Question

The distance from origin to point (x,y,z) is:​

Answer 1

Answer:

The distance from the origin to a point (x, y, z) on the surface is

d =

p

x

2 + y

2 + z

2 where y

2 = 36 + xz, so we minimize

d

2 = x

2 + 36 + xz + z

2 = f (x, z). Then fx = 2x + z, fz = x + 2z, and

fx = 0, fz = 0 ⇒ x = 0, z = 0, so the only critical point is (0, 0).

D (0, 0) = (2)(2) − 1 = 3 > 0 with fxx (0, 0) = 2 > 0, so this is a minimum.

Thus y

2 = 36 + 0 ⇒ y = ±6 and the points on the surface closest to the

origin are (0, ±6, 0).