Maximizing a volume with fixed perimeter

Lagrange. $\nabla(xyz)=\lambda\nabla(2(xy+yz+zx))$:
$$yz=2\lambda(y+z),\quad xz=2\lambda(x+z),\quad xy=2\lambda(x+y).$$

Solve. Dividing the first two gives $y/x=z/y$, so $x=y=z$ at the critical point.

Conclusion. Maximum volume when $x=y=z=\sqrt{A/6}$, giving $V=(A/6)^{3/2}$.