Question

the function $f(x)$ is defined below. what is the end behavior of $f(x)$?
$f(x) = -96 + 10x^4 + x^5 + 24x^3 - 16x^2 - 112x$
answer
\\(\circ\\) as $x \to \infty, f(x) \to \infty$ and as $x \to -\infty, f(x) \to -\infty$
\\(\circ\\) as $x \to \infty, f(x) \to \infty$ and as $x \to -\infty, f(x) \to \infty$
\\(\circ\\) as $x \to \infty, f(x) \to -\infty$ and as $x \to -\infty, f(x) \to -\infty$
\\(\circ\\) as $x \to \infty, f(x) \to -\infty$ and as $x \to -\infty, f(x) \to \infty$

Explanation:

Step1: Identify Leading Term

The leading term of a polynomial is the term with the highest degree. For \( f(x)=-96 + 10x^{4}+x^{5}+24x^{3}-16x^{2}-112x \), the highest degree is \( 5 \) (from \( x^{5} \)) and the leading term is \( x^{5} \) (coefficient \( 1 \), degree \( 5 \)).

Step2: Analyze End Behavior

For a polynomial \( a_nx^n+\dots+a_0 \):

• If \( n \) is odd:
• If \( a_n>0 \), as \( x

ightarrow\infty \), \( f(x)
ightarrow\infty \); as \( x
ightarrow-\infty \), \( f(x)
ightarrow-\infty \).

• If \( a_n<0 \), as \( x

ightarrow\infty \), \( f(x)
ightarrow-\infty \); as \( x
ightarrow-\infty \), \( f(x)
ightarrow\infty \).

Here, \( n = 5 \) (odd) and \( a_n=1>0 \). So:

• As \( x

ightarrow\infty \), \( x^{5}
ightarrow\infty \), so \( f(x)
ightarrow\infty \).

• As \( x

ightarrow-\infty \), \( x^{5}
ightarrow-\infty \) (since odd power of negative is negative), so \( f(x)
ightarrow-\infty \).

Answer:

as \( x
ightarrow \infty, f(x)
ightarrow \infty \) and as \( x
ightarrow -\infty, f(x)
ightarrow -\infty \) (the first option in the choices)