Suppose we are given functions $f$ and $g$ such that $f(x)\ge 0, g(x)\ge 0$$\forall x \in [a,b]$ and $f,g$ are differentiable on $(a,b)$. Suppose $f'(x)=u(x)v(x)$ and $g'(x)=u(x)w(x)$ such that $u(x)\ge 0$, $v(x)\ge 0$ while $w(x)> 0$ (i.e only $w$ is strictly greater than zero). The generalized mean value theorem states that there exists a point $c$ such that $(f(b)-f(a))g'(c)=(g(b)-g(a))f'(c)$. However can we make the stronger claim that if $g(b)\neq g(a)$ there exists a point $c^*$ which satisfies $$\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{v(c^*)}{w(c^*)}$$
i.e can we cancel the $u$?
If this claim is untrue, can someone give an exception to this claim? Is this question somehow equivalent to L'Hospital?