Question 175741
Let's call your two numbers, A and B.
"difference is 14" 
1.{{{A-B=14}}}
"product is a minimum"
2.{{{f=A*B}}}
We want to minimize f.
First, substitute for A and make f only a function of B. 
1.{{{A-B=14}}}
{{{A=14+B}}}
{{{f(B)=(14*B)*B=14B+B^2}}}
To minimize, take the derivative wrt B and set it equal to 0.
{{{df/dB=14+2B=0}}}
{{{B=-7}}}
Then from 1,
{{{A=14+B}}}
{{{A=14-7}}}
{{{A=7}}}