TITLE: Open map and topology
QUESTION [6 upvotes]: If $ f:\mathbb{R} \to \mathbb{R}$,  $f(x)= x^3+3x^2+ax+3$, for what $a$ is $f$ an open map?
I was thinking the following:
It suffices to show $f$ maps basic open set (interval) to an open set in $\mathbb{R}$. Since $f$ is a continuous map because it is a polynomial, $f$ should map an open interval into a connected subset of $\mathbb{R}$ which is of the form of an interval. But what are the other conditions needed to make sure it is an open map?

REPLY [2 votes]: This is an application of a common question which states: "prove a continuous monotonic function is an open map" or "prove a strictly monotonic onto function has a continuous inverse". 
Essentially, you want the derivative always positive or always negative, and you can find conditions on $a$ for that.