why Quasiconvex function is not concave?

Question

A quasiconvex function is a function whose all sublevel set are convex. I am curious to know whether a quasiconvex function is a concave function.

Answer 1

In general, no. The function $x \mapsto x^2$ is quasiconvex but not concave.

The function $x \mapsto x$ is quasiconvex and concave.

Answer 2

The function $x \mapsto x^3$ is quasiconvex, quasiconcave, but neither convex nor concave.