Answer
The required pair of numbers is $\left( 10,10 \right)$ and their product is $\underline{100}$.
Work Step by Step
Let us assume the two numbers be x and y. Then,
$x+y=20$ ….. (I)
Now,
$\begin{align}
& x+y=20 \\
& y=20-x \\
\end{align}$
And calculate xy,
$\begin{align}
& xy=x\left( 20-x \right) \\
& xy=20x-{{x}^{2}} \\
\end{align}$
Which is a downwards opening parabola, which attains its maximum value at $\frac{-b}{2a}$ , where $b=20$ and $a=-1$.
$\begin{align}
& x=\frac{-20}{-2} \\
& x=10 \\
\end{align}$
Putting the value of $x$ in the equation (I) we get,
$\begin{align}
& 10+y=20 \\
& y=20-10 \\
& y=10
\end{align}$
The maximum product is
$10\times 10=100$.
Hence, the product is maximum when both numbers are 10 and the maximum product is 100.
