Find the critical points of the following constrained optimization problem

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

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 g(x) = (2x_1 + x_2 + x_3 - 4)

Lagrangian is

L(x) = f(x) + g(x)

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

Then grad L = 0 is equivalent to


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

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

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


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



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

with constraint

(4) 2x_1 + x_2 + x_3 = 4


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

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

so \lambda = -2

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

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}<br />
\end{pmatrix}  = \begin{pmatrix} 4 & 0 & 0 \\ 0 & 2 & 0 \\0 & 0 & 3 \end{pmatrix}

and

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.