Find the critical points of the following constrained optimization problem

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

$\displaystyle g(x_1, x_2, x_3) = 2x_1 + x_3 + x3 = 4$

and check that they are non-degenerate. Determine the local minima and maxima.

My work:

Let $\displaystyle g(x) = (2x_1 + x_2 + x_3 - 4)$

Lagrangian is

$\displaystyle L(x) = f(x) + g(x)$

$\displaystyle = 2x_1^2 + x_2^2 + x_3^2 + \lambda (2x_1 + x_2 + x_3 - 4)$

Then grad $\displaystyle L = 0$ is equivalent to

(1) $\displaystyle \frac{\partial L}{\partial x_1} = 4x_1 + 2 \lambda = 0 $

=====> $\displaystyle 2(2x_1 + \lambda) = 0$

=====> $\displaystyle 2x_1 + \lambda = 0$

(2) $\displaystyle \frac{\partial L}{\partial _2} = 2x_1 + \lambda = 0 $

(3) $\displaystyle \frac{\partial L}{\partial x_3} = 2x_2 + \lambda = 0 $

with constraint

(4) $\displaystyle 2x_1 + x_2 + x_3 = 4 $

(1), (2), and (3) implies that $\displaystyle x_1 = x_2 = x_3$

so with constraint (4) we have $\displaystyle x_1 = x_2 = x_3 = 1$

so $\displaystyle \lambda = -2$

Now I'm supposed to use the Hessian of L which I think is

$\displaystyle H_L = \begin{pmatrix} L_{x_1x_1} & L_{x_1x_2} & L_{x_1x_3}\\ L_{x_2x_1} & L_{x_2x_2} & L_{x_2x_3}\\ L_{x_3x_1} & L_{x_3x_2} & L_{x_3x_3}

\end{pmatrix}$ $\displaystyle = \begin{pmatrix} 4 & 0 & 0 \\ 0 & 2 & 0 \\0 & 0 & 3 \end{pmatrix}$

and

$\displaystyle B = \bigtriangledown g = \begin{pmatrix} 2 \\ 1 \\1 \end{pmatrix}$

This is where I get lost.....

I think I need to find the determinate of the bordered Hessian and find a tangent vector to somehow check if it is non-degenerate and determine the local minima and maxima.

Can anyone please help?

Thanks in advance.