(i) State (without proof) Rolle's Theorem.

(ii) State and prove the Mean Value Theorem.

(iii) Let $f, g:[a, b] \rightarrow \mathbb{R}$ be continuous, and differentiable on $(a, b)$ with $g^{\prime}(x) \neq 0$ for all $x \in(a, b)$. Show that there exists $\xi \in(a, b)$ such that

$$\frac{f^{\prime}(\xi)}{g^{\prime}(\xi)}=\frac{f(b)-f(a)}{g(b)-g(a)}$$

Deduce that if moreover $f(a)=g(a)=0$, and the limit

$$\ell=\lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)}$$

exists, then

$$\frac{f(x)}{g(x)} \rightarrow \ell \text { as } x \rightarrow a$$

(iv) Deduce that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable then for any $a \in \mathbb{R}$

$$f^{\prime \prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)+f(a-h)-2 f(a)}{h^{2}} .$$