Find the points on the surface y^2 = 81 + xz that are closest to the origin. (x, y, z) = (smaller y-value) (x, y, z) = (larger y-value)

You want to minimize subject to . The distance of any point on the given surface to the origin is actually , but it's easy to show that and share the same critical points.

The Lagrangian is

with critical points when

If , then

which tells us about critical points at (-9, 0, -9), (-9, 0, 9), (9, 0, -9), and (9, 0, 9).

If , then

which gives 2 more critical points, (0, -9, 0) and (0, 9, 0).

Let . Then

so the closest points are (0, -9, 0) and (0, 9, 0).

step-by-step explanation: