Let $[a, b] \subseteq \mathbb{R}$ be a D544: Closed real interval such that
| (i) | \begin{equation} a < b \end{equation} |
| (ii) | $f : [a, b] \to \mathbb{R}$ is a D5614: Differentiable real function on $(a, b)$ |
Then
| (1) | \begin{equation} |f(b) - f(a)| \leq \sup_{x \in (a, b)} |f'(x)| |b - a| \end{equation} |
| (2) | \begin{equation} \exists \, m, M \in \mathbb{R} : m \leq f' \leq M \quad \implies \quad m \leq \frac{f(b) - f(a)}{b - a} \leq M \end{equation} |