Find the local maxima and minima of the following problem by introducing two Lagrange multipliers:

$\displaystyle f(x_1,x_2,x_3) = x_1 + 2x_2 + 2x_3$

subject to

$\displaystyle x_1^2 + x_2^2 + 4x_3^2 = 2$ and $\displaystyle x_1^2 + (x_2 - 1)^2 + 4_3^2 = 3$