Let $[a, b] \subseteq \mathbb{R}$ be a D544: Closed real interval such that
(i) | \begin{equation} a < b \end{equation} |
(ii) | $f, g : [a, b] \to \mathbb{R}$ are each a D5231: Standard-continuous real function on $[a, b]$ |
(iii) | $f, g$ are each a D5614: Differentiable real function on $(a, b)$ |
Then
\begin{equation}
\exists \, x \in (a, b) :
f'(x) (g(b) - g(a))
= g'(x) (f(b) - f(a))
\end{equation}