Why would distnce magically decrease? What's a Tangent and what's a Normal?
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?
I`m not sure how finding the normal to the surface would provide anything... just a vector, not a distance, you don`t have any points on the plane.
Without a specific problem, it's difficult to discuss the approach and solution.
Could it be something like this?
Find the minimum distance from the point (3, 1, -2)
. . to the plane 2x + 3y - 6z + 7 .= .0.
Let P(x,y,z) be a point on the plane.
Its distance from Q(3,1,-2) is given by:
. . . . . . . . . ____________________________
. . . d . = . √(x - 3)^2 + (y - 1)^2 + (z + 2)^2
or: .D . = . (x - 3)^2 + (y - 1)^2 + (z + 2)^2
The constraint is: .2x + 3y - 6z + 7 .= .0
The function to minimize is:
. . f(x,y,L) . = . (x - 3)^2 + (y - 1)^2 + (z + 2)^2 + L(2x + 3y - 6z + 7)