Verified Answer :

caylus

Hello,

Let's assume x the greatest number
                       y the smallest

x-y=48
P(x)=xy=x(x-48)=x²-2*24x+24²-24²=(x-24)²-576

Minimum if the vertex when x=24 and y=-24

Minimum product=-576
The two numbers are 24 and -24 but i don't know how to simplify!

meerkat18

Let x and y be the two numbers. If the difference of the numbers is 48,
                                    y - x = 48
                                      y = 48 + x 
For the product of the numbers, 
                        P = x y = x (48 + x) = 48x + x²
To determine the minimum product, differentiate the equation, equate to zero and solve for x.
                   dP = 0 = 48 + 2x             ; x = -24
The value of y is equal to 24
Thus, the minimum product is -576 and the numbers are -24 and 24.