Answer
The required pair of numbers is $\left( 8,-8 \right)$ , and their product is $-64$.
Work Step by Step
Let us assume that the two numbers be x and y. Then,
$x-y=16$ ..... (I)
Now,
$\begin{align}
& x-y=16 \\
& y=x-16 \\
\end{align}$
And calculate xy,
$\begin{align}
& xy=x\left( x-16 \right) \\
& xy={{x}^{2}}-16x \\
\end{align}$
Which is an upwards opening parabola, which attains its minimum value at $\frac{-b}{2a}$ , where $b=-16$ and $a=1$.
$\begin{align}
& x=\frac{-b}{2a} \\
& =\frac{-\left( -16 \right)}{2} \\
& =8
\end{align}$
Putting in the value of $x$ in the equation (I) we get:
$\begin{align}
& 8-y=16 \\
& -y=16-8 \\
& y=-8
\end{align}$
Now, the minimum product is:
$\begin{align}
& xy=8\times \left( -8 \right) \\
& =-64
\end{align}$
Hence, the product is minimum when the numbers are 8 and -8 and the minimum product is -64.
