I was currently asked to explain why the following optimization has a solution: Given $x=(x_1,x_2) \in [0,1]^2$, consider $$\min_{z\in [0,1]^2} \max\big\{ x_1x_2^2(z_1-x_1), x_1^2x_2(z_2-x_2)\big\}.\tag{1}$$ Since $z \mapsto x_1x_2^2(z_1-x_1)$ and $z\mapsto x_1^2x_2(z_2-x_2)$ are continuous from $\mathbb R^2$ to $\mathbb R$, where $x\in [0,1]^2$ is fixed, I know that $z\mapsto \max\{ x_1x_2^2(z_1-x_1), x_1^2x_2(z_2-x_2)\}$ is continuous too. Thus, due to the fact that $[0,1]^2$ is a compact subset of $\mathbb R^2$, problem $(1)$ attains a solution.
However, I am now looking to find all solutions of optimization problem $(1)$.
I have shown that $z \mapsto x_1x_2^2(z_1-x_1)$ and $z\mapsto x_1^2x_2(z_2-x_2)$ are convex functions. Since $[0,1]^2$ is convex, the minimum is attained at the boundary of $[0,1]^2$. Therefore I believe that we have $$ \min_{z\in [0,1]^2} \max\big\{ x_1x_2^2(z_1-x_1), x_1^2x_2(z_2-x_2)\big\} = \max\big\{-x_1^2x_2^2,-x_1^2x_2^2 \big\} = - x_1^2x_2^2.$$
So, my question is: Are my explanations and my result correct?