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To determine the intervals where the function \( f(x) = -2x^2 + 4 \ln x \) is increasing or decreasing, let's proceed by finding its first derivative \( f'(x) \) and analyzing the sign of \( f'(x) \) over different intervals.

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

1. The derivative of \( -2x^2 \) with respect to \( x \) is \( -4x \).
2. The derivative of \( 4 \ln x \) with respect to \( x \) is \( \frac{4}{x} \).

Thus, the first derivative \( f'(x) \) is:
\[
f'(x) = -4x + \frac{4}{x}
\]

### Step 2: Simplify the Derivative
Rewrite \( f'(x) \) in a single fraction form:
\[
f'(x) = \frac{-4x^2 + 4}{x} = \frac{4(1 - x^2)}{x}
\]
This simplifies further to:
\[
f'(x) = \frac{4(1 - x)(1 + x)}{x}
\]

### Step 3: Determine Critical Points
To find the critical points, set \( f'(x) = 0 \):
\[
\frac{4(1 - x)(1 + x)}{x} = 0
\]
This equation is zero when:
1. \( x = 1 \)
2. \( x = -1 \) (although \( \ln x \) implies \( x > 0 \), so \( x = -1 \) is not in the domain of \( f(x) \))

So the only critical point in the domain \( x > 0 \) is \( x = 1 \).

### Step 4: Test Intervals Around the Critical Point
Examine the sign of \( f'(x) \) in the intervals \( (0, 1) \) and \( (1, \infty) \):

1. For \( x \in (0, 1) \):
   - Choose \( x = 0.5 \): \( f'(0.5) = \frac{4(1 - 0.5)(1 + 0.5)}{0.5} = \frac{4 \cdot 0.5 \cdot 1.5}{0.5} = 6 \) (positive)

2. For \( x \in (1, \infty) \):
   - Choose \( x = 2 \): \( f'(2) = \frac{4(1 - 2)(1 + 2)}{2} = \frac{4 \cdot (-1) \cdot 3}{2} = -6 \) (negative)

### Conclusion
The function \( f(x) \) is:
- **Increasing** on the interval \( (0, 1) \)
- **Decreasing** on the interval \( (1, \infty) \)

### Answer
The correct answer is:
**A.** The function is increasing on the open interval(s) \( (0, 1) \) and decreasing on the open interval(s) \( (1, \infty) \).

Would you like further details on this solution or have any questions?

Here are five related questions you might find helpful:
1. How do you find the critical points of a function?
2. What is the significance of the derivative in determining increasing and decreasing intervals?
3. How does the natural logarithm affect the domain of a function?
4. Can a function change from increasing to decreasing without a critical point?
5. How do we handle functions with undefined points when finding intervals?

**Tip:** Always check the domain of the function before finding critical points, as this can affect the intervals you consider for increasing and decreasing behavior.

Find the intervals on which f is increasing and decreasing. f(x) = -2x^2 + 4 ln x

Learn how to find the intervals where the function f(x) = -2x^2 + 4 ln x is increasing and decreasing by using calculus. This problem involves finding the first derivative, identifying critical points, and applying the first derivative test. Suitable for students in Grade 12 or college-level calculus.

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