Math Problem Statement

study the sign of (x-1)^2(lnx-1)

Solution

To study the sign of the expression (x1)2(lnx1)(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: (x1)2and(lnx1)(x - 1)^2 \quad \text{and} \quad (\ln x - 1)

a. (x1)2(x - 1)^2

This is a squared term, so:

  • (x1)20(x - 1)^2 \geq 0 for all xRx \in \mathbb{R}.
  • The term is zero when x=1x = 1, i.e., (x1)2=0(x - 1)^2 = 0.

Thus, (x1)2(x - 1)^2 is always non-negative and is zero only at x=1x = 1.

b. lnx1\ln x - 1

We need to analyze the sign of lnx1\ln x - 1, which depends on lnx\ln x.

  • lnx=1\ln x = 1 when x=ex = e (since lne=1\ln e = 1).
  • For 0<x<10 < x < 1, lnx\ln x is negative, so lnx1<0\ln x - 1 < 0.
  • For 1<x<e1 < x < e, lnx\ln x is positive but less than 1, so lnx1<0\ln x - 1 < 0.
  • For x=ex = e, lnx1=0\ln x - 1 = 0.
  • For x>ex > e, lnx>1\ln x > 1, so lnx1>0\ln x - 1 > 0.

Thus, the sign of lnx1\ln x - 1 is:

  • Negative for 0<x<e0 < x < e.
  • Zero at x=ex = e.
  • Positive for x>ex > e.

2. Sign of the Product

The sign of the product (x1)2(lnx1)(x - 1)^2 (\ln x - 1) is determined by the signs of the two factors:

  • For x=1x = 1, the term (x1)2=0(x - 1)^2 = 0, so the entire expression is zero.
  • For 0<x<10 < x < 1, (x1)2>0(x - 1)^2 > 0 and lnx1<0\ln x - 1 < 0, so the product is negative.
  • For 1<x<e1 < x < e, (x1)2>0(x - 1)^2 > 0 and lnx1<0\ln x - 1 < 0, so the product is negative.
  • For x=ex = e, (x1)2>0(x - 1)^2 > 0 and lnx1=0\ln x - 1 = 0, so the product is zero.
  • For x>ex > e, (x1)2>0(x - 1)^2 > 0 and lnx1>0\ln x - 1 > 0, so the product is positive.

3. Conclusion

The sign of (x1)2(lnx1)(x - 1)^2 (\ln x - 1) is:

  • Negative for 0<x<e0 < x < e (except x=1x = 1).
  • Zero at x=1x = 1 and x=ex = e.
  • Positive for x>ex > 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 lnx\ln x and powers of xx?
  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 (x1)3(lnx1)(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 xx 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