Which point on the surface 1/x + 1/y + 1/z = 1 xyz > 0 is closest to the origin?

Question

Please show work so that I can follow along

Answer 1

d = (x² + y² + z²)^1/2; F(x, y, z, λ) = (x² + y² + z²)^1/2 - λ(1/x + 1/y + 1/z - 1); F_x = x/(x² + y² + z²)^1/2 - λ/x² = 0;

F_y = y/(x² + y² + z²)^1/2 - λ/y² = 0; F_z = z/(x² + y² + z²)^1/2 - λ/z²; F_λ = - 1/x - 1/y - 1/z + 1 = 0.

x³ = y³ = z³ = λ(x² + y² + z²)^1/2; x = y = z; 3/x = 1; x = y = z = 3. Point (3, 3, 3)

Distance d = 3√3. To prove that this is closest point to origin, we take some different popint (6,2, 3)

1/6 + 1/2 + 1/3 = 1. Distance d = √36 + 4 + 9 = 7 > 3√3