Proving Inequalities with Positive Real Numbers

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SUMMARY

The discussion focuses on proving the inequality \((c + \frac{1}{c})^2 + (d + \frac{1}{d})^2 \geq \frac{25}{2}\) for positive real numbers \(c\) and \(d\) such that \(c + d = 1\). Participants explore various methods, including expansion and the Binomial theorem, to tackle the problem. A key insight shared is that if \(c > d\) and \(c = 1 - d\), then it can be deduced that \(c > 0.5\), which aids in proving the inequality. The discussion also hints at a generalization for \(n\) variables.

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  • Understanding of basic algebraic inequalities
  • Familiarity with the Binomial theorem
  • Knowledge of properties of positive real numbers
  • Experience with mathematical proofs and generalizations
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  • Learn about generalizations of inequalities for multiple variables
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Mathematicians, students studying algebra, and anyone interested in advanced inequality proofs and their generalizations.

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1. c, d positive real numbers and c+d=1 prove that (c+1/c)2+(d+1/d)2>=25/2
2. Prove the previous also formulate and prove a generalization for n
3. I first tried to expand but got stuck, I thought of using Binomial but also got stuck. I tried to do (a+b)2 were a=(c+1/c) and b=(d+1/d) but also nothing
Any help is appreciated
 
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Well, if c is greater than d, and c=1-d, then 1-d > d. Add d to both sides and divide by 2 to get that .5>d. So you know that .5>d, and similar processes lead you to the fact that c>.5 . I have a feeling that this might help you, but I'm not sure.
 

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