What is the easiest and simplest way to bound $xy$ Let $-1\le x\le 2$ and $-2\le y\le 3$. I have to bound $xy$. I can do it case by case. E.g
Case1
\begin{cases}
0\le -x\le 1\\
0\le -y\le2
\end{cases} Taking the product term by term yields $\boxed{0\le xy\le 2}$
Case2
\begin{cases}
0\le -x\le 1\\
0\le y\le3
\end{cases} Again I take the product to have $\boxed{-3\le xy\le0}$
Case3
\begin{cases}
0\le x\le 2\\
0\le -y\le2
\end{cases} then $\boxed{-4\le xy\le 0}$
Case 4
\begin{cases}
0\le x\le 2\\
0\le y\le3
\end{cases} then $\boxed{0\le xy\le6}$
Combining all these stuffs gives finally $-4\le xy\le 6$. 
Does someone have an alternative method, more simpler and not as long as this thing above.

In this case, it is easy to see that the least value of the product $xy$ is either $-1×3$ or $-2×2.$ Similarly, the greatest value of this product in the rectangle is either $-1×-2$ or $2×3,$ so that we have $$-4\le xy\le 6.$$
