State and prove Rolle's theorem.

Let $f$ and $g$ be two continuous, real-valued functions on a closed, bounded interval $[a, b]$ that are differentiable on the open interval $(a, b)$. By considering the determinant

$$\phi(x)=\left|\begin{array}{ccc} 1 & 1 & 0 \\ f(a) & f(b) & f(x) \\ g(a) & g(b) & g(x) \end{array}\right|=g(x)(f(b)-f(a))-f(x)(g(b)-g(a))$$

or otherwise, show that there is a point $c \in(a, b)$ with

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

Suppose that $f, g:(0, \infty) \rightarrow \mathbb{R}$ are differentiable functions with $f(x) \rightarrow 0$ and $g(x) \rightarrow 0$ as $x \rightarrow 0$. Prove carefully that if the $\operatorname{limit}_{x \rightarrow 0} \frac{f^{\prime}(x)}{g^{\prime}(x)}=\ell$ exists and is finite, then the limit $\lim _{x \rightarrow 0} \frac{f(x)}{g(x)}$ also exists and equals $\ell$.