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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?

1. $f$ has no local maximum
2. For every $n$, $f$ has a local maximum at $x = 0$
3. $f$ has no local extremum if $n$ is odd and has a local maximum at $x = 0$ when $n$ is even
4. $f$ has no local extremum if $n$ is even and has a local maximum at $x = 0$ when $n$ is odd.
in Calculus | 60 views
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Option d?