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This seems more like a basic level general math question, but since it was asked in my calc III/linear class on my final exam and possibly involves derivatives, I decided to post here.

The original question:

Given f(x,y) = x^2 + y^2, find the point on the surface that is closest to the point (3,0,0).

I sketched the figure (a parabloid) relative to the point and saw that it was obvious that the point on the surface which was closest to (3,0,0) was on the xz-plane.

I drew a projection of the xz-plane with the point, and then realized that the shortest line would be the one which was perpindicular to the curve z=x^2.

It's probably got something to do with relating the derivative of the curve to the line, but I'm not sure how. I ended up saying that the slope of the curve = 2x (dy/dx), so the slope of the line from (3,0,0) to the point on z=x^2 must be 1/(2x).

However, that's where I got stuck. It's really frustrating because it seems like it's so obvious, but the answer keeps eluding me!

How could I go about solving this problem? Also, what's the more generalized way of doing this in three dimensions? I'm sure that mine is not the quickest or most general form, considering I had a special case where the point lied on an axis.

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# Given a point and a curve: how to find perpindicular POI?

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