Answer
$\dfrac{L^3}{3\sqrt 3}$
Work Step by Step
Use Lagrange Multipliers Method:
$\nabla f(x,y,z)=\lambda \nabla g(x,y,z)$
Volume of a box is given by $f(x,y,z)=V=xyz$
This yields $\nabla f=\lt yz,xz,xy \gt$ and $\lambda \nabla g=\lambda \lt 2x,2y,2z \gt$
Using the constraint condition we get, $yz=\lambda 2x, xz=\lambda 2y,xy=\lambda 2z$
After solving, we get $x=y=z$
Since, $g(x,y,z)=x^2+y^2+z^2=L^2$ yields $x^2+x^2+x^2=L^2$
$x=y=z=\dfrac{L}{\sqrt 3}$
Hence,
Volume of a box is given by $f(x,y,z)=V=xyz=(\dfrac{L}{\sqrt 3})^3=\dfrac{L^3}{3\sqrt 3}$
