Explanation

(c)
Let $x + y = u$ and $x - y = v$
$\Rightarrow \ \ \ \ \ \ x = \frac{u + v}{2}$ and $y = \frac{u - v}{2}$
$\therefore\ \ \ f(u,\ v) = \left( \frac{u + v}{2} \right)\left( \frac{u - v}{2} \right)$
The arithmetic mean of $f(u,v)$ and $f(v,\ u)$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \frac{f(u,\ v) + f(v,\ u)}{2}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \frac{\frac{u + v}{2}\left( \frac{u - v}{2} \right) + \left( \frac{u + v}{2} \right)\left( \frac{v - u}{2} \right)}{2} = 0$