jmc

Question

Let P be a point on the graph of the equation xyz = 1 in three-dimensional space. Find the minimum possible distance between P and the origin.

Accepted Answer

Let P = (x,y,z), so xyz = 1. We want to minimize x^2 + y^2 + z^2, which is equivalent to minimizing x^2 + y^2 + z^2. By AM-GM, x^2 + y^2 + z^2 3 [3]x^2 y^2 z^2 = 3,so x^2 + y^2 + z^2 3. Equality occurs when x = y = z = 1, so the minimum distance is 3.