partial differentiation - maximum point - $3 $ variables

Question

I'm asked to maximise the volume of a rectangular cuboid inscribed in the ellipsoid

$4x^2 + 4y^2 + 9z^2 = 36$

and i'm given a hint that the rectangular cuboid has volume $8xyz$

.

How do I go about finding the maximum for this question?

Answer 1

Hint: Try to use Lagrange Multiplier

Answer 2

Apply Arithmetic Mean - Geometric Mean inequality we have: $ 36 = 4x^2+4y^2+9z^2 \ge 3\sqrt[3]{144(xyz)^2}$. From this you can conclude. Observe that the max occurs at $2x = 2y = 3z$.