1. The problem statement, all variables and given/known data Find the greatest and least distances from the point P(2,1,-2) to the sphere x^2 + y^2 + z^2 = 1 using Lagrange Multipliers. (The question had an = before the z^2 which was a typo and I think it should of been a + so just saying in case I made a wrong correction.) 2. Relevant equations Partial differentiation. Lagrange Multiplier equation. 3. The attempt at a solution I tried visualizing this geometrically and I see that it is a sphere centered at (0,0,0) with a radius of 1 and I need to find the shortest line from the point to the surface of the sphere so I was thinking of taking a vector <2,1,-2> and dotting it with the gradient of f(x,y,z) and finding x, y and z values such that the dot product is 0. I'm not entirely sure if this approach is good (please, tell me if it is or isn't since I'm curious) but I have to do it using Lagrange Multipliers so I'm left thinking: "What's the constraint explicitly?" I'm having a bit of trouble understanding what the greatest distance is as well. Is it the line from the point to an edge of the surface area of the sphere facing the point (which is a circle) such that we maximize the distance by making the line joining the point to the sphere be the hypotenuse of a triangle? A push in the right direction or any help whatsoever would be greatly appreciated! Thanks in advance!