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To determine the intervals on which the function \( f(x) = -2x^2 + 4 \ln x \) is increasing or decreasing, we need to find the derivative \( f'(x) \) and analyze its sign over the domain of \( f(x) \).

### Step 1: Differentiate \( f(x) \)
Given:
\[
f(x) = -2x^2 + 4 \ln x
\]

The derivative \( f'(x) \) is:
\[
f'(x) = -4x + \frac{4}{x}
\]

### Step 2: Set \( f'(x) = 0 \) to find critical points
\[
-4x + \frac{4}{x} = 0
\]
Multiply through by \( x \) (assuming \( x > 0 \) since \( \ln x \) requires \( x > 0 \)):
\[
-4x^2 + 4 = 0
\]
\[
-4(x^2 - 1) = 0
\]
\[
x^2 = 1
\]
Thus, \( x = 1 \) (only positive \( x \) values are valid here due to \( \ln x \)).

### Step 3: Determine the sign of \( f'(x) \) on intervals around \( x = 1 \)
We need to examine the sign of \( f'(x) = -4x + \frac{4}{x} \) in the intervals \( (0, 1) \) and \( (1, \infty) \):

1. For \( x \in (0, 1) \):
   - Choose a test point, say \( x = 0.5 \):
   \[
   f'(0.5) = -4(0.5) + \frac{4}{0.5} = -2 + 8 = 6 > 0
   \]
   So, \( f'(x) > 0 \) on \( (0, 1) \), indicating \( f(x) \) is **increasing** on \( (0, 1) \).

2. For \( x \in (1, \infty) \):
   - Choose a test point, say \( x = 2 \):
   \[
   f'(2) = -4(2) + \frac{4}{2} = -8 + 2 = -6 < 0
   \]
   So, \( f'(x) < 0 \) on \( (1, \infty) \), indicating \( f(x) \) is **decreasing** on \( (1, \infty) \).

### Conclusion
- **Increasing interval**: \( (0, 1) \)
- **Decreasing interval**: \( (1, \infty) \)

Would you like further details on any step, or do you have any questions?

---

Here are some related questions to explore further:

1. What is the behavior of the function \( f(x) \) as \( x \to 0^+ \) and \( x \to \infty \)?
2. How do we find local maxima and minima using the first derivative test?
3. Could we use the second derivative to confirm the concavity of \( f(x) \) at \( x = 1 \)?
4. What does the graph of \( f(x) = -2x^2 + 4 \ln x \) look like, and where does it cross the axes?
5. How would we analyze the behavior of \( f(x) \) if it included a domain beyond \( x > 0 \)?

**Tip:** Always check the domain of the function before determining intervals of increase or decrease, especially when dealing with logarithmic functions.

Find the intervals on which f is increasing and decreasing.

f(x) = -2x^2 + 4 ln x

Learn how to find intervals where the function f(x) = -2x^2 + 4 ln x is increasing or decreasing. This solution involves differentiation and the first derivative test, focusing on logarithmic and polynomial functions. Suitable for students in Grades 11-12 or introductory college calculus.

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