Consider the problem
$$\left\{\begin{matrix} \min & x^2 -2xy + 2xz +y^2 + 4yz + z^2 + c_1x + c_2y + c_3z \\ s.t & g(x,y,z)=-x^2 -4xy - 4xz -2y^2 -4yz - 2z^2 + x -y+z+4 =0 \\ \; & h(x,y,z)=-x-y+z+5 \geq 0 \end{matrix}\right. $$
For which values of $c_1, c_2$ and $c_3$ you can guarantee that p=(1, 2, -2) is a local minimum? Repeat the same question, trying to find the values that you can guarantee it is not a local minimum and you can't decide if it is or not.
My attempt I first tried to check the first order necessary condition. Since both the restrictions are active at this point:
$$ \nabla f(p) + \lambda \nabla g(p) + \mu \nabla (-h)(p) = 0$$ for $\mu \geq 0$. Getting the gradients and substituting we have
$$\left\{\begin{matrix} -6+c_1-\lambda+\mu=0\\ -6+c_2-5\lambda+\mu = 0\\ 6+c_3-3\lambda-\mu =0 \end{matrix}\right. $$
From first and third, and second and third we have
$$\lambda = \frac{c_1+c_3}{4} = \frac{c_2+c_3}{8}$$
I've tried to get $\mu$ in function of the c's, but this just help to find when it is not a minimum. Also, I couldnt use it to check the second order conditions.
Maybe what I got is wrong, but I am stuck. I would be glad if someone could help me.
Thanks in advance!