Question 14 Describe the long run behavior of $$ f(p)=-5 p^{5}-2 p^{3}-4 p^{2}+2 $$ As $$ p \rightarrow-\infty, f(p) \rightarrow $$ ? © As $$ p \rightarrow \infty, f(p) \rightarrow $$ ? © Question Help: $$ \square $$ Message instructor Add Work

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The function is given by

$$
f(p) = -5p^5 - 2p^3 - 4p^2 + 2.
$$

Since the polynomial is of degree 5, the term $$-5p^5$$ will dominate the behavior as $$|p| \rightarrow \infty$$.

**As $$ p \rightarrow -\infty $$:**

- The leading term is $$-5p^5$$.
- When $$ p $$ is very negative, $$ p^5 $$ (an odd degree power) is also negative.
- Multiplying by $$-5$$ gives $$-5p^5 \rightarrow -5(-\infty) = \infty$$.

Thus,

$$
\text{As } p \rightarrow -\infty, \quad f(p) \rightarrow \infty.
$$

**As $$ p \rightarrow \infty $$:**

- Again, consider the leading term $$-5p^5$$.
- When $$ p $$ is very positive, $$ p^5 $$ is positive.
- Multiplying by $$-5$$ yields $$-5p^5 \rightarrow -\infty$$.

Thus,

$$
\text{As } p \rightarrow \infty, \quad f(p) \rightarrow -\infty.
$$
As $$ p \rightarrow -\infty $$, $$ f(p) \rightarrow \infty $$. As $$ p \rightarrow \infty $$, $$ f(p) \rightarrow -\infty $$.

Question 14 Describe the long run behavior of \( f(p)=-5 p^{5}-2 p^{3}-4 p^{2}+2 \) As \( p \rightarrow-\infty, f(p) \rightarrow \) ? © As \( p \rightarrow \infty, f(p) \rightarrow \) ? © Question Help: \( \square \) Message instructor Add Work

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