Searching on the internet, I have seen that this problem can be solvable using the method of Lagrange multipliers, being the Cartesian equation of the ellipsoid the one constraint. But I cannot understand how the people came to know the points of the parallelepiped, especially because I am awful visualizing tridimensional forms, and I can't understand why the equation of the quadric is a constraint either.
For example, when the equation of the ellipsoid is x²/a² + y²/b² + z²/c² = 1, people wrote that the points of the parallelepiped are (±x, ±y, ±z). This is very confusing. Why are these the points? However, the conclusion that the volume is 8|x||y||z|, assuming that those points are really true, is easy to grasp.