(a) State and prove Taylor's theorem with the remainder in Lagrange's form.

(b) Suppose that $e: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function such that $e(0)=1$ and $e^{\prime}(x)=e(x)$ for all $x \in \mathbb{R}$. Use the result of (a) to prove that

$$e(x)=\sum_{n \geqslant 0} \frac{x^{n}}{n !} \quad \text { for all } \quad x \in \mathbb{R}$$

[No property of the exponential function may be assumed.]