Let's determine the maximum possible value of the expression $x^{2} y-y^{2} x$, where $x$ and $y$ independently run through the interval $[0,1]$. $(\mathbf{H})$
If $x=0$, then $x^{2} y-y^{2} x=0$.

If $x \neq 0$, then

$$
x^{2} y-y^{2} x=x^{3}\left(\frac{y}{x}-\left(\frac{y}{x}\right)^{2}\right)=x^{3}\left(\frac{1}{4}-\left(\frac{y}{x}-\frac{1}{2}\right)^{2}\right) \leq \frac{x^{3}}{4} \leq \frac{1^{3}}{4}=\frac{1}{4}
$$

and when $x=1, y=\frac{1}{2}$, $x^{2} y-y^{2} x=\frac{1}{4}$ holds.

Therefore, the maximum value of the expression $x^{2} y-y^{2} x$ is $\frac{1}{4}$.

Based on the work of Zoltán Gyenes (Budapest, ELTE Apáczai Csere J. Gymnasium, 1st year).