To study the sign of the expression \((x - 1)^2 (\ln x - 1)\), we need to analyze the factors individually and determine under which conditions the product is positive, negative, or zero.

### 1. Factor Analysis
The given expression has two factors:
\[
(x - 1)^2 \quad \text{and} \quad (\ln x - 1)
\]

#### a. \( (x - 1)^2 \)
This is a squared term, so:
- \( (x - 1)^2 \geq 0 \) for all \( x \in \mathbb{R} \).
- The term is zero when \( x = 1 \), i.e., \( (x - 1)^2 = 0 \).

Thus, \( (x - 1)^2 \) is always non-negative and is zero only at \( x = 1 \).

#### b. \( \ln x - 1 \)
We need to analyze the sign of \( \ln x - 1 \), which depends on \( \ln x \).

- \( \ln x = 1 \) when \( x = e \) (since \( \ln e = 1 \)).
- For \( 0 < x < 1 \), \( \ln x \) is negative, so \( \ln x - 1 < 0 \).
- For \( 1 < x < e \), \( \ln x \) is positive but less than 1, so \( \ln x - 1 < 0 \).
- For \( x = e \), \( \ln x - 1 = 0 \).
- For \( x > e \), \( \ln x > 1 \), so \( \ln x - 1 > 0 \).

Thus, the sign of \( \ln x - 1 \) is:
- Negative for \( 0 < x < e \).
- Zero at \( x = e \).
- Positive for \( x > e \).

### 2. Sign of the Product
The sign of the product \((x - 1)^2 (\ln x - 1)\) is determined by the signs of the two factors:

- For \( x = 1 \), the term \( (x - 1)^2 = 0 \), so the entire expression is zero.
- For \( 0 < x < 1 \), \( (x - 1)^2 > 0 \) and \( \ln x - 1 < 0 \), so the product is negative.
- For \( 1 < x < e \), \( (x - 1)^2 > 0 \) and \( \ln x - 1 < 0 \), so the product is negative.
- For \( x = e \), \( (x - 1)^2 > 0 \) and \( \ln x - 1 = 0 \), so the product is zero.
- For \( x > e \), \( (x - 1)^2 > 0 \) and \( \ln x - 1 > 0 \), so the product is positive.

### 3. Conclusion
The sign of \((x - 1)^2 (\ln x - 1)\) is:
- **Negative** for \( 0 < x < e \) (except \( x = 1 \)).
- **Zero** at \( x = 1 \) and \( x = e \).
- **Positive** for \( x > e \).

Would you like more details on any part of this analysis? Here are some related questions to deepen your understanding:

1. How do we differentiate functions involving \( \ln x \) and powers of \( x \)?
2. What is the significance of the natural logarithm function in real-world applications?
3. How would the sign change if the expression had been \( (x - 1)^3 (\ln x - 1) \)?
4. Can we find the derivative of the given expression to analyze its critical points?
5. How do different intervals of \( x \) affect the behavior of logarithmic expressions?

**Tip:** Always remember that squared terms are non-negative, which helps simplify analyzing the sign of expressions.

Explore how to determine the sign of the expression (x-1)^2(lnx-1) across different intervals of x. This problem covers logarithmic functions, algebraic expressions, and factor analysis, suitable for advanced high school students (Grades 11-12).

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