[Math] Sufficient condition for global maximum of strictly quasi-concave functions (unconstrained)

Question

Suppose $f(x)$ is a function from $\mathbb{R}$ to $\mathbb{R}$ and $f$ is strictly quasi-concave. If $x^*$ is a point such that $f'(x^*)=0$, then can we say that $x^*$ is a global maximum of this function? What about local maximum then?

Thanks in advance!!!

Answer 1

Take $f(x) = (\operatorname{sgn} x) x^2$. Then $f$ is strictly monotonic (hence strictly quasi-convex) and $f'(0) = 0$, but $f$ has no local $\max$ or $\min$.