Prove/Disprove: Inequality for x ≥ 0

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Discussion Overview

The discussion centers around the inequality \( (1+x)^{1+x} \geq 1+x+x^2 \) for \( x \geq 0 \). Participants explore methods of proof, including derivative comparisons and Taylor series expansions, while seeking to establish the validity of the inequality.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the truth of the inequality and suggests comparing derivatives as a method of analysis.
  • Another participant mentions having a proof that utilizes the first two derivatives and references the Taylor expansion of \( (x+1)^{x+1} \), expressing interest in alternative proofs.
  • A different participant expresses confidence that the method involving derivatives is the best approach to prove the inequality.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the truth of the inequality, and multiple approaches to proving or disproving it are discussed without resolution.

Contextual Notes

Some assumptions about the behavior of the functions involved may be missing, and the discussion does not clarify the conditions under which the derivatives are compared or the validity of the Taylor expansion used.

acarchau
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Prove/disprove the following:
For [itex]x \geq 0[/itex], [itex](1+x)^{1+x} \geq 1+x+x^2[/itex].
 
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Well, first of all, do you think it's true?

Second, compare derivatives.
 
adriank said:
Well, first of all, do you think it's true?

I have a proof which uses the first two derivatives, btw the polynomial consists of the first few three terms of the taylor expansion of [itex](x+1)^{x+1}[/itex]. I was curious if someone could come out with a better proof.
 
Well, I think that's the best way to prove the inequality then.
 

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