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It looks like you're working on a calculus problem involving intervals where a function \( f(x) = \frac{-8x^2}{(x-6)^2} \) is increasing or decreasing, as well as identifying local minima or maxima.

From the information on the screen, here’s a breakdown of the answer requirements and some clarifications:

1. **Increasing Interval**: It appears you initially tried to input the interval as \((-\infty, 0) \cup (12, 0)\), but the platform returned an error that the "Left endpoint must be less than right endpoint." In interval notation, the lower value should always come before the higher value, so this entry is invalid. The increasing interval seems tricky; you’ll need to carefully examine \(f'(x)\) to determine where it’s positive (indicating an increase).

2. **Decreasing Interval**: You entered \((0, 6) \cup (6, 12)\), which is marked as correct. This indicates that on these intervals, the derivative \( f'(x) \) is negative, confirming a decrease in \( f(x) \).

3. **Local Maxima and Minima**: The values for potential local extrema were entered as 12 and 0. However, both entries were marked incorrect. To determine the actual points of local extrema, you would need to examine the critical points and analyze the behavior of \( f(x) \) around these points. This might involve using the first derivative test or examining the second derivative \( f''(x) \).

Would you like a detailed step-by-step on finding the correct intervals for increasing/decreasing behavior and identifying the precise locations of local maxima and minima for this function?

Here are five follow-up questions that could expand on this:

1. How do we calculate the derivative \( f'(x) \) for rational functions like this one?
2. What tests can we use to determine if a critical point is a maximum, minimum, or neither?
3. Why does the function have asymptotes, and how do they affect the intervals of increase or decrease?
4. How can we use interval notation correctly to avoid endpoint issues in inputs?
5. How do we interpret the derivative’s sign (positive or negative) over various intervals?

**Tip:** Always double-check endpoint order and consider using open intervals for points of discontinuity (like vertical asymptotes).

Determine the intervals where the function f(x) = -8x^2 / (x - 6)^2 is increasing or decreasing, and identify the local minima or maxima points.

Learn how to determine the intervals where the function f(x) = -8x^2 / (x - 6)^2 is increasing or decreasing. This solution covers critical points, the first derivative test, and interval notation, helping calculus students identify local minima and maxima accurately.

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