= Solution
For a regular constraint $g=0$, an interior constrained extremum satisfies
$$
\nabla f=\lambda\nabla g,
$$
together with the constraint; boundary cases must also be checked.
Here maximize $f=xyz$ subject to $x^2+y^2+z^2=L^2$. The <Lagrange multiplier> equations give
$$
yz=2\lambda x,\qquad xz=2\lambda y,\qquad xy=2\lambda z.
$$
At a positive maximizer these imply $x=y=z=L/\sqrt3$. Boundary points have zero volume, so the maximum is
$$
\boxed{\frac{L^3}{3\sqrt3}}.
$$
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