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To analyze this problem, let’s break down each part of the given question.

### 1. Identifying the Extrema of \( f(x) \)
The function provided is:
\[
f(x) = \frac{12x - 3 - 3x^2}{x}
\]

The question asks to identify the extrema on the interval \( (0, 3) \) by examining the graph. From the options given, the fourth choice is selected:
> "There is only a maximum on \( (0, 3) \) since \( f \) is only decreasing after \( x = 1 \) and is only increasing before \( x = 1 \)."

Looking at the graph:
- There is a clear maximum at around \( x = 1 \).
- The function increases up to \( x = 1 \) and then decreases from \( x = 1 \) to \( x = 3 \).
- There are no points where the graph changes from decreasing to increasing, which would indicate a minimum.

Thus, this choice seems correct.

### 2. Determining the Extreme Value(s) of \( f \)
To determine the extreme values, we need to find the exact maximum value of \( f(x) \) at \( x = 1 \).

Let's evaluate \( f(1) \):
\[
f(1) = \frac{12(1) - 3 - 3(1)^2}{1} = \frac{12 - 3 - 3}{1} = \frac{6}{1} = 6
\]

Therefore, the extreme value of \( f \) on \( (0, 3) \) is:
\[
6
\]

### Final Answer
- **Extreme value(s) of \( f \):** \( 6 \)

Would you like more details, or do you have any specific questions about this solution?

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Here are some related questions to deepen your understanding:

1. How would you find the derivative of \( f(x) \) to confirm the maximum point mathematically?
2. What role does the interval \( (0, 3) \) play in determining the extrema of \( f(x) \)?
3. How would the presence of asymptotes impact the behavior of \( f(x) \) outside \( (0, 3) \)?
4. What would happen to the extrema if the interval changed to \( (0, 2) \) or \( (1, 3) \)?
5. How does the choice of graphing utility affect the precision of identifying maximum and minimum points?

**Tip:** To analyze extrema, consider using the first and second derivative tests for a more rigorous approach, especially if the function behavior is complex or involves asymptotes.

Identify the extrema of f(x) = (12x - 3 - 3x^2) / x on the interval (0, 3) using the graphing utility. Determine the extreme value(s) of f.

Learn how to identify the extrema of the function f(x) = (12x - 3 - 3x^2) / x on the interval (0, 3) using graphing. This problem explores key calculus concepts like finding critical points and using derivatives to determine maximum and minimum values.

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