Consider the set $\{(x,y) \in \Bbb R_+ \times \Bbb R \text{ s.t. } y\leq \ln x - e^x\}$. This set is:
A) A linear subspace of $\mathbb R^2$
B) Convex
C) Convex & a linear subspace of $\mathbb R^2$
D) neither convex nor linear subspace of $\mathbb R^2$
I got the part that the set should be convex. I drew a rough graph of the set and it can be viewed as the negative of the distance between the graphs of $e^x$ and $\ln(x)$. So $y$ being negative, first increases at a decreasing rate, reaches a max(still negative) and then decreases at an increasing rate, giving us a concave shaped graph in the fourth quadrant. The entire space below this graph and the graph itself is what the set mentioned in the above question is. What I don't understand although, is if this is a linear subspace. how to find that out?
Hint: What's the relationship between convex functions and convex sets?
In $\Bbb R^2$, the only linear subspaces are all of $\Bbb R^2$, lines through the origin, and the set $\{(0,0)\}$. It should be clear that this set is none of these things.