How Fermat found extrema

We present a short elementary proof of the well-known criterion for a cubic polynomial to have three real roots. The proof is based on Fermat's approach to calculus for polynomials. This approach illustrates the idea of a derivative rigorously but without technical $\varepsilon$-$δ$ language. The note is accessible to high-school students. An English version is followed by a more detailed Russian version.

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