Explanation:
Let's call the two numbers x and y, where x is the larger number.
According to the problem, we know that: x - y = 46
To minimize the product xy, we can use the fact that for any two given numbers, their product is minimized when they are as close as possible to each other.
So, we want to find values of x and y that satisfy the equation above and are as close as possible to each other.
If we add y to both sides of the equation, we get:
x = y + 46
Now, we can substitute this expression for x into the equation xy to get: xy = y(y+46)
Simplifying this expression, we get: xy = y^2 + 46y
To minimize this quadratic expression, we can use calculus by taking its derivative with respect to y and setting it equal to zero:
d/dy (y^2 + 46y) = 2y + 46 = 0
Solving for y, we get: y = -23
Substituting this value of y back into the equation x = y + 46, we get: x = -23 + 46 = 23
So the two numbers that will minimize the product are -23 and 23, since their difference is 46 and they are as close as possible to each other