12. Ashlley was asked by her Math teacher to find two numbers which differ by 8 and whose product is a minimum. Determine the numbers. Show all work.

Step 1 ：Let the two numbers be x and y, such that y - x = 8.

Step 2 ：Express y in terms of x: y = x + 8.

Step 3 ：Find the product of x and y: xy = x(x + 8).

Step 4 ：Differentiate the product with respect to x: \(\frac{d(xy)}{dx} = 2x + 8\).

Step 5 ：Find the critical points by setting the derivative equal to 0: \(2x + 8 = 0\) => x = -4.

Step 6 ：Check if the critical point is a minimum by analyzing the second derivative: \(\frac{d^2(xy)}{dx^2} = 2\), which is positive, indicating a minimum.

Step 7 ：Substitute the value of x into the equation for y: y = -4 + 8 = 4.

Step 8 ：\(\boxed{\text{The two numbers are -4 and 4.}}\)