- Identify the leading term \(-x^5\) of the polynomial \(f(x) = -x^5 - 5x^3 + 12x^2 + 36x\).
- Recognize that for a polynomial with an odd degree and a negative leading coefficient, the graph points upward toward positive infinity as \(x \to -\infty\) and downward toward negative infinity as \(x \to \infty\).
- Confirm the behavior analytically by evaluating the limit of the leading term.
- Final results: As \(x \to -\infty\), \(f(x) \to \boxed{\infty}\). As \(x \to \infty\), \(f(x) \to \boxed{-\infty}\).
Explanation
- Analyzing the Leading Term
The function given is a fifth-degree polynomial: \(f(x) = -x^5 - 5x^3 + 12x^2 + 36x\). To determine the end behavior, we focus on the leading term of the polynomial, which is \(-x^5\). For large values of \(|x|\), the term with the highest power dominates the behavior of the entire function, as the other terms grow much more slowly relative to \(x^5\).
- Evaluating the Limit as \(x \to \infty\)
To evaluate the limit as \(x \to \infty\), we can factor out the leading term:
\(f(x) = -x^5 \left( 1 + \frac{5}{x^2} - \frac{12}{x^3} - \frac{36}{x^4} \right)\).
As \(x \to \infty\), the fractions \(\frac{5}{x^2}\), \(\frac{12}{x^3}\), and \(\frac{36}{x^4}\) all approach \(0\).
Therefore, the expression inside the parentheses approaches \(1\).
The behavior is then dominated by \(-x^5\). Since \(x^5\) grows to infinity as \(x \to \infty\), \(-x^5\) approaches \(-\infty\).
Thus, \(\lim_{x \to \infty} f(x) = -\infty\).
- Evaluating the Limit as \(x \to -\infty\)
To evaluate the limit as \(x \to -\infty\), we use the same factored form:
\(f(x) = -x^5 \left( 1 + \frac{5}{x^2} - \frac{12}{x^3} - \frac{36}{x^4} \right)\).
As \(x \to -\infty\), the fractions still approach \(0\).
The expression inside the parentheses approaches \(1\).
Now, examine the term \(-x^5\). As \(x \to -\infty\), \(x^5\) approaches \(-\infty\).
Multiplying by \(-1\), we get \(-(-\infty) = \infty\).
Thus, \(\lim_{x \to -\infty} f(x) = \infty\).
- Final Conclusion
Combining the results, we have:
As \(x \to -\infty\), \(f(x) \to \infty\).
As \(x \to \infty\), \(f(x) \to -\infty\).
The final answers are \(\boxed{\infty}\) and \(\boxed{-\infty}\).
Examples
In economics, analyzing the end behavior of a cost function (like a polynomial representing total cost versus production quantity) helps businesses understand the long-term sustainability or extreme trends of their operations. If a cost function grows indefinitely as production increases, managers can identify the point where scaling is no longer profitable, just as we identified the end behavior of the polynomial function \(f(x)\). Understanding how functions behave as inputs move toward extreme values is fundamental to modeling physical, financial, and biological systems.