To determine the end behavior of the polynomial function [tex]\( f(x) = 3x^6 + 30x^5 + 75x^4 \)[/tex], we'll focus on the leading term. The leading term dominates the behavior of the function as [tex]\( x \)[/tex] approaches positive and negative infinity.
### Step-by-Step Solution:
1. Identify the Leading Term:
The polynomial [tex]\( f(x) = 3x^6 + 30x^5 + 75x^4 \)[/tex] has the leading term [tex]\( 3x^6 \)[/tex] because it has the highest degree (the highest power of [tex]\( x \)[/tex]).
2. Determine the Behavior of the Leading Term as [tex]\( x \)[/tex] Approaches Positive Infinity ([tex]\( x \rightarrow \infty \)[/tex]):
Since the term [tex]\( 3x^6 \)[/tex] is positive (as 3 is a positive coefficient) and [tex]\( x^6 \)[/tex] (an even power of [tex]\( x \)[/tex]) grows very large positively when [tex]\( x \)[/tex] is large and positive:
[tex]\[
\lim_{x \to \infty} 3x^6 = \infty
\][/tex]
Thus, as [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex].
3. Determine the Behavior of the Leading Term as [tex]\( x \)[/tex] Approaches Negative Infinity ([tex]\( x \rightarrow -\infty \)[/tex]):
The term [tex]\( x^6 \)[/tex] is always positive regardless of whether [tex]\( x \)[/tex] is positive or negative because it is raised to an even power. Therefore, as [tex]\( x \)[/tex] becomes very large negatively, [tex]\( x^6 \)[/tex] still becomes very large positively:
[tex]\[
\lim_{x \to -\infty} 3x^6 = \infty
\][/tex]
Thus, as [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex].
### Conclusion:
Based on the analysis of the leading term [tex]\( 3x^6 \)[/tex]:
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y (= f(x)) \rightarrow \infty \)[/tex]
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y (= f(x)) \rightarrow \infty \)[/tex]
Thus, the correct statement for the end behavior of [tex]\( f(x) \)[/tex] is:
As [tex]\( x \rightarrow -\infty, y \rightarrow \infty \)[/tex], and as [tex]\( x \rightarrow \infty, y \rightarrow \infty \)[/tex].
