Question

the function $f(x)$ is defined below. what is the end - behavior of $f(x)$?
$f(x)=216 - 36x-138x^{2}-3x^{4}-39x^{3}$
answer attempt 1 out of 2
the degree of the polynomial is, and the leading coefficient is
choose which pair of limits below represents the end - behavior:
$lim_{x
ightarrow-infty}f(x)=infty,lim_{x
ightarrowinfty}f(x)=infty$
$lim_{x
ightarrow-infty}f(x)=-infty,lim_{x
ightarrowinfty}f(x)=infty$
$lim_{x
ightarrow-infty}f(x)=infty,lim_{x
ightarrowinfty}f(x)=-infty$
$lim_{x
ightarrow-infty}f(x)=-infty,lim_{x
ightarrowinfty}f(x)=-infty$

Explanation:

Step1: Identify degree of polynomial

The highest - power of \(x\) in \(f(x)=216 - 36x-138x^{2}-3x^{4}-39x^{3}\) is \(4\), so the degree \(n = 4\) (even).

Step2: Identify leading coefficient

The leading term is \(-3x^{4}\), so the leading coefficient \(a=-3\) (negative).

Step3: Determine end - behavior

For a polynomial \(y = a x^{n}\) with \(n\) even and \(a<0\), as \(x\to-\infty\), \(y = f(x)\to-\infty\) and as \(x\to\infty\), \(y = f(x)\to-\infty\). That is \(\lim_{x\to-\infty}f(x)=-\infty\) and \(\lim_{x\to\infty}f(x)=-\infty\).

Answer:

The degree of the polynomial is even, and the leading coefficient is \(- 3\). The pair of limits that represents the end - behavior is \(\lim_{x\to-\infty}f(x)=-\infty\), \(\lim_{x\to\infty}f(x)=-\infty\)