<s>[INST] 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.
 [/INST] 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 $$-4\le xy\le 6.$$
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