Given the surface z=xy+1. Prove that on this surface there exists a point closest to the origin, and find that point.
Justify your solution.
Thank you very much!
Since z=xy+1, thus any point on surface is (x,y,xy+1)
The distance of this point from the origin is,
sqrt(x^2+y^2+(xy+1)^2) = sqrt(x^2+y^2+x^2y^2+2xy+1)
We need to minimize this.
Instead let us minize its square, that is to remove the square root.
We need to minimize,
g(x,y)=x^2+y^2+x^2y^2+2xy+1
Find its partial derivatives and make them zero.