $a<c<b$.
Consider the function: $$g(x)=f(x)(x-a)(b-x)$$ Note that $g$ is also continuous on $[a,b]$ and differentiable on $(a,b)$, $g(a)=g(b)$, find $g'(x)$ and apply Rolle's theorem.