I had a question on my final exam that I can't seem to get my mind off...

Unfortunately I don't remember the equations that were given but the general question was:

How is the maximum/minimum value (found with Lagrange multipliers) related to the distance from a point (the max/min) to a plane?

The only reasoning I've managed to do is this:

the plane is the constraint function, so as the distance between the point and plane goes to zero, you get closer to the max/min value of the objective function.

To me this seems like too basic of reasoning to be on the final.

Any other thoughts?