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Answer :
To determine the end behavior of the polynomial function [tex]\( f(x) = 3x^6 + 30x^5 + 75x^4 \)[/tex], we need to focus on the highest degree term of the polynomial, which is [tex]\( 3x^6 \)[/tex].
### Step-by-Step Analysis
1. Identify the Leading Term:
- The leading term of the polynomial is [tex]\( 3x^6 \)[/tex]. This term will dominate the behavior of the polynomial as [tex]\( x \)[/tex] becomes very large or very small.
2. Determine the End Behavior:
- As [tex]\( x \to -\infty \)[/tex]:
- Since the leading term is [tex]\( 3x^6 \)[/tex], which has an even power of [tex]\( x \)[/tex], [tex]\( (x^6) \)[/tex] will always be positive, regardless of whether [tex]\( x \)[/tex] is negative or positive.
- Therefore, as [tex]\( x \to -\infty \)[/tex], [tex]\( 3x^6 \to \infty \)[/tex].
- So, [tex]\( f(x) \to \infty \)[/tex].
- As [tex]\( x \to \infty \)[/tex]:
- Similarly, as [tex]\( x \to \infty \)[/tex], [tex]\( 3x^6 \to \infty \)[/tex].
- Consequently, [tex]\( f(x) \to \infty \)[/tex].
3. Conclusion:
- The end behavior of the polynomial function is:
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
- As [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
Given this analysis, none of the options provided in the question match the true end behavior of the polynomial, which is that for both [tex]\( x \to -\infty \)[/tex] and [tex]\( x \to \infty \)[/tex], the function [tex]\( f(x) \)[/tex] approaches infinity.
### Step-by-Step Analysis
1. Identify the Leading Term:
- The leading term of the polynomial is [tex]\( 3x^6 \)[/tex]. This term will dominate the behavior of the polynomial as [tex]\( x \)[/tex] becomes very large or very small.
2. Determine the End Behavior:
- As [tex]\( x \to -\infty \)[/tex]:
- Since the leading term is [tex]\( 3x^6 \)[/tex], which has an even power of [tex]\( x \)[/tex], [tex]\( (x^6) \)[/tex] will always be positive, regardless of whether [tex]\( x \)[/tex] is negative or positive.
- Therefore, as [tex]\( x \to -\infty \)[/tex], [tex]\( 3x^6 \to \infty \)[/tex].
- So, [tex]\( f(x) \to \infty \)[/tex].
- As [tex]\( x \to \infty \)[/tex]:
- Similarly, as [tex]\( x \to \infty \)[/tex], [tex]\( 3x^6 \to \infty \)[/tex].
- Consequently, [tex]\( f(x) \to \infty \)[/tex].
3. Conclusion:
- The end behavior of the polynomial function is:
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
- As [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
Given this analysis, none of the options provided in the question match the true end behavior of the polynomial, which is that for both [tex]\( x \to -\infty \)[/tex] and [tex]\( x \to \infty \)[/tex], the function [tex]\( f(x) \)[/tex] approaches infinity.
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