$ Suppose\ x,\ y\ are\ positive\ real\ numbers.\ Show\ that\ \left(x^2-y^2\ \right)\left(\frac{1}{y}-\frac{1}{x}\right)\ge 0 $

My attempt:

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I tried proving it using AM-GM inequality. Is my approach correct? If not, how can I prove it using AM-GM inequality?

  • $\begingroup$ The square of any real number is non-negative, so I don't see why you need to prove that $(x-y)^2 \ge 0\,$. $\endgroup$ – dxiv May 10 '17 at 20:16
  • $\begingroup$ So the work you went through using AM-GM is actually how the inequality is typically proved. $(x-y)^2$ is always a positive quantity, so you can just use this fact without appealing to AM-GM $\endgroup$ – helloworld112358 May 10 '17 at 20:17
  • $\begingroup$ I understand that it is obvious but I how do I formally explain my steps? $\endgroup$ – user444945 May 10 '17 at 20:18
  • $\begingroup$ At the level where you know and are allowed to use AM-GM, I presume you don't need a proof that $z^2 \ge 0\,$ for all $\,\forall z \in \mathbb{R}\,$. $\endgroup$ – dxiv May 10 '17 at 20:20
  • $\begingroup$ What about (x + y) / xy $\endgroup$ – user444945 May 10 '17 at 20:21

It's just $$\frac{(x-y)^2(x+y)}{xy}\geq0$$

  • $\begingroup$ That's what the OP wrote, but tried to prove $(x-y)^2 \ge 0$ with AM-GM. $\endgroup$ – dxiv May 10 '17 at 20:18
  • $\begingroup$ @dxiv I did not see it. Now I see that AM-GM here it's not necessary. $\endgroup$ – Michael Rozenberg May 10 '17 at 20:21
  • $\begingroup$ Could you start from the beginning and explain your steps? Thanks. $\endgroup$ – user444945 May 10 '17 at 20:22
  • $\begingroup$ @Josh Mitkitzel Now I see that you got it in your seventh line. $\endgroup$ – Michael Rozenberg May 10 '17 at 20:25
  • $\begingroup$ Do you think thats a good enough explanation? $\endgroup$ – user444945 May 10 '17 at 20:26

If $x = y$, the inequality holds.

If $y > x$, then $x^2 < y^2$ (hence $x^2 - y^2 < 0$) and $1/y < 1/x $ (hence $1/y - 1/x < 0$) and the inequality holds.

Do the same for $y < x$.

  • $\begingroup$ I don't quite understand. Could you show me some steps? $\endgroup$ – user444945 May 10 '17 at 20:30
  • $\begingroup$ @JoshMitkitzel You have a product of the numbers $x^2 - y^2$ and $1/y - 1/x$, and you must show that the product is not smaller than $0$. This will be true if both terms are $ > 0$ or both terms are $<0$ or at least one of the terms $= 0$. $\endgroup$ – Antoine May 10 '17 at 20:32

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