Let x and y be the numbers; let x be the lesser number and y be the greater number, so y-x = 5.


Consider the mean value a =   and half the distance between x and y,  b =  =  = 2.5


Then, obviously,  x = a - 2.5  and  y = a + 2.5, so we can write


    xy = (a-2.5)*(a+2.5) = .


From this formula, it is clear that xy is minimum when a= 0, i.e. when x+y = 0;  in other words, the numbers x and y 
shoud be opposite: y = -x.


Thus we get that the minimum of the product xy  is achieved when x = -y.


Together with y-x = 5 it implies that y - (-y) = 5,  or  2y = 5, y = 5/2 = 2.5.


Thus the minimum value of xy is  at x= -2.5,  y= 2.5,  and the value of xy is   = -6.25 then.   


ANSWER.  The minimum is achieved when x= -2.5, y= 2.5.

         The minimum value of the product is  then -6.25.