State and prove the Lagrangian sufficiency theorem.

Use the Lagrangian sufficiency theorem to find the minimum of $2 x_{1}^{2}+2 x_{2}^{2}+x_{3}^{2}$ subject to $x_{1}+x_{2}+x_{3}=1$ (where $x_{1}, x_{2}$ and $x_{3}$ are real).