If the diagonal of a rectangular box measures L, then what is its maximum volume ( in terms of L ) ?
$L = \sqrt{x^2 + y^2 + z^2}$, then $L^2 = x^2 + y^2 + z^2$.
By AM-GM, $$L^2 \ge 3V^{\frac{2}{3}}$$.
Therefore the maximum volume is $ V = \frac{L^2}{3}\sqrt\frac{L^2}{3}$.
Is this correct?