I have a question with regards to lagrange multiplers. The question is as follows: 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). Now, it is clear to me that the distance from any point to (3,1,-1) is simply using the R^3 metric, d=$$\sqrt((x-3)^2+(y-1)^2+(z+1)^2$$ and I can make the algebra easier by considering $$d^2$$. However, my question is as follows: How can I be sure that the maximum or minimum occurs when $$\nabla(d^2(x,y,z))=\lambda\nabla(g(x,y,z))$$ where $$g(x,y,z)=x^2+y^2+z^2$$? I would like an explanation in terms of tangent and normal spaces please? As i'm unable to find one.
Furthermore, it is to my knowledge that the lagrange multiplier works when the contours of the function f(x,y) just touches the constraint g(x,y)=c. However, is it possible to have contours of f(x,y) where the contours increase and decrease and wouldn't then the lagrange multiplier not work?