Derivative Problem

**Markeur** · January 27th 2011, 05:33 AM

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

$\frac{f(x)-f(0)}{g(x)-g(0)} = \frac{f^{'}(\theta (x))}{g^{'}(\theta (x))}$ .

Show that

$\lim_{x \to 0^+} \frac{\theta (x)}{x} = \frac{1}{2}$ .

**Markeur** · January 30th 2011, 08:39 AM

Hey so has anyone managed to solve this?

The hint given is that differentiate each side of

$[f(x)-f(0)]g^{'}(\theta (x)) = [g(x)-g(0)]f^{'}(\theta (x))$

with respect to x, collect the terms that involve $\theta ^{'}(x)$ on one side, divide both sides by x, and then take the limit of each side as $x \rightarrow 0^{+}$ .

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

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