I encountered the following function $$f(x) = x^{3} + x^{2} + 1 $$ and was asked to check for quasiconcavity.
On inspecting the first and second order derivatives I found that the function increases in the interval $(- \infty, -\frac{2}{3})$ , decreases in the interval $ (-\frac{2}{3} , 0) $ and finally increases in $ (0 , +\infty ) $. Hence, $f(x)$ is not increasing always in its domain. This gives us the result that the function is not quasiconcave, or is it?
The function $f$ is quasiconcave in $\mathbb{R}$ iff for any $x<y$ and for any $t\in [x,y]$, $$f(t)\geq \min(f(x),f(y) \tag{*}.$$ Now, according to your analysis, $f$ has a local minimum at $x_0=0$ (which is a global minimum in $[-2/3,+\infty)$). Hence $f$ is not quasiconcave because $(*)$ does not hold for $t=0$, $x=-2/3$ and $y=1$.