If you are not limited to positive numbers or integers, you would be locating the vertex of the parabola using the parabolic equation:
y = ax^2 + bx + c
In this case we have:
y = x(x+9)
y = x^2 + 9x
The x coordinate of the vertex is found by using the equation:
In this case b is 9 and a is 1, so
-9/2 = -4.5
The minimum value will be when x = -4.5 and x + 9 = 4.5, so:
-4.5*4.5 = -20.25
-20.25 is the minimum product.
Assuming we aren't dealing with negative or fractional numbers, the smallest product would have to be 0. The numbers 9 and 0 have a difference of 9 and when multiplied together the product is 0.
The minimum product of two numbers has to be determined where the difference of the numbers is 9.
If one of the numbers is represented by x, the other number is x - 9.
The product of the two numbers is x*(x - 9) = x^2 - 9x
Now if we plot y = x^2 - 9x we get the following graph:
Take the derivative of y = x^2 - 9x
dy/dx = 2x - 9
equate this to 0 and solve for x
2x - 9 = 0
x = 4.5
At x = 4.5, y = (4.5)^2 - 9*(4.5) = -20.25
The minimum value of the product is -20.25