- Find the derivative \(f'(x) = -3x^2 - 8x + 3\).
- Solve \(f'(x) = 0\) to find critical points at \(x = -3\) and \(x = \frac{1}{3}\).
- Determine that the function decreases on the intervals \((-\infty, -3)\) and \((\frac{1}{3}, \infty)\).
- Evaluate which ordered pairs fall within these intervals: \(\boxed{(-4, -12), (1, -2), (2, -18)}\).
Explanation
- Finding the Intervals of Decrease
The function is given by \(f(x) = -x^3 - 4x^2 + 3x\). To determine where the function is decreasing, we must find the interval(s) where its first derivative \(f'(x)\) is negative. Taking the derivative, we have:
\[f'(x) = \frac{d}{dx}(-x^3 - 4x^2 + 3x) = -3x^2 - 8x + 3\]
The critical points occur where \(f'(x) = 0\):
\[-3x^2 - 8x + 3 = 0\]
Multiplying by \(-1\), we get \(3x^2 + 8x - 3 = 0\). We can factor this quadratic equation:
\[(3x - 1)(x + 3) = 0\]
This gives critical points at \(x = \frac{1}{3}\) and \(x = -3\). Since the original function is a cubic with a negative leading coefficient, it follows a general shape that decreases, increases, and decreases again. Specifically, the derivative \(-3x^2 - 8x + 3\) is a downward-opening parabola, so it is negative for \(x < -3\) and \(x > \frac{1}{3}\). Therefore, the function \(f(x)\) is decreasing on the intervals \((-\infty, -3)\) and \((\frac{1}{3}, \infty)\).
- Evaluating the Ordered Pairs
We now evaluate the given points to see if they lie on the graph of \(f(x) = -x^3 - 4x^2 + 3x\) and fall within the identified intervals of decrease \((x < -3\) or \(x > \frac{1}{3})\).
- \((-1, -6)\): \(x = -1\). This does not fall in the intervals \((-\infty, -3) \cup (\frac{1}{3}, \infty)\).
- \((2, -18)\): \(f(2) = -(2)^3 - 4(2)^2 + 3(2) = -8 - 16 + 6 = -18\). It is on the graph, and \(x = 2 > \frac{1}{3}\). (Correct)
- \((0, 0)\): \(f(0) = 0\). This does not fall in the intervals \((-\infty, -3) \cup (\frac{1}{3}, \infty)\).
- \((1, -2)\): \(f(1) = -(1)^3 - 4(1)^2 + 3(1) = -1 - 4 + 3 = -2\). It is on the graph, and \(x = 1 > \frac{1}{3}\). (Correct)
- \((-3, -18)\): This is a critical point where the function changes behavior. It is not part of the decreasing intervals.
- \((-4, -12)\): \(f(-4) = -(-4)^3 - 4(-4)^2 + 3(-4) = 64 - 64 - 12 = -12\). It is on the graph, and \(x = -4 < -3\). (Correct)
- Final Conclusion
Based on the analysis, the ordered pairs located on the graph where \(f(x)\) is decreasing are \((2, -18)\), \((1, -2)\), and \((-4, -12)\). Final check of the points shows they are indeed on the function and within the identified intervals. The set of correct points is \(\{(-4, -12), (1, -2), (2, -18)\}\).
Examples
Understanding where a function is increasing or decreasing is essential for analyzing trends in data. For instance, an economist might model the profit of a new product over time using a cubic function. By identifying the intervals where the profit function is decreasing, they can pinpoint the period when the product's market saturation or increased competition began to impact revenue, helping the business make informed strategic decisions to pivot or adjust pricing.