Final Answer:
The two numbers with a difference of 122 and whose product is a minimum are -61 and 61.
Step-by-step explanation:
To find two numbers with a difference of 122 and whose product is minimum, let's denote the two numbers as x and y, where y = x + 122.
The product of two numbers, P = x * (x + 122).
To minimize the product, let's express the product in terms of a single variable:
P = x^2 + 122x.
To find the minimum value of P, we'll differentiate it with respect to x and set it to zero:
dP/dx = 2x + 122 = 0.
Solving for x gives x = -61. Substituting this into y = x + 122, we get y = 61.
Therefore, the two numbers that fulfill the condition are -61 and 61, having a difference of 122, and their product is minimized at -61 * 61, which equals -3721. This solution complies with the condition while minimizing the product of the two numbers.