Answer
The required pair of numbers is $\left( 7,7 \right)$ , and their product is $49$.
Work Step by Step
Let the required numbers be x and y. Then,
$x+y=14\text{ }$
The above equation can be written as,
$\begin{align}
& x+y=14 \\
& y=14-x\text{ }.............................\left( 1 \right)
\end{align}$
Now, the product of the assumed numbers will be
$P=xy$
Using equation (2) above,
$\begin{align}
& P\left( x \right)=x\left( 14-x \right) \\
& P\left( x \right)=14x-{{x}^{2}}
\end{align}$
Which is a downwards opening parabola since the coefficient of ${{x}^{2}}$ is negative, and thereby the function has a maximum value.
Then, the coordinates of the vertex of the parabola will be the coordinates of the maxima of the parabola. Thus, the maximum value occurs at,
$\begin{align}
& x=\frac{-b}{2a} \\
& =\frac{-14}{-2} \\
& =7
\end{align}$
Substituting the value of x in the equation (1) we get,
$\begin{align}
& y=14-x \\
& y=14-7 \\
& y=7
\end{align}$
And, the maximum product is:
$7\times 7=49$
Thus, the product is maximum when both numbers are 7 and then the maximum product is 49.
