Find three real numbers x, y, and z whose sum is 6 and the sum of whose squares is as small as possible. g

We're minimizing subject to . Using Lagrange multipliers, we have the Lagrangian



with partial derivatives



Set each partial derivative equal to 0:



Subtracting the second equation from the first, we find



Similarly, we can determine that and by taking any two of the first three equations. So if determines a critical point, then



So the smallest value for the sum of squares is when .