Question

Use lagrange multipliers to find the shortest distance, d, from the point (4, 0, −5) to the plane x y z = 9.

Answer 1

One function you would be trying to minimize is
f(x, y, z) = d² = (x - 4)² + y² + (z + 5)²

Your values for x, y, z, and λ would be correct, but
d² = (20/3 - 4)² + (8/3)² + (-7/3 + 5)²
d² = (8/3)² + (8/3)² + (8/3)²
d² = 64/3
d = 8/sqrt(3) = 8sqrt(3)/3