1. ## Derivative Problem

Let $\displaystyle f$ and $\displaystyle g$ be functions with continuous second derivatives on $\displaystyle [0,1]$ and $\displaystyle f^{'}(0)g^{''}(0)-f^{''}(0)g^{'}(0) \neq 0$. Define a function $\displaystyle \theta$ for $\displaystyle x \in (0,1)$ so that $\displaystyle \theta (x)$ is one of the values that satisfies the generalised mean value theorem,
$\displaystyle \frac{f(x)-f(0)}{g(x)-g(0)} = \frac{f^{'}(\theta (x))}{g^{'}(\theta (x))}$.
Show that
$\displaystyle \lim_{x \to 0^+} \frac{\theta (x)}{x} = \frac{1}{2}$.

2. Hey so has anyone managed to solve this?

The hint given is that differentiate each side of
$\displaystyle [f(x)-f(0)]g^{'}(\theta (x)) = [g(x)-g(0)]f^{'}(\theta (x))$
with respect to x, collect the terms that involve $\displaystyle \theta ^{'}(x)$ on one side, divide both sides by x, and then take the limit of each side as $\displaystyle x \rightarrow 0^{+}$.

Despite the hint, I'm still stuck. Is there anyone who can help? Thanks.