Studying properties of an integral $f:\mathbb R \to \mathbb R, \: f(x) =\int _0^x\:e^{-t^3}\left(t^3+t^2-t-1\right)dt$
Firstly, I have to find $min(f(x))$. The possible answers are:
A) $f(0)$
B) $f(1)$
C) $f(-1)$
D) $f(3)$
E) $f$ tends to $- \infty$
In order to do that, I have rewritten the function like this
$f\left(x\right)=\int _0^x\:e^{-t^3}\left(t+1\right)^2\left(t-1\right)dt$
and I studied the sign of the function $e^{-t^3}\left(t+1\right)^2\left(t-1\right)$ but there was nothing conclusive. Can someone teach me how to think this kind of problem ?
Secondly, for the same function, I have to mark the correct answer for" $G_f$ has":
A) a horisontal asymptote to $\infty$
B) asymptotes to $\infty$ and $-\infty$
C) a horisontal asymptote and an oblique asymptote
D) two oblique asymptotes
E) no asymptotes
Any hint to this problems would be greatly appreciated.

The derivative is $f'(x)=e^{-x^3}(x+1)^2(x-1)$ so it only vanishes at $1$ and $-1$.
However $f$ is decreasing on $(-\infty,1]$ and increasing on $[1,\infty)$, because the derivative is positive in a neighborhood of $-1$ (except only at $-1$).
For the question about asymptotes, note that
$$
\lim_{x\to\infty}f'(x)=0
$$
and
$$
\lim_{x\to-\infty}f'(x)=-\infty
$$
