For each real number x, x(1 − x) ≤
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please solve the following discrete math quesation please
![**Problem Statement:** Our friend, Kaden, offered the following “proof” for the given proposition. Find his error, and describe, in detail, why he’s wrong.
**Instructions:** Determine if the proposition is true or false. If it’s true, then write a formal proof of it, and if it’s false, then provide a counterexample that shows it is false.
**Proposition:** For each real number \( x \), \( x(1-x) \leq \frac{1}{4} \).
**Proof:** A proof by contradiction will be used. So we assume the proposition is false. This means that there exists a real number \( x \) such that \( x(1-x) > \frac{1}{4} \). If we multiply both sides of this inequality by 4, we obtain \( 4x(1-x) > 1 \). However, if we let \( x = 3 \), we then see that
\[
4x(1-x) > 1
\]
\[
4 \cdot 3(1-3) > 1
\]
\[
-12 > 1
\]
The last inequality is clearly a contradiction and so we have proved the proposition.
---
**Error Explanation:**
The error in the proof occurs when selecting \( x = 3 \). The value \( x = 3 \) is not in the domain of consideration for the inequality \( x \in \mathbb{R} \) where \( 0 \leq x \leq 1 \). The function \( x(1-x) \) is a quadratic function that achieves its maximum value at \( x = 0.5 \), which is \( 0.25 \) or \( \frac{1}{4} \).
The choice of \( x = 3 \) lies outside of this relevant domain, making the contradiction invalid. The proposition is true within the domain \( 0 \leq x \leq 1 \).
To properly prove the proposition, consider the domain and evaluate \( x(1-x) \) within this interval. The maximum of the quadratic function can be found by completing the square or using calculus to verify that the maximum value within the domain is indeed \( \frac{1}{4} \), occurring at \( x = 0.5 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe1065c73-a773-4439-b194-4cac2295760b%2F3ee291b4-4856-4242-a7c8-31e372d20ddf%2Fcjfo895_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:** Our friend, Kaden, offered the following “proof” for the given proposition. Find his error, and describe, in detail, why he’s wrong.
**Instructions:** Determine if the proposition is true or false. If it’s true, then write a formal proof of it, and if it’s false, then provide a counterexample that shows it is false.
**Proposition:** For each real number \( x \), \( x(1-x) \leq \frac{1}{4} \).
**Proof:** A proof by contradiction will be used. So we assume the proposition is false. This means that there exists a real number \( x \) such that \( x(1-x) > \frac{1}{4} \). If we multiply both sides of this inequality by 4, we obtain \( 4x(1-x) > 1 \). However, if we let \( x = 3 \), we then see that
\[
4x(1-x) > 1
\]
\[
4 \cdot 3(1-3) > 1
\]
\[
-12 > 1
\]
The last inequality is clearly a contradiction and so we have proved the proposition.
---
**Error Explanation:**
The error in the proof occurs when selecting \( x = 3 \). The value \( x = 3 \) is not in the domain of consideration for the inequality \( x \in \mathbb{R} \) where \( 0 \leq x \leq 1 \). The function \( x(1-x) \) is a quadratic function that achieves its maximum value at \( x = 0.5 \), which is \( 0.25 \) or \( \frac{1}{4} \).
The choice of \( x = 3 \) lies outside of this relevant domain, making the contradiction invalid. The proposition is true within the domain \( 0 \leq x \leq 1 \).
To properly prove the proposition, consider the domain and evaluate \( x(1-x) \) within this interval. The maximum of the quadratic function can be found by completing the square or using calculus to verify that the maximum value within the domain is indeed \( \frac{1}{4} \), occurring at \( x = 0.5 \).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 5 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
