Answer
$-49$
Work Step by Step
Let the smaller of the numbers be x.
The other is x+14
The product, expressed as a function of x,
$f(x)=x(x+14)=x^{2}+14x$
is in the form $f(x)=ax^{2}+bx+c$.
Its graph opens up, since $a=1 > 0,$ and has a minimum point.
The vertex is a minimum point and it occurs at
$x=-\displaystyle \frac{b}{2a}=-\frac{14}{2(1)}=-7$
The other number is $-7+14=7$,
and the minimum product of such numbers is
$f(-7)=(-7)(7)=-49$
