Question

question
the function $f(x)$ is defined below. what is the end behavior of $f(x)$?
$f(x) = -576x^2 - 48x^5 - 6912x + 4x^6 - 6912 + 896x^3 + 48x^4$
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 the leading term

The leading term of a polynomial is the term with the highest degree. For the polynomial \( f(x) = -576x^2 - 48x^5 - 6912x + 4x^6 - 6912 + 896x^3 + 48x^4 \), we look at the exponents of \( x \) in each term. The degrees are: for \( -576x^2 \) it's 2, for \( -48x^5 \) it's 5, for \( -6912x \) it's 1, for \( 4x^6 \) it's 6, for the constant term \( -6912 \) it's 0, for \( 896x^3 \) it's 3, and for \( 48x^4 \) it's 4. So the highest degree is 6, and the leading term is \( 4x^6 \).

Step2: Analyze the leading term's end behavior

The general rule for the end behavior of a polynomial \( a_nx^n + \dots + a_1x + a_0 \) (where \( a_n
eq 0 \)) is:

• If \( n \) is even:
• If \( a_n > 0 \), as \( x \to \infty \), \( f(x) \to \infty \) and as \( x \to -\infty \), \( f(x) \to \infty \) (because raising a negative number to an even power gives a positive number, and multiplying by a positive leading coefficient keeps it positive).
• If \( a_n < 0 \), as \( x \to \infty \), \( f(x) \to -\infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \).
• If \( n \) is odd:
• If \( a_n > 0 \), as \( x \to \infty \), \( f(x) \to \infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \).
• If \( a_n < 0 \), as \( x \to \infty \), \( f(x) \to -\infty \) and as \( x \to -\infty \), \( f(x) \to \infty \).

In our case, the leading term is \( 4x^6 \), so \( n = 6 \) (which is even) and \( a_n = 4 \) (which is positive). So we apply the rule for even \( n \) and positive \( a_n \): as \( x \to \infty \), \( f(x) \to \infty \) and as \( x \to -\infty \), \( f(x) \to \infty \) (since \( (-\infty)^6 \) is positive, and multiplying by 4 keeps it positive).

Answer:

as \( x \to -\infty \), \( f(x) \to \infty \) and as \( x \to \infty \), \( f(x) \to \infty \) (the third option in the list, which is "as \( x \to -\infty \), \( f(x) \to \infty \) and as \( x \to \infty \), \( f(x) \to \infty \)")