How Fermat found extrema

TL;DR

This paper offers an accessible, elementary proof of the criterion for a cubic polynomial to have three real roots, based on Fermat's calculus approach, suitable for high-school students.

Contribution

It introduces a simplified, rigorous Fermat-based method for determining cubic roots, avoiding advanced calculus language.

Findings

Proof is accessible to high-school students.

Method rigorously illustrates derivative concepts.

Criterion for three real roots is clearly established.

Abstract

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.