Problem

Consider the part of the surface $x^3yz^2 = 2$ in the first octant ($x > 0, y > 0, z > 0$).

(a) Use Lagrange multipliers to find the point on the surface $x^3yz^2 = 2$ that is closest to the origin.

(b) Let $f(x,y) = x^2 + y^2 + \frac{2}{x^3y}$. Find all critical points of $f$ (points with $f_x = f_y = 0$). How is this related to part (a)?

(c) Find $f_{xx}, f_{xy}, f_{yy}$ and $D = f_{xx}f_{yy} - [f_{xy}]^2$ at the critical point.