If x and y are integers and |x - y| = 12, what is the minimum possible value of xy?

Question

Sahil Verma · Accepted Answer

We're given that |x - y| = 12. This means that the difference between x and y is 12, but we don't know which is larger. So we have two possible scenarios:x - y = 12 or y - x = 12
Let's consider the product xy. We can rewrite this as:xy = x(x - 12) when x > yxy = x(x + 12) when y > x
Let's look at the first case: xy = x(x - 12)We can expand this: xy = x&sup2; - 12xThis is a quadratic function in terms of x. Its graph would be a parabola opening upwards.
To find the minimum value of this function, we can differentiate it and set it to zero:d/dx(x^2 - 12x) = 2x - 12Set this to zero: 2x - 12 = 0Solve for x: x = 6
This means that when x = 6, y must be -6 (because x - y = 12).
Let's check the second case: xy = x(x + 12)Following the same process, we'd find that the minimum occurs when x = -6 and y = 6.
In both cases, the minimum value of xy is the same:xy = 6 * (-6) = -36
Therefore, the minimum possible value of xy is -36.