I was looking for the solutions for these two problems:
Find the point on the plane $x+2y+3z= 13$ closest to the point(1,1,1).
Find the point on the sphere $x^2+y^2 +z^2 = 4$ farthest from the point(1,-1,1).
The solutions appear to use the same process using Lagrange multiplies. I.e. Find the gradient, of $f$ and $g$. Use the equation $\Delta f=\lambda \Delta g$ and solve for $\lambda$. Plug back in to find values of $x,y,z$ . Then evaluate $f$ at that point.
So does it make any difference finding the closest point or the farthest point?