Math Problem Statement
study the sign of (x-1)^2(lnx-1)
Solution
To study the sign of the expression , 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:
a.
This is a squared term, so:
- for all .
- The term is zero when , i.e., .
Thus, is always non-negative and is zero only at .
b.
We need to analyze the sign of , which depends on .
- when (since ).
- For , is negative, so .
- For , is positive but less than 1, so .
- For , .
- For , , so .
Thus, the sign of is:
- Negative for .
- Zero at .
- Positive for .
2. Sign of the Product
The sign of the product is determined by the signs of the two factors:
- For , the term , so the entire expression is zero.
- For , and , so the product is negative.
- For , and , so the product is negative.
- For , and , so the product is zero.
- For , and , so the product is positive.
3. Conclusion
The sign of is:
- Negative for (except ).
- Zero at and .
- Positive for .
Would you like more details on any part of this analysis? Here are some related questions to deepen your understanding:
- How do we differentiate functions involving and powers of ?
- What is the significance of the natural logarithm function in real-world applications?
- How would the sign change if the expression had been ?
- Can we find the derivative of the given expression to analyze its critical points?
- How do different intervals of affect the behavior of logarithmic expressions?
Tip: Always remember that squared terms are non-negative, which helps simplify analyzing the sign of expressions.
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Math Problem Analysis
Mathematical Concepts
Logarithmic Functions
Algebraic Expressions
Factor Analysis
Sign Analysis
Formulas
(x - 1)^2
ln x - 1
Theorems
Properties of Logarithmic Functions
Properties of Squared Terms
Suitable Grade Level
Grades 11-12
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