 Solution

Assume distinct positive separations, and first label them so $A<B$. Let $x=R_0^6$, $a=A^6$ and $b=B^6$. The positive contrast isIts derivative has a particularly simple sign:It is positive below $x=\sqrt{ab}$ and negative above, so this is the global maximum. Both end limits $x\to 0$ and $x\to \infty$ give zero contrast. ThusTo recover the printed expression, factor its sixth-power radicand:Its sixth root is indeed $\sqrt{AB}$. The cancellation requires $A\ne B$. If $A=B$, every choice gives zero contrast and there is no unique optimum; the printed expression then has a removable algebraic $0/0$ without a unique maximization conclusion. For unordered $A,B$, maximize absolute contrast and replace the numerator of the maximum by $|B^3-A^3|$. A negative signed difference for $A>B$ is minimized, not maximized, by the same interior point. These qualifications are needed to make the statement precise.