Optimization greatest possible volume Problem

[nrm]

Question: [ A rectangular box, whose edges are parallel to the coordinate axes, is inscribed in the ellipsoid 96x^2 + 4y^2 + 4z^2 = 36, What is the greatest possible volume for such a box ]

I realize that the volume of the box: V = (2x)(2y)(2z) = 8xyz
Thus far I've solved for z^2 in the equation of the ellipsoid and then squared the volume so that I could make the substitution easier
V^2 = 64(x^2)(y^2)(9-24x^2-y^2)
Then I've taken the partial derivates of this to look cor critical points, but here I get an algebraic nightmare and can't find critical points. I'm wondering if my initial steps are correct, it's the only thing I could think of doing.

Any help would be great. thank you

 

[mathmike]

what did you get for the partials

 

[nrm]

Partial with respect to x:
1152x(y^2)-6144(x^3)(y^2)-128x(y^4)

y
1152(x^2)y-3072(x^4)y-256(x^2)(y^3)

 

[HallsofIvy]

The partial derivative of
$$64(x^2)(y^2)(9-24x^2-y^2)$$
with respect to x is, by the product rule,
$$128xy^2(9- 24x^2- y^2)- 3072x^3y$$
set that equal to 0 and you should be able to do a lot of cancelling.

I would do this problem with "Lagrange multipliers" but you may not have had that yet.