Find the critical points of the following constrained optimization problem

subject to

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

My work:

Let

Lagrangian is

Then grad is equivalent to

(1)

=====>

=====>

(2)

(3)

with constraint

(4)

(1), (2), and (3) implies that

so with constraint (4) we have

so

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

and

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.