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To solve for where the function \( f(x) = x \log(x) \) is increasing and decreasing on the interval \( (0, \infty) \), let's analyze the first derivative.

1. **Compute the first derivative \( f'(x) \):**
   \[
   f(x) = x \log(x)
   \]
   Using the product rule, where \( u = x \) and \( v = \log(x) \), we get:
   \[
   f'(x) = u'v + uv' = 1 \cdot \log(x) + x \cdot \frac{1}{x} = \log(x) + 1
   \]

2. **Set \( f'(x) = 0 \) to find critical points:**
   \[
   \log(x) + 1 = 0
   \]
   \[
   \log(x) = -1
   \]
   Solving for \( x \), we get:
   \[
   x = e^{-1} = \frac{1}{e}
   \]

   So, \( x = \frac{1}{e} \) is the critical point.

3. **Determine intervals of increase and decrease:**
   - For \( x < \frac{1}{e} \), choose a test point (e.g., \( x = 0.1 \)):
     \[
     f'(0.1) = \log(0.1) + 1 < 0
     \]
     This implies \( f(x) \) is decreasing on \( \left( 0, \frac{1}{e} \right) \).
   - For \( x > \frac{1}{e} \), choose a test point (e.g., \( x = 1 \)):
     \[
     f'(1) = \log(1) + 1 = 1 > 0
     \]
     This implies \( f(x) \) is increasing on \( \left( \frac{1}{e}, \infty \right) \).

**Conclusion:**
- The function \( f(x) = x \log(x) \) is:
  - **Decreasing on \( \left( 0, \frac{1}{e} \right) \)**
  - **Increasing on \( \left( \frac{1}{e}, \infty \right) \)**

Would you like further details on any of these steps?

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### Related Questions
1. What is the behavior of \( f(x) = x \log(x) \) near \( x = 0 \)?
2. How can we verify the concavity of \( f(x) = x \log(x) \) using the second derivative?
3. What happens to \( f(x) = x \log(x) \) as \( x \to \infty \)?
4. Can we determine any points of inflection for \( f(x) = x \log(x) \)?
5. How would the intervals change if the function were \( f(x) = x \log(x) + c \), where \( c \) is a constant?

### Tip
For functions involving logarithmic terms, always check that the argument of the logarithm is positive within the interval of interest.

Consider the function f(x) = x log(x) on (0, ∞). The function f is increasing on (___, ___) and decreasing on (___, ___).

Explore how to find the intervals of increase and decrease for the function f(x) = x log(x) on (0, ∞). Learn how to use the first derivative test, find critical points, and interpret intervals of behavior for calculus problems.

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