By definition, (x 1 , . . . , x n ) is a critical point of V if and only if the gradient ∇V (x 1 , . . . , x n ) is zero. But, the gradient of V is zero if and only if the gradient of each g i is zero. Therefore, (x 1 , . . . , x n ) is a critical point of V if and only if each coordinate x i is a critical point of g i .