Answer
The required pair of number is $\left( 12,-12 \right)$ , and their product is $\underline{-144}$.
Work Step by Step
Let us assume that the two numbers be x and y. Then,
$x-y=24$ (I)
Now,
$\begin{align}
& x-y=24 \\
& y=x-24 \\
\end{align}$
Calculate xy,
$\begin{align}
& xy=x\left( x-24 \right) \\
& xy={{x}^{2}}-24x \\
\end{align}$
Which is an upwards opening parabola, which attains its minimum value at $\frac{-b}{2a}$ , where $b=-24$ and $a=1$.
$\begin{align}
& x=\frac{-b}{2a} \\
& =\frac{-\left( -24 \right)}{2} \\
& =12
\end{align}$
Putting in the value of $x$ in the equation (I) we get:
$\begin{align}
& 12-y=24 \\
& -y=24-12 \\
& y=-12
\end{align}$
The minimum product is:
$\begin{align}
& xy=12\times \left( -12 \right) \\
& =-144
\end{align}$
Hence, the product is minimum when the numbers are 12 and -12 and the minimum product is $~-144$.
