Consider the function $h(x, y, z) = x^2 − 2y^2 + 3z^2$, and its level surface S through the point $(1, 1, 1)$. Determine the shortest distance from S to the origin by considering the function $f(x, y, z) = x^ 2 + y^ 2 + z^ 2$ and writing down the Lagrangian for finding the minimum of $f$ subject to $(x, y, z)$ lying on S. Determine the critical points and by comparison of the values of $f$ at these points determine the minimum distance.
So I've set up the Lagrangian, $L = x^2 + y^2 + z^2 - \lambda(x^2 + 2y^2 + 3z^2 -2)$
and found the partial derivatives with respect to each variable:
$L_x = 2x - 2\lambda x = 0$
$L_y = 2y + 4\lambda y = 0$
$L_z = 2z - 6\lambda z = 0$
$L_{\lambda} = x^2 -2y^2 +3z^2 - 2 = 0$
Firstly, am I correct in saying the level Surface S is given by subbing in the point (1,1,1), and secondly where do I go from here to find the minimum distance? Thanks.