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 $\varepsilon$-$\delta$ language. The note is accessible to high-school students. An English version is followed by a more detailed Russian version. 
A. Skopenkov
math.HO