Find the points on the sphere $x^2 + y^2 + z^2 = 4$ that are closest to, and farthest from the point (3, 1, -1). Find the points on the sphere $x^2 + y^2 + z^2 = 4$ that are closest to, and farthest from the point $(3, 1, -1)$.
I identified that this is a constrained optimisation problem which I will solve using an auxiliary function and a Lagrange multiplier however I am struggling to continue from there as I am unsure which is the constraining function.
Any help would be greatly appreciated!

$(2/\sqrt{11})(3,1,-1)$ is the closest point and $(-2/\sqrt{11})(3,1,-1)$ is the most distant point.
If you have to use Lagrange multipliiers....
minimize/maximize: $(x-3)^2 + (y-1)^2 + (z+1)^2$  (this is the distance squared from x,y,z to your point.)
constrained by: $x^2 + y^2 + z^2 = 4$
$F(x,y,z,\lambda) = (x-3)^2 + (y-1)^2 + (z+1)^2 - \lambda(x^2 + y^2 + z^2 - 4)$
Now find, $\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z},\frac{\partial F}{\partial \lambda}$ and set them all equal to $0,$  and solve the system of equations.  
