Find the points on the surface y2 = 25 + xz that are closest to the origin.

You are essentially minimizing subject to . (The distance between the origin and any point on the given surface is , but and share the same critical points.)

Via Lagrange multipliers, we have Lagrangian



with partial derivatives (set equal to 0)










We assume , which means .



In the first case, we have



which means one of must be positive, and the other is negative. From we have , so



So we get two critical points, (-5, 0, 5) and (5, 0, -5).

In the second case, if , we get



which leads us to



i.e. we have two additional critical points (0, 5, 0) and (0, -5, 0).

At each of these points, we get respective distances from the origin of , so the two closest points to the origin on the surface are (0, 5, 0) and (0, -5, 0).