1. The problem statement, all variables and given/known data Find the max and min values of f(x,y,z) = x3 - y3 + 6z2 on the sphere x2 + y2 + z2 = 25. 2. Relevant equations I will use λ to denote my Lagrange multipliers. 3. The attempt at a solution So clearly there is no interior to examine since we are on the boundary of the sphere. Thus we form our constraint equation F = f + λg where g = x2 + y2 + z2 - 25. After taking all the required derivatives and considering when the derivatives are zero we get the system of four equations : x(3x + 2λ) = 0 y(-3y + 2λ) = 0 2z(6+λ) = 0 25 - x2 - z2 = y2 Solving the first and second equations immediately we get x = 0 and y = 0. Now solving the fourth equation we get z = ±5 and then solving the third equation when z = ±5 yields λ = -6. Also z = 0 is another solution to the third equation. So I'm a bit stuck here. I have two maximum values occurring at (0,0,±5), but I'm apparently supposed to get minimum values at (-5,0,0) and (0,5,0). So I'm guessing I would want to consider two cases? When x=0 and when y=0 to get the minimums?