Proving quasiconvexity of $x^3 + y^3$

Question

I was studying quasiconvex functions, and suddenly I have stumbled upon the function $f(x,y)= x^3+y^3$.

Is it quasiconvex? From the graph it seems like.

Answer 1

It is not quasiconvex nor is it quasiconcave.

In general, for $a,b,c>0$ and $a>b$, consider local extrema of $x^3+y^3$ along the line $ax+by=c$. substituting the constraint gives us a cubic in $x$ and cubics of a real variable always have range $\mathbb{R}$. The cubic you get will always have two distinct critical points, a local max, and a local min, and so it is not quasiconvex or quasiconcave. Therefore, picking two points with appropriate $x$ on such a line will violate the definitions of both quasiconvexity and quasiconcavity for $f$.