State the Lagrangian sufficiency theorem.

Use Lagrange multipliers to find the optimal values of $x_{1}$ and $x_{2}$ in the problem: maximize $x_{1}^{2}+x_{2} \quad$ subject to $\quad x_{1}^{2}+\frac{1}{2} x_{2}^{2} \leqslant b_{1}, \quad x_{1} \geqslant b_{2}$ and $x_{1}, x_{2} \geqslant 0$ for all values of $b_{1}, b_{2}$ such that $b_{1}-b_{2}^{2} \geqslant 0$.