Maximum Abscissa for Loop of Folium of Descartes

Theorem
Consider the folium of Descartes defined in parametric form as:


 * $\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$


 * FoliumOfDescartes.png

The point on the loop at which the $x$ value is at a maximum occurs when $t = \sqrt [3] {\dfrac 1 2}$, corresponding to the point $P$ defined as:
 * $P = \tuple {2^{2/3} a, 2^{1/3} a}$

Proof
We calculate the derivative of $x$ $t$:

Thus $x$ is stationary when:

From Behaviour of Parametric Equations for Folium of Descartes according to Parameter, it is clear from the geometry that $x$ is a local maximum for this value of $t$.

Then we have:

and:

Hence the result.