We have the following Method of Lagrange Multipliers in Three Variables, which is similar to what we have for two variables. Suppose we are looking for extreme values of a

differentiable function f(x, y, z) subject to a constraint g(x, y, z) = 0, with g differentiable and Vg +0 when g(x, y, z) = 0. We look for values of æ, y, z, and A satisfying Vf(x, y, z) = \Vg(x,y, z) and g(x, y, z) = 0. For each point (x, y, z) we have as a solution, take f(x, y, z) and compare the resulting values to find our maximum and minimum values. Using the method of Lagrange multipliers in three variables, find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid 9 x^{2}+36 y^{2}+4 z^{2}=36