Convexity-Quasiconvexity differences and implications

Question

So we have the following theorem:

Let $f : A ⊆ \mathbb{R}^n $ convex (concave) on $A$ convex and let $x_0$ be a critical point. Then x_0 is a global minimum (maximum) point for $f$ in A.

What would change, if $f$ was quasiconcave/quasiconvex?

Answer 1

The theorem would be false. Consider:

$$ f(x) = x^3 $$

Then $f$ is strictly quasi-concave, strictly quasi-convex, and there is a unique critical point which is neither a local nor global minima (nor maxima).

Edit: One example of the use of quasi-concavity in maximization problems is from game theory. Suppose for some agent $i$, their utility $u_i(s_i, s_{-i})$ depends upon their choice of strategy ($s_i)$ and the choice of strategy of all the other opponents ($s_{-i}$, where $-i$ may be read as 'not $i$'). For a given choice of strategy $s_{-i}$ by the opponents, define: $$ BR_i(s_{-i}) = \textrm{arg max}_{\tilde{s}} u_i(\tilde{s}, s_{-i}) $$ to be the utility-maximizing choices of response (the 'best response' correspondence). Insofar as the set of strategies of $i$ is convex and for each $s_{-i}$ $u_i$ is quasiconcave in $s_i$, $BR_i$ will have convex values as a correpsondence. This, coupled with a number of nice continuity properties it will generally possess, allow us to easily guarantee the existence of equilibria.

In essence, on a convex domain, quasiconcavity implies the set of maximizers is convex (resp. quasiconvexity and minimizers).

Answer 2

For quasi-convex (-concave) functions the sets of global minima (maxima) are convex.