Question

theorem: for any real numbers, x and y, max (x,y) = (1/2) one of the cases in the proof ot the theorem uses the assumtpion |x-y|=x-y. select the case that corresponds to this argument

Answer 1

The case |x - y| = x - y corresponds to when x is greater than or equal to y, thus proving the max(x, y) equals x for this scenario in the theorem.

Step-by-step explanation:

The theorem in question states that for any real numbers x and y, max(x,y) = (1/2)(x + y + |x - y|). When the assumption |x - y| = x - y is used, it corresponds to the case where x is greater than or equal to y. This is because the absolute value of a real number is equal to the number itself if it is positive (or zero) and is the negation of the number if it is negative. Hence, if x ≥ y, then x - y is non-negative, and |x - y| = x - y. In this case, max(x,y) is x itself.

In terms of proving this, we apply the definition of the maximum function to get:

max(x,y) = (1/2)(x + y + x - y)

max(x,y) = (1/2)(2x)

max(x,y) = x

This proves the theorem for the case when x ≥ y.