I am working on the question displayed below. I know that the method of Lagrange Multipliers is used to find the solution for optimization problems constrained to one or more equalities and when our constraints also have inequalities, we need to extend the method to the KKT conditions. But I don't know where to start.
Let $\mathcal{L}=x+y+\lambda(x^2-(2-y)^3) + \mu y$
Then the necessary conditions of KKT are:
$\frac{d \mathcal{L}}{dx}=1+2\lambda x=0$ and
$\frac{d \mathcal{L}}{dy}=1-3\lambda(2-y)^2 + \mu=0 $ and
$\lambda \geq 0, \ \mu \geq 0$ and
$\lambda(x^2-(2-y)^3)=0$ and
$\mu y=0$
Now you can consider several cases. For instance you can consider the case that $x^2-(2-y)^2=0$ but $y \neq 0$ (so that only one of the constraints holds with equality). This case is the most plausible, since if $y=0$ would hold, our solution would become quite trivial, so probably not correct (you can verify this by also considering that case later, but I won't do that).
You then have to solve the system of equations
$\begin{cases} \mu=0\\ 1+2\lambda x=0\\ 1-3\lambda(2-y)^2=0\\ x^2-(2-y)^2=0 \end{cases}$
Can you take it from here?