Prove that if a, 6, and c are positive real numbers with ab = c, then a < yc or 6 < Vc.
Prove that if a, 6, and c are positive real numbers with ab = c, then a < yc or 6 < Vc.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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can someone answer this asap, please
![**Prove that if \(a\), \(b\), and \(c\) are positive real numbers with \(ab = c\), then \(a \leq \sqrt{c}\) or \(b \leq \sqrt{c}\).**
To prove this, let's analyze the given condition \(ab = c\).
1. Consider the contrapositive: suppose both \(a > \sqrt{c}\) and \(b > \sqrt{c}\).
2. Then we have:
\[
a > \sqrt{c} \quad \text{and} \quad b > \sqrt{c}
\]
3. Multiplying these inequalities gives:
\[
ab > \sqrt{c} \cdot \sqrt{c} = c
\]
4. However, this contradicts the given condition \(ab = c\).
5. Therefore, it must be the case that \(a \leq \sqrt{c}\) or \(b \leq \sqrt{c}\).
This proves the required inequality.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba208968-0ccc-4a94-8921-5ea260cefe19%2Fd421eb7e-ee28-403e-bff9-23787d819f25%2Fro9volg_processed.png&w=3840&q=75)
Transcribed Image Text:**Prove that if \(a\), \(b\), and \(c\) are positive real numbers with \(ab = c\), then \(a \leq \sqrt{c}\) or \(b \leq \sqrt{c}\).**
To prove this, let's analyze the given condition \(ab = c\).
1. Consider the contrapositive: suppose both \(a > \sqrt{c}\) and \(b > \sqrt{c}\).
2. Then we have:
\[
a > \sqrt{c} \quad \text{and} \quad b > \sqrt{c}
\]
3. Multiplying these inequalities gives:
\[
ab > \sqrt{c} \cdot \sqrt{c} = c
\]
4. However, this contradicts the given condition \(ab = c\).
5. Therefore, it must be the case that \(a \leq \sqrt{c}\) or \(b \leq \sqrt{c}\).
This proves the required inequality.
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