Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$. It it true that for every $c\in[a,b]$, there exists some $(a_0, b_0)$ such that
$$f'(c) = \frac{f(b_0)-f(a_0)}{b_0 - a_0}\,? $$
2026-03-27 21:54:39.1774648479
A question related to the mean value theorem
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No, not necessarily. Consider for example $f(x)=x^3$. Then $f'(0)=0$, but there are no $a$ and $b$ such that $$\frac{a^3-b^3}{a-b}=0$$ without having $a=b$.