Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. (If an answer does not exist, enter DNE.) $ \ \ f(x, y, z) = xyz \ ; \ \ x^2 + 2y^2 + 3z^2 = 96$
What I have gotten to: $\Delta f = \ <yz, xz, xy>$ and $\Delta g = \ λ<2x, 4y, 6z>$
set them equal and get: $x^2 = 2y^2$ and $z^2 = \frac{2}{3} y^2$ .
Then: $$ \begin{cases}x^2 + 2y^2 +3z^2 = 96 \\ 6y^2 = 96 \\ y = \pm16 \end{cases} $$
Plugging $y$ into $z^2$ and $x^2$ equations above, you get the points: $(\pm16\sqrt{2}, \ \pm 16, \ \pm16\sqrt{2/3} ) \ .$
Plugging the points back into $f$, I got a maximum of $32\sqrt{3}$ and a minimum of $-32\sqrt{3}$.
Where did I go wrong?
Thank you in advance!