Solving Utility Maximization with Lagrangian
You have added inequality constraints $x_1 \geq 0$ and $x_2 \geq 0$.
Hence, you now have Kuhn-Tucker conditions for complimentary slackness:
$$
\mu_1 x_1 = 0 \text{ and } \mu_2x_2 = 0
$$
So whenever $x_i &gt; 0$, by the complimentary slackness condition, $\mu_i = 0$.
See Karush-Kuhn-Tucker conditions. It also contains regularity conditions you ask for.