Find two numbers differing by 46 whose product is as small as possible.

Let the numbers be x and y. They differ by 46. Then x = y + 46.

Their product is P = xy. Since x = y + 46, P = xy = (y + 46) y, or

P = y^2 + 46y. You could graph this and then identify the coordinates of the vertex, which would give you the minimum value of P.

Or you could differentiate P (y) with respect to y, set the result = to 0, and solve for y:

2y + 46 = 0; y = - 23. x = y + 46, or + 23.

The vertex of the graph of this parabola represents the minimum value of the product P. It is (23, 46), and 46 is the smallest possible product here.