Question

the function f(x) is defined below. what is the end - behavior of f(x)?

f(x)=-1610x + 55x^4+9800 - 55x^3-1795x^2 + 5x^5

answer attempt 1 out of 3

the degree of the polynomial is, and the leading coefficient is

choose which pair of limits below odd

limits the end - behavior:

lim_{x\to - \infty}f(x)=\infty, lim_{x\to\infty}f(x)=-\infty

lim_{x\to - \infty}f(x)=-\infty, lim_{x\to\infty}f(x)=\infty

lim_{x\to - \infty}f(x)=\infty, lim_{x\to\infty}f(x)=-\infty

lim_{x\to - \infty}f(x)=-\infty, lim_{x\to\infty}f(x)=-\infty

Explanation:

Step1: Identify degree of polynomial

The degree of a polynomial is the highest - power of the variable. For \(f(x)=5x^{5}+55x^{4}-55x^{3}-1795x^{2}-1610x + 9800\), the degree \(n = 5\) (since the highest power of \(x\) is 5).

Step2: Identify leading coefficient

The leading coefficient is the coefficient of the term with the highest - power of the variable. For \(f(x)=5x^{5}+55x^{4}-55x^{3}-1795x^{2}-1610x + 9800\), the leading coefficient \(a = 5\) (the coefficient of \(x^{5}\)).

Step3: Determine end - behavior using degree and leading coefficient rules

For a polynomial \(y = a x^{n}\), when \(n\) is odd and \(a>0\): \(\lim_{x
ightarrow-\infty}f(x)=-\infty\) and \(\lim_{x
ightarrow\infty}f(x)=\infty\).

Answer:

The degree of the polynomial is 5, and the leading coefficient is 5. The pair of limits that represents the end - behavior is \(\lim_{x
ightarrow-\infty}f(x)=-\infty,\lim_{x
ightarrow\infty}f(x)=\infty\)