To determine the end behavior of the polynomial function [tex]\( f(x) = 1536 + 3724x^3 + 4x^6 + 5248x + 6616x^2 + 916x^4 + 100x^5 \)[/tex], we need to analyze the term with the highest degree, as it dominates the behavior of the polynomial for large positive or negative values of [tex]\( x \)[/tex].
The term with the highest degree in this polynomial is [tex]\( 4x^6 \)[/tex]. Let's consider the end behavior step-by-step:
1. Identify the term with the highest degree:
The highest degree term is [tex]\( 4x^6 \)[/tex].
2. Examine the leading coefficient and the exponent of the highest degree term:
- The coefficient of [tex]\( x^6 \)[/tex] is [tex]\( 4 \)[/tex], which is positive.
- The exponent is [tex]\( 6 \)[/tex], which is even.
3. Determine the behavior as [tex]\( x \to \infty \)[/tex]:
- Since the leading coefficient is positive and the exponent is even, as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], [tex]\( 4x^6 \)[/tex] becomes very large and positive.
- Hence, as [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
4. Determine the behavior as [tex]\( x \to -\infty \)[/tex]:
- With [tex]\( x \)[/tex] being very negative and raised to an even power, [tex]\( (-x)^6 \)[/tex] still results in a positive value.
- Thus, as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( 4x^6 \)[/tex] also becomes very large and positive.
- Hence, as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
Therefore, the end behavior of the polynomial function [tex]\( f(x) \)[/tex] is as follows:
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex].
So, the correct answer is:
[tex]\[
\text{as } x \rightarrow \infty, f(x) \rightarrow \infty \text{ and as } x \rightarrow -\infty, f(x) \rightarrow \infty
\][/tex]
