Consider the function
$f(x)=\bigg(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots+\frac{x^n}{n!}\bigg)e^{-x}$,
where $n\geq4$ is a positive integer. Which of the following statements is correct?
- $f$ has no local maximum
- For every $n$, $f$ has a local maximum at $x = 0$
- $f$ has no local extremum if $n$ is odd and has a local maximum at $x = 0$ when $n$ is even
- $f$ has no local extremum if $n$ is even and has a local maximum at $x = 0$ when $n$ is odd.