I would like to understand under what conditions a cubic function $f(x)=ax^3+bx^2+cx+d$ can be considered quasiconcave or quasiconvex. Specifically, I am interested in finding conditions on the coefficients π , π , π that make the function either quasiconvex or quasiconcave. I have considered the following:
- Definition of a quasiconvex function: a function is quasiconvex if, for every $ \alpha \in \mathbb{R}, $the set $\{x \mid f(x)\leq\alpha\}$ is convex
- Condition for a quasiconvex function: Suppose $f:$ $R^{n} \to R$ is differentiable,then $f$ is quasiconvex if and only if dom$f$ is convex and for all $x,y\in $ dom$f$,$$ f(y)\leq f(x) \implies \nabla f(x)^{T}(y-x) \leq0 $$ I attempted to solve the problem using the definitions above, but it seems quite challenging to compute directly. Is there a more elegant or simpler way to determine the conditions on π , π , π for a cubic function to be quasiconvex or quasiconcave?
Any suggestions or references would be greatly appreciated. Thanks!
