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?

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.