Question

for a polynomial function ( f ), ( lim_{x
ightarrow -infty}f(x)=infty ) and ( lim_{x
ightarrow infty}f(x)=-infty ). which of the following must be true about ( f )?
a the degree of ( f ) is even, and the leading coefficient is negative.
b the degree of ( f ) is even, and the leading coefficient is positive.
c the degree of ( f ) is odd, and the leading coefficient is negative.

Explanation:

Step1: Recall the end - behavior of polynomial functions

For a polynomial function \(f(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_0\) (\(a_n
eq0\)), the end - behavior is determined by the leading term \(a_nx^n\).
If \(n\) is even:

• When \(a_n>0\), \(\lim_{x

ightarrow-\infty}f(x)=\lim_{x
ightarrow\infty}f(x)=\infty\)

• When \(a_n < 0\), \(\lim_{x

ightarrow-\infty}f(x)=\lim_{x
ightarrow\infty}f(x)=-\infty\)
If \(n\) is odd:

• When \(a_n>0\), \(\lim_{x

ightarrow-\infty}f(x)=-\infty\) and \(\lim_{x
ightarrow\infty}f(x)=\infty\)

• When \(a_n < 0\), \(\lim_{x

ightarrow-\infty}f(x)=\infty\) and \(\lim_{x
ightarrow\infty}f(x)=-\infty\)

Step2: Analyze the given limits

We are given that \(\lim_{x
ightarrow-\infty}f(x)=\infty\) and \(\lim_{x
ightarrow\infty}f(x)=-\infty\).
Since the limits as \(x
ightarrow-\infty\) and \(x
ightarrow\infty\) have opposite signs, the degree \(n\) of the polynomial \(f(x)\) must be odd.
Let \(f(x)=a_nx^n+\cdots\). When \(x
ightarrow-\infty\), \(x^n=-|x|^n\) (for \(n\) odd) and when \(x
ightarrow\infty\), \(x^n = |x|^n\).
We know that \(\lim_{x
ightarrow-\infty}a_nx^n=a_n(-1)^n\lim_{x
ightarrow\infty}|x|^n\) and \(\lim_{x
ightarrow\infty}a_nx^n=a_n\lim_{x
ightarrow\infty}|x|^n\).
Since \(\lim_{x
ightarrow-\infty}f(x)=\infty\) and \(\lim_{x
ightarrow\infty}f(x)=-\infty\) and \(n\) is odd (\((-1)^n=-1\)), we must have \(a_n<0\)

Answer:

C. The degree of \(f\) is odd, and the leading coefficient is negative.