\begin{align*} xy+2yz+3zx&=xy+2y(1-x-y)+3(1-x-y)x\\ &=-3x^2+(3-4y)x-2y^2+2y\\ &\le \frac{4(-3)(-2y^2+2y)-(3-4y)^2}{4(-3)}\color{blue}{\left(at^2+bt+c\le \frac{4ac-b^2}{4a}(a<0)\right)}\\ &=\frac{3}{4}-\frac{2}{3}y^2\\ &\le \frac{3}{4}. \end{align*}\\
