Question

use the leading coefficient test to determine the end behavior of the graph of the given polynomial function.
f(x)=3x^7 + 3x^6+6x^4 + 3

a. falls left & rises right.
b. rises left & rises right.
c. rises left & falls right.
d. falls left & falls right.
e. none of the above.

Explanation:

Step1: Identify degree and leading coefficient

The polynomial function is $f(x)=3x^{7}+3x^{6}+6x^{4}+3$. The degree $n = 7$ (odd) and the leading - coefficient $a = 3$ (positive).

Step2: Apply leading - coefficient test rules

For a polynomial $y = a_nx^n+\cdots+a_0$, if $n$ is odd and $a_n>0$, as $x\to-\infty$, $y\to-\infty$ (falls left) and as $x\to+\infty$, $y\to+\infty$ (rises right).

Answer:

A. Falls left & rises right.