The importance of the Treatise on equations by Sharaf al-Dīn al-Ṭūsī (12th century) has been brought to our attention by R. Rashed (1974, 1986), who underlined the analytical aspects of this essentially algebraic work. Following Rashed, this article concentrates on one of these analytical concepts, namely the maximum of a polynomial expression f(x) of degree 3. The purpose is to clarify the techniques that led al-Ṭūsī, when computing the maximum of f(x), to systematically display algebraic equations equivalent to f(x) = 0. By demonstrating that al-Ṭūsī's essentially algebraic proofs were also based on analytical procedures, we show that the presence of these equations was not fortuitous, but resulted from a correct understanding of the maximum of f(x). The accompanying geometrical representations were primarily for illustration.