Convex set in ${ℝ}^2$ proof

Question

How can I know and prove that whether the set B = {$(x,y)∈ℝ^2:{x}^3≤y$} is convex or not?

Thanks in advance.

Answer 1

This set is the set of all points above the graph for $y=x^3$. Looking at the graph, one can see that $(0,0)$ and $(-1,-1)$ are in your set, but $(-0.5,-0.5) = 0.5(0,0)+(1-0.5)(-1,-1))$ is not, since $(-0.5)^3 = -0.125 > -0.5$.