Question

Given that $x>y>z>0$. Which of the following is necessarily true?

• A. la$(x, y, z) <$ le$(x, y, z)$
• B. ma$(x, y, z) <$ la$(x, y, z)$
• C. ma$(x, y, z) <$ le$(x, y, z)$
• D. None of these

Hint:

When a function is defined as the average of two numbers, it is always less than the maximum of those two numbers.

Answer 1

The Correct Option is C

Definitions: la$(x, y, z) = \min(x+y, y+z)$, le$(x, y, z) = \max(x-y, y-z)$, ma$(x, y, z) = \frac{1}{2} [\text{le}(x, y, z) + \text{la}(x, y, z)]$. Given $x>y>z>0$: - $x - y>0$ and $y - z>0$, so le$(x, y, z)$ is the larger of these differences. - la$(x, y, z)$ is the smaller of $(x+y)$ and $(y+z)$ → clearly $y+z<x+y$ so la$(x, y, z) = y+z$. Since ma is the average of le and la, it must be less than the larger of the two, i.e., less than le. Hence, ma$(x, y, z) <$ le$(x, y, z)$.