Can the Inequality x^x + y^y < (x+y)^(x+y) be Proven Algebraically?

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Discussion Overview

The discussion revolves around the inequality x^x + y^y < (x+y)^(x+y) for x, y ≥ 1, exploring whether it can be proven algebraically. Participants consider various approaches, including calculus and algebraic expansion.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the universality of the inequality for all x, y ≥ 1 and suggests using derivatives to analyze the slopes of both sides of the inequality.
  • Another participant proposes expanding the right side of the inequality, asserting that the terms will be greater than x^x and y^y due to the positivity of the remaining terms when x, y > 1.
  • A later reply emphasizes a preference for an algebraic solution over graphical methods, asserting that the problem can be solved algebraically and is not a transcendental equation.

Areas of Agreement / Disagreement

Participants express differing views on the methods to approach the problem, with no consensus on a definitive algebraic proof or the applicability of the inequality across all specified values of x and y.

Contextual Notes

Some assumptions regarding the parameters of x and y are not fully explored, and the discussion includes various methods without resolving the mathematical steps involved in proving the inequality.

bill01
Is it possible to prove this:
x^x + y^y < (x+y)^(x+y) for every x,y >=1 ?
 
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Well, let me think, since I am not really sure if this would work for all parameters of x and y... never mind, you have x,y >1! What you could do is take the derivative of both equations to measure it's change in slope, and if (x+y)^(x+y)is greater, then it will have a change in slope that is greater then the other equation. But I am not sure if that is what you want.
 
Iam just proving it,just expand the Right side (x+y)^(x+y), u get x multiplied by x+y times which is obviously greater than x^x and same in case of y and remaining terms of expansion are positive as x,y>1 and no negative terms in expansion. hope it helps.
 
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Thanks for the answers, but I would prefer an algebraic solution.
I did what Raul said with the graph but I would like an algebraic sol.
I believe that it is solved algebraically and it is not a transcendental equation.
 
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