Use Lagrange multipliers to find the points?

Question

Use Lagrange multipliers to find the points on the ellipsoid x^2+y^2-3y+yz-6z+z^2=-5 that are closest and farthest from the xy-plane.

Answer 1

x² + y² -3y +yz -6z +z² = 5

 

Solve for z

 

It turns out there are 2 solutions:

 

SOLUTION 1: z = [(-4x² -3y² +56)^½ -y +6]/2

 

SOLUTION 2: z= -[(-4x² -3y² +56)^½ +56) +y -6]/2