find all points on the surface z^2=xy+1 that are closet to the origin?

Question

Calculus 3

Answer 1

the points of this surface are {(x,y,√(xy+1)) in R^3: x,y in R}:

We want to minimize the distance, this can be done (with less difficulty) by minimizing the distance squared. Therefore we want to minimize D(x,y)=x^2+y^2+xy+1. Taking the partials, we get D_x(x,y)=2x+y and D_y(x,y)=2y+x. Setting these equal to zero, we get:
2x+y=0
x+2y=0

(by taking second partials, we see that this is truly a minimum and not a maximum or saddle point).

It is easy to see that the is obtained when x=y=0. This corresponds to the point (0,0,1). Thus (0,0,1) is the closest point to the origin.

Answer 2

nah, don't feel like it right now.......next time say please.