Find the max and min values of f(x, y,z)=x+y-z on the sphere x^2+y^2+z^2=81

Using Lagrange multipliers, we have the Lagrangian



with partial derivatives (set equal to 0)






Substituting the first three equations into the fourth allows us to solve for :



For each possible value of , we get two corresponding critical points at .

At these points, respectively, we get a maximum value of and a minimum value of .

Maybe the answer is 20mm. all i really did was just multiply, which is obvious.