Maximizing function in 2 variables with constraint in 3

Question

\begin{gather*} \max &{3x_1-x_2+x_2^3} \\ s.t. &x_1+x_2+x_3 \leq 0 \\ &-x_1+2x_2+x_3^2=0 \end{gather*} I calculated the partial derivatives: $$\frac{\partial f}{\partial x_1}= 3$$ $$\frac{\partial f}{\partial x_2}= -1+3x_2^2$$ $$\frac{\partial f}{\partial x_3}= 0$$ In order to use the KKT condition, but how should I proceed with the 3rd derivative equal null? And should I convert it in minimization problem?

Answer 1

Solve the last equation for $x_3$ and plug it into the inequality constraint. You want the negative square root to make the inequality easy to satisfy.