i) State Rolle's theorem.

Let $f, g:[a, b] \rightarrow \mathbb{R}$ be continuous functions which are differentiable on $(a, b)$.

ii) Prove that for some $c \in(a, b)$,

$$(f(b)-f(a)) g^{\prime}(c)=(g(b)-g(a)) f^{\prime}(c) .$$

iii) Suppose that $f(a)=g(a)=0$, and that $\lim _{x \rightarrow a+} \frac{f^{\prime}(x)}{g^{\prime}(x)}$ exists and is equal to $L$.

Prove that $\lim _{x \rightarrow a+} \frac{f(x)}{g(x)}$ exists and is also equal to $L$.

[You may assume there exists a $\delta>0$ such that, for all $x \in(a, a+\delta), g^{\prime}(x) \neq 0$ and $g(x) \neq 0 .]$

iv) Evaluate $\lim _{x \rightarrow 0} \frac{\log \cos x}{x^{2}}$.