for a polynomial function ( f ), ( lim_{x\rightarrow -infty}f(x)=infty ) and ( lim_{x\rightarrow…

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

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

Answer

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\neq0)), the end - behavior is determined by the leading term (a_nx^n). If (n) is even:

  • When (a_n>0), (\lim_{x\rightarrow-\infty}f(x)=\lim_{x\rightarrow\infty}f(x)=\infty)
  • When (a_n < 0), (\lim_{x\rightarrow-\infty}f(x)=\lim_{x\rightarrow\infty}f(x)=-\infty) If (n) is odd:
  • When (a_n>0), (\lim_{x\rightarrow-\infty}f(x)=-\infty) and (\lim_{x\rightarrow\infty}f(x)=\infty)
  • When (a_n < 0), (\lim_{x\rightarrow-\infty}f(x)=\infty) and (\lim_{x\rightarrow\infty}f(x)=-\infty)

Step2: Analyze the given limits

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