What is the end behavior of the graph of [tex]f(x)=x^5-8x^4+16x^3[/tex]?

A. [tex]f(x) \rightarrow -\infty[/tex] as [tex]x \rightarrow -\infty[/tex]; [tex]f(x) \rightarrow -\infty[/tex] as [tex]x \rightarrow +infty[/tex]

B. [tex]f(x) \rightarrow -\infty[/tex] as [tex]x \rightarrow -\infty[/tex]; [tex]f(x) \rightarrow +\infty[/tex] as [tex]x \rightarrow +\infty[/tex]

C. [tex]f(x) \rightarrow +\infty[/tex] as [tex]x \rightarrow -\infty[/tex]; [tex]f(x) \rightarrow -\infty[/tex] as [tex]x \rightarrow +\infty[/tex]

D. [tex]f(x) \rightarrow +\infty[/tex] as [tex]x \rightarrow -\infty[/tex]; [tex]f(x) \rightarrow +\infty[/tex] as [tex]x \rightarrow +\infty[/tex]



Answer :

To determine the end behavior of the polynomial function [tex]\( f(x) = x^5 - 8x^4 + 16x^3 \)[/tex], we need to analyze the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches very large positive and negative values.

### Step 1: Analyze the Leading Term
For polynomials, the highest degree term (leading term) dictates the end behavior. In this case, the leading term is [tex]\( x^5 \)[/tex]:

[tex]\( f(x) = x^5 - 8x^4 + 16x^3 \)[/tex]

### Step 2: As [tex]\( x \to +\infty \)[/tex]
- When [tex]\( x \)[/tex] is a very large positive number, [tex]\( x^5 \)[/tex] dominates and [tex]\( x^5 \)[/tex] is positive.
- Therefore, [tex]\( f(x) \)[/tex] will also be a very large positive number.

So, we have:
[tex]\[ f(x) \rightarrow +\infty \text{ as } x \rightarrow +\infty \][/tex]

### Step 3: As [tex]\( x \to -\infty \)[/tex]
- When [tex]\( x \)[/tex] is a very large negative number, [tex]\( x^5 \)[/tex] dominates and [tex]\( x^5 \)[/tex] is negative.
- Therefore, [tex]\( f(x) \)[/tex] will also be a very large negative number.

So, we have:
[tex]\[ f(x) \rightarrow -\infty \text{ as } x \rightarrow -\infty \][/tex]

### Conclusion:
The end behavior of the graph can be described as follows:
[tex]\[ f(x) \rightarrow -\infty \text{ as } x \rightarrow -\infty \][/tex]
[tex]\[ f(x) \rightarrow +\infty \text{ as } x \rightarrow +\infty \][/tex]

Thus, the correct answer is:
[tex]\[ f(x) \rightarrow -\infty \text{ as } x \rightarrow -\infty ; f(x) \rightarrow +\infty \text{ as } x \rightarrow +\infty \][/tex]