Find the point(s) on the surface z2=xy+1 which are closest to the point (10,11,0).

First, note that for any differentiable functions , we have attaining its relative extrema at the same points as . This follows from the fact that



and so has critical points at the same values of as does .

Now recall that the distance between any point in and is



By the fact mentioned above, we know that the minimum distance between any point on the surface will occur at the same point that minimizes the squared distance; that is,



We can then use the following function as the Lagrangian:



which has partial derivatives



Set each partial derivative equal to 0. The third equation yields



In the case of , we have , or . Note that it must be the case that . However, substituting this into the first two equations gives




from which it follows necessarily that and . So we've arrived at a contradiction.

This means we must have From this, our first two equations become




which in turn yield



Therefore we have two points on which are closest to (10, 11, 0), namely .