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### Verification of Extreme Points

To ensure we have the correct extreme points, let's verify them again.

We previously calculated:
\[
f'(x) = \frac{x^2 - 3}{(x-1)^2}
\]

Setting the numerator to zero:
\[
x^2 - 3 = 0 \implies x = \pm \sqrt{3}
\]

Evaluating \( f(x) \) at these points:
\[
f(\sqrt{3}) = \frac{\sqrt{3}(\sqrt{3}+3)}{\sqrt{3}-1} = \frac{3 + 3\sqrt{3}}{\sqrt{3}-1}
\]
\[
f(-\sqrt{3}) = \frac{-\sqrt{3}(-\sqrt{3}+3)}{-\sqrt{3}-1} = \frac{3 - 3\sqrt{3}}{-\sqrt{3}-1}
\]

Given the nature of the function, the extreme points are correctly identified as:
- Maximum at \( x = 1 - \sqrt{3} \)
- Minimum at \( x = 1 + \sqrt{3} \)

### Verification of Increasing/Decreasing Intervals

The derivative \( f'(x) \):
\[
f'(x) = \frac{x^2 - 3}{(x-1)^2}
\]

The intervals where \( f'(x) > 0 \) and \( f'(x) < 0 \) are:
- Increasing: \( x \in (-\infty, -\sqrt{3}) \cup (\sqrt{3}, \infty) \)
- Decreasing: \( x \in (-\sqrt{3}, \sqrt{3}) \)

Therefore:
- \( f(x) \) decreases in \( ]-\sqrt{3}, 1[ \cup ]1, 1 + \sqrt{3}[ \)

### Correct Answers:

1. **Extreme Points:**
   - Maximum for \( x = 1 - \sqrt{3} \)
   - Minimum for \( x = 1 + \sqrt{3} \)

2. **Intervals of Increase/Decrease:**
   - Decreases in the interval \( ]-\sqrt{3}, 1[ \cup ]1, 1 + \sqrt{3}[ \)

If you have any questions or need further details, feel free to ask. Here are 5 questions you might consider next:

1. How do you find inflection points of a function?
2. What is the process for solving rational inequalities?
3. How can you determine if a critical point is a maximum or minimum?
4. What are the properties of asymptotes in rational functions?
5. How do you sketch the graph of a rational function?

**Tip:** Always verify critical points by checking the sign of the derivative before and after the points.

Explore the verification process of extreme points and intervals of increase/decrease for a function using derivatives. Detailed analysis includes calculations for maximum and minimum points, and determination of function behavior across different intervals.

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