Show that $x_1^2+x_2^2+(x_1-x_2)^3 \rightarrow \min$ has no solution How can I show (preferably using the Bolzano-Weierstrass theorem), that $x_1^2+x_2^2+(x_1-x_2)^3 \rightarrow \min$ has no solution? I can see that it is true, but how can I show it?

Change variables to $\xi := x_1-x_2$ and $\eta :=x_1+x_2$. Write the poly in these terms, and it will become clear it has no lower bound.
