Final answer:
To find two numbers whose difference is 112 and whose product is a minimum, we can use quadratic functions. By finding the vertex of the quadratic function, we can determine that the minimum product occurs when the numbers are -28 and -28.
Step-by-step explanation:
To find two numbers whose difference is 112 and whose product is a minimum, we need to understand the concept of quadratic functions. Let's assume the two numbers are x and y, where x > y. Since their difference is 112, we can write the equation:
x - y = 112
To minimize the product of x and y, we need to maximize their sum. So, let's write an equation for the sum:
Sum = x + y
Now, we can express x in terms of y using the first equation:
x = y + 112
Substituting this value of x in the equation for the sum:
Sum = (y + 112) + y = 2y + 112
To find the minimum product, we need to find the minimum value of this sum. Since y is a positive number, the minimum value occurs when y is minimized. So, we can find the minimum by finding the vertex of the quadratic function:
The x-coordinate of the vertex is given by:
x = -b/2a
In this case, a = 2 and b = 112. Substituting these values, we get:
x = -112/2(2) = -56/2 = -28
So, the minimum product occurs when the numbers are -28 and -28. Therefore, option (a) 56, 56 is the correct answer.