Finding the maximum and minimum on a constraint.

Question

Let $f(x,y,z) = 2x + y$. Find the absolute maximum and minimum on the constraint $x+y+z=1$ So we know that $\nabla f(x,y,z) = \lambda \nabla g(x,y,z)$ where $g(x,y,z) = x+y+z-1$. Calculating, we receive $\nabla f(x,y,z) = (2,1,0)$ and $\lambda \nabla g(x,y,z) = (\lambda , \lambda , \lambda )$ This creates the system of linear equations $$2=\lambda \\ 1=\lambda \\ 0=\lambda \\ x+y+z=1$$ I'm not quite sure what this implies. We can see that $$\lambda \nabla g(x,y,z) = (0, 0, 0) = \nabla f(x,y,z)$$ However we calculated $\nabla f(x,y,z) = (2,1,0)$ Does this imply that there is no maximum or minimum on this constraint?