Lagrange Multiplier Question and my attempt

There must be some constraints on $x,y,z$

Without using Lagrange Multiplier

Assuming $x,y,z\gt0$

Using AM-GM

$$\frac{x+y+z}{3}\ge\sqrt[3]{xyz}$$

$$x+y+z=a\ge3\sqrt[3]{xyz}$$ $$\frac{a^3}{27}\ge xyz$$ and equality occurs when $x=y=z$

You have derived the correct result with the Lagrange Multiplier method:

$xy = yz = zx = -\lambda$

Here $\lambda$ is a non-zero parameter. From this it follows that $x, y$ and $z$ can not be zero. The only solution is $x = y = z = \sqrt{-\lambda}$. And therefore $x = y = z = a/3$.