Let $a < b,$ $[a,b]$ be a closed real interval, and $f:[a,b]\to\mathbb R$ a continuous function with $f(a)=f(b)$. Further, let $f$ be differentiable on the open real interval $]a,b[.$ Then there is an $\xi\in ]a,b[$ with $f’(\xi)=0.$

In particular, between any two roots of $f$ there is a root of $f’$.

This theorem is named after Michael Rolle.

| | | | | created: 2017-07-31 21:02:58 | modified: 2017-07-31 21:29:01 | by: *bookofproofs* | references: [581]

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[581] **Forster Otto**: “Analysis 1, Differential- und Integralrechnung einer VerĂ¤nderlichen”, Vieweg Studium, 1983