Answer
The numbers are 12 and 12 and the maximum product is 144.
Work Step by Step
Let x be one of the two numbers.
So, $24-x $ will be the other number.
The product function is given by:
$\begin{align}
& f\left( x \right)=x\left( 24-x \right) \\
& f\left( x \right)=24x-{{x}^{2}} \\
\end{align}$
The x-coordinate of the vertex for the equation $ f\left( x \right)=24x-{{x}^{2}}$ is:
$\begin{align}
& x=-\frac{b}{2a} \\
& x=-\frac{-24}{2\left( 1 \right)} \\
& x=-\frac{-24}{2} \\
& x=12 \\
\end{align}$
Therefore,
$\begin{align}
& f\left( 12 \right)=24\left( 12 \right)-{{\left( 12 \right)}^{2}} \\
& f\left( 12 \right)=288-144 \\
& f\left( 12 \right)=144 \\
\end{align}$
Thus, the vertex is (12, 144).
Hence, the maximum product is 144 obtained when x is 12, so other number is,
$\begin{align}
& 24-x=24-12 \\
& =12
\end{align}$
