Consider the optimization problem
\begin{equation} \min_{(x_1,x_2)\in\Omega}f(x_1,x_2), \end{equation}
where $f(x_1,x_2)=(1+x_1^2)x_2$, $\Omega=\{(x_1,x_2):x_1^3\leq x_2\}$.
Identify all feasible directions at (0,0).
I could show that the directions $(d_1,d_2)^T$ with $d_1\leq0$ and $d_2\geq0$ are feasible and the ones with $d_1\in\mathbb{R}$ and $d_2<0$ are not feasible. However, I can't show anything for the ones with $d_1\geq0$ and $d_2\geq0$.
The constraint is given by $g(x_1,x_2) \le 0$, where $g(x_1,x_2) := x_1^3-x_2$. We have $\nabla g(x_1,x_2) = (3x_1^2,-1)$. Since, $g(0,0) = 0$, the feasible directions $d = (d_1,d_2)$ at $(0,0)$ satisfy $d^T\nabla g(0,0) \le 0$. This gives $$d^T\nabla g(0,0) = d_1 0 - d_2 \le 0.$$ Hence all feasible $d=(d_1,d_2)$ require $d_2 \ge 0$. When $d_2 > 0$, this is sufficient (first-order change dominates in the vicinity of $(0,0)$). When $d_2 = 0$, we must have $d_1 \le 0$.