 Solution

For positive edge lengths the volume is $V=abc$ and the surface area constraint is $2(ab+ac+bc)=S$. Introduce a Lagrange multiplier $\mu$:The stationary point equations areMultiplying these by $a,b,c$, respectively, and equating their left sides gives $\mu c(a-b)=0$ and $\mu b(a-c)=0$. Because all edge lengths are positive, the equations imply $\mu >0$; hence $a=b=c$. The constraint gives $6a^2=S$.

To establish a global maximum, rather than only a stationary point, apply the arithmetic-geometric mean inequality to $ab,ac,bc$:Equality holds exactly when $ab=ac=bc$, or $a=b=c$. Therefore