Optimisation - first order necessary conditions, directions, Chong & Zak Q. 6.16

Question

For this question from Introduction to optimisation by Chong & Zak :

The answer is :

I'm unsure about what's been done for part (b) here.

From part (a) it seems that for $d$ we have $d= [d_1, d_2]$ for $d_1, d_2 \in \mathbb{R}$.

I don't understand where $d_2 \leq 2d_1$ has come from for part (b) though

Answer 1

Here is a quick sketch:

In order to find the (local) cone of feasible directions at the origin, we can take the tangent lines along the boundary lines that define $\Omega$, namely, $x_2=0$ and $x_2 = x_1^2 + 2x_1$. These are $x_2=0$ and $x_2 = 2x_1 + 2$ as shown in the figure. We see that all directions $d=(d_1, d_2)$ with $d_2 \leq 2d_1$ are feasible directions at the origin.