Answer
Numbers: $-7$; $7$;
Minimum product: $-49$
Work Step by Step
Since the difference between the two numbers is $14$, let's denote them by $x$ and $x+14$.
Now let's consider the product function:
$$f(x)=x(x+14)=x^2+14x.$$
We need to find $x$ so that $f$ reaches its minimum.
Bring the function to the vertex form $f(x)=a(x-h)^2+k$:
$$\begin{align*}
f(x)&=(x^2+14x+49)-49\\
&=(x+7)^2-49.
\end{align*}$$
Identify the constants $a$, $h$, $k$:
$$\begin{align*}
a&=1\\
h&=-7\\
k&=-49.
\end{align*}$$
Determine the vertex of the function:
$$(h,k)=(-7,-49).$$
The function $f$ is a quadratic function with positive leading coefficient. This means that its graph is a parabola opening upward, which has its minimum in the vertex. Since the vertex is $(-7,-49)$, it means the value of $x$ for which the minimum is reached is $x=-7$. The two numbers are:
$$\begin{align*}
x&=-7\\
x+14&=-7+14=7.
\end{align*}$$
The minimum product is $-49$.
