Hi Gia huy Ho. I have a question here. How can you apply Cauchy's inequality when you haven't proved that 1/x is non-negative?

Question

https://vn.answers.yahoo.com/question/index?qid=20150814022033AARpoOb

Answer 1

The question asked for the maximum value of f(x) = x/(x^3 + 2). I am not sure whether it meant a global maximum or a local maximum. AM-GM gets you the global maximum for non-negative values of the variable. In this particular case the global and local maxima are the same for non-negative x. As it happens, x = 1 giving f(x) = 1/3 is not a global maximum for f(x), since f(-1.5) = 1.09 > 1/3. I don't think there is any substantive disagreement between Josh and Gia. The AM-GM inequality did make its appearance in Cauchy's Cours d'Analyse and, because of the seminal importance and influence of the Cours, was widely known at the time as Cauchy's inequality, especially in the non-English-speaking world. Josh is clearly thinking of what is now widely known as Cauchy-Schwarz, which applies to all real numbers, and he couldn't see how this was used to deduce your key step. Of course he couldn't see this, as Gia didn't use this result, but used instead AM-GM, which he called the Cauchy inequality. I think if the two of you want an argument, the argument should be whether AM-GM should be called the Cauchy inequality or the AM-GM inequality.

PS:

It was a bit dodgy, though, using AM - GM to find a local maximum to dodge having to use calculus, for you had to make the assumption that the local max was in the non-negative domain of the function in order to be able satisfy the requirements for AM - GM

Answer 2

You can't; 1/x could certainly be negative, in which case the step



x^2 + 1/x + 1/x >= 1/3



may fail. Indeed, the LHS is zero at -2^(1/3) [since one has x^3 = -2 there], and is negative between 0 and there.



In any case, Cauchy-Schwarz doesn't apply here. Or more accurately, it's not at all clear how you'd want to apply it in a way to get rid of the x's, and I don't see it. As far as I can tell, the accepted answer you linked just got lucky that the max occurs at x=1 and used otherwise incorrect reasoning to justify a correct answer. Melvyn's answer is essentially correct though and probably should have been the accepted answer.

Answer 3

No, Josh Swanson, i mean AM-GM inequality, not Cauchy-Schwarz inequality. In Vietnam, we call it Cauchy's inequality, which is found in his Cours d'analyse. Cauchy's inequality is equivalent to AM-GM inequality, but Cauchy-Schwarz is different. AM-GM inequality's condition is that the list of the number should be NON-negative numbers. So, x^2 and 1/x should be non-negative number. Sorry, i make you guys confused this problem.