I have to consider the nonlinear optimization problem
$$\min_{x\in\mathbb{R}^2}\ -\frac1{10}(x_1-4)^2-x_2^2\\ \text{s.t.}\ 1-x_1^2-x_2^2\leq0\\ x_1^2-4\leq0$$
and I have to compute all local minima using the necessary and sufficient optimality conditions.
When I consider the 3D graph of this function my intuition tells me that there exist no local minima.
Anyway I saw that LICQ holds for all $x_1\neq0$ and $x_2\neq 0$, and if I apply the KKT-conditions I get $$-\frac15(x_1^*-4)-2\mu_1x_1^*+2\mu_2x_1^*=0\\ 2x_2^*(\mu_1-1)=0\\ 1-x_1^{*2}-x_2^{*2}\leq0\\ x_1^{*2}-4\leq0\\\mu_1\geq0\\\mu_2\geq0\\\mu_1(1-x_1^{*2}-x_2^{*2})=0\\ \mu_2(x_1^{*2}-4)=0$$ for a possible KKT-point, which is not very helpful.
Do I have to proof that there exist no local minima?