(Lagrange Multiplier method and partial differentiation).

Find the maximum value of a rectangular box that can be inscribed in an ellipsoid
x^2 /4 + y^2 /64 + z^2 /81 = 1
with sides parallel to the coordinate to the coordinate axes.

My work: http://f.imgtmp.com/LNWh5.jpg

I feel like I should have the correct answer but I did something wrong and I would appreciate it if someone could point out my mistake as well as tell me why it's wrong.

Question

(Lagrange Multiplier method and partial differentiation).

Find the maximum value of a rectangular box that can be inscribed in an ellipsoid
x^2 /4 + y^2 /64 + z^2 /81 = 1
with sides parallel to the coordinate to the coordinate axes.

My work: http://f.imgtmp.com/LNWh5.jpg

I feel like I should have the correct answer but I did something wrong and I would appreciate it if someone could point out my mistake as well as tell me why it's wrong.