We are to maximise $x^{2}y-y^{2}x$, where $x,y \in [0,1]$. I've tried using AM-GM to find another (easier to maximise) expression, which gave me $xy(x-y) \le \frac{1}{2}(x^{2}+y^{2}) (x-y)$ but that doesn't appear to help.
The next part of the question asks to maximise $x^{2}y+y^{2}z+z^{2}x-x^{2}z-y^{2}x-z^{2}y$, where $x,y,z \in [0,1]$. I would imagine that this is simply triple the maximum for the first expression.
Any help would be appreciated.
NOTE: I shouldn't be using any calculus.