The function $$ f(x) $$ is defined below. What is the end behavior of $$ f(x) $$ ? $$ f(x)=80 x^{2}+640-400 x-5 x^{3} $$ Answer as $$ x \rightarrow-\infty, f(x) \rightarrow-\infty $$ and as $$ x \rightarrow \infty, f(x) \rightarrow-\infty $$ as $$ x \rightarrow-\infty, f(x) \rightarrow-\infty $$ and as $$ x \rightarrow \infty, f(x) \rightarrow \infty $$ ansen $$ x \rightarrow-\infty, f(x) \rightarrow \infty $$ and as $$ x \rightarrow \infty, f(x) \rightarrow-\infty $$ as $$ x \rightarrow-\infty, f(x) \rightarrow \infty $$ and as $$ x \rightarrow \infty, f(x) \rightarrow \infty $$

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UpStudy AI · Accepted Answer

We start with the function  
$$
f(x) = 80x^{2} + 640 - 400x - 5x^{3}.
$$

It is helpful to rewrite the function in standard form (highest degree to lowest degree):
$$
f(x) = -5x^{3} + 80x^{2} - 400x + 640.
$$

The end behavior of a polynomial function is determined by its leading term. In this case, the leading term is  
$$
-5x^{3}.
$$

Since the degree is 3 (an odd degree) and the coefficient is $$-5$$ (a negative number):

- As $$ x \rightarrow \infty $$, the term $$-5x^3$$ dominates and since $$ x^3 \rightarrow \infty $$, we have  
  $$
  -5x^3 \rightarrow -\infty.
  $$
  Thus,  
  $$
  f(x) \rightarrow -\infty.
  $$

- As $$ x \rightarrow -\infty $$, note that $$ x^3 \rightarrow -\infty $$ and multiplying by $$-5$$ gives  
  $$
  -5x^3 \rightarrow \infty.
  $$
  Therefore,  
  $$
  f(x) \rightarrow \infty.
  $$

Thus, the end behavior of the function is:

- As $$ x \rightarrow -\infty $$, $$ f(x) \rightarrow \infty $$
- As $$ x \rightarrow \infty $$, $$ f(x) \rightarrow -\infty $$

The correct answer is:  
as $$ x \rightarrow -\infty, f(x) \rightarrow \infty $$ and  
as $$ x \rightarrow \infty, f(x) \rightarrow -\infty $$.
As $$ x $$ approaches negative infinity, $$ f(x) $$ approaches positive infinity, and as $$ x $$ approaches positive infinity, $$ f(x) $$ approaches negative infinity.

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Answer
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