1. The problem statement, all variables and given/known data The plane 4x − 3y + 8z = 5 intersects the cone z^2 = x^2 + y^2 in an ellipse. Use LaGrange Multipliers to find the highest and lowest points on the ellipse. 2. Relevant equations Lagrange Multiplier 3. The attempt at a solution I guess I lack an understanding of Lagrange multiplier to begin solving this problem. Normally, I would be given some kinda of function express in term of x, y, and z and the plane would be the constraint for the function. Then I would just solve using Lagrange multiplier. Here what is my function and what is my constrain? I was thinking of using the cone as my function, but then I would have 0 = x^2 + y^2 - z^2 which is a surface and not a function of x, y, and z or is it? And the plane is the constrain, but then I wouldn't be just looking for the max and min of the ellipse intersection anymore, it would be the max and min of the cone with the plane as the constrain. I'm guessing both the plane and the cone are constrains in this problem because their intersection is the actual constrain, then what is my function to maximize/minimize? Thanks in advance for any help/explanation.