Find interval of $ab$

Question

We know $-1\le a \le 1$ and $-1\le b \le 1$ . Now find the interval of $ab$

In general , I want to know if we have intervals of $a$ and $b$ , what we can say about $ab$ interval .

Answer 1

Hint: $-1 \le a \le 1 \;\land\; -1 \le a \le 1 \iff |a| \le 1 \;\land\; |b| \le 1 \implies |ab| = |a| \cdot |b| \le1\,$ therefore $-1 \le ab \le 1\,$. Since the bounds are attained for $a=1, b=\pm1$ it follows that the interval is $[-1,1]$.

Answer 2

Here, $-1\le ab\le 1$.

If $a_0\le a\le a_1$ and $b_0\le b\le b_1$, then several cases have to be considered depending on the signs of the bounds. However, it turns out that we simply have $$\min\{a_0b_0,a_1b_0,a_0b_1,a_1,b_1\} \le ab\le\max\{a_0b_0,a_1b_0,a_0b_1,a_1,b_1\}$$