showing strictly qusi convex?

Question

$f= 3x-2x^2+x^3+2x^4$ is quasi convex? I tried to show by definition , but in the middle of writing , I can't continued. Is there any theorem?

Answer 1

Yes, the specified function is quasiconvex. From Wikipedia:

Any monotonic function is both quasiconvex and quasiconcave. More generally, a function which decreases up to a point and increases from that point on is quasiconvex (compare unimodality).

The derivative of $f$ is $f'(x)=3-4x+3x^2+8x^3$, which has only one real root ($\approx-1.12019$). Hence, $f$ decreases monotonically from $x=-\infty$ to $x\approx-1.12019$, then increases monotonically from $x\approx-1.12019$ to $x=\infty$. Per the above fact, $f$ is quasiconvex.