State the Lagrange sufficiency theorem.

Find the maximum of $\log (x y z)$ over $x, y, z>0$ subject to the constraint

$$x^{2}+y^{2}+z^{2}=1$$

using Lagrange multipliers. Carefully justify why your solution is in fact the maximum.

Find the maximum of $\log (x y z)$ over $x, y, z>0$ subject to the constraint

$$x^{2}+y^{2}+z^{2} \leqslant 1$$