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

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