= Cauchy mean value theorem
{c}
= Cauchy's mean value theorem
{c}
{synonym}
For real functions continuous on $[a,b]$ and differentiable on $(a,b)$, there exists $\xi\in(a,b)$ with
$$
[f(b)-f(a)]g'(\xi)=[g(b)-g(a)]f'(\xi).
$$
Apply <Rolle's theorem> to a linear combination of the functions with equal endpoint values. If $g'$ never vanishes, $g(b)\ne g(a)$ and the equality can be divided into a ratio. This form directly proves the zero-over-zero endpoint case of <L'Hôpital's rule>, without continuity of the derivatives.
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