ThmDex – An index of mathematical definitions, results, and conjectures.
Real mean value inequality

Let $[a, b] \subseteq \mathbb{R}$ be a D544: Closed real interval such that
 (i) $$a < b$$ (ii) $f : [a, b] \to \mathbb{R}$ is a D5614: Differentiable real function on $(a, b)$
Then
 (1) $$|f(b) - f(a)| \leq \sup_{x \in (a, b)} |f'(x)| |b - a|$$ (2) $$\exists \, m, M \in \mathbb{R} : m \leq f' \leq M \quad \implies \quad m \leq \frac{f(b) - f(a)}{b - a} \leq M$$
Proofs
Proof 0
Let $[a, b] \subseteq \mathbb{R}$ be a D544: Closed real interval such that
 (i) $$a < b$$ (ii) $f : [a, b] \to \mathbb{R}$ is a D5614: Differentiable real function on $(a, b)$
Result R1073: Mean value theorem shows that there exists $c \in (a, b)$ such that $$f(b) - f(a) = f'(c)(b - a)$$ Applying result R1108: Monotonicity of supremum, we then have $$|f(b) - f(a)| = |f'(c)||b - a| \leq |b - a| \sup_{x \in (a, b)} |f'(x)|$$ This concludes the first claim. Suppose then that $f'$ is bounded with a lower bound $m \in \mathbb{R}$ and an upper bound $M \in \mathbb{R}$. Then results
imply $$f(b) - f(a) = f'(c) (b - a) \leq M (b - a)$$ as well as $$f(b) - f(a) = f'(c) (b - a) \geq m (b - a)$$ $\square$