I am studying Higher Algebra by Hall and Knight and not much explanation is given on any article. So, I had some doubts on this article.
To find the greatest value of $a^mb^nc^p….$ when $a+b+c+…..$ is constant; $m,n,p…..$ being positive integers.
Since $m,n,p…..$ are constants the expression $a^mb^nc^p….$ will be greatest when $\left(\frac{a}{m}\right)^{m}\left(\frac{b}{n}\right)^{n}\left(\frac{c}{p}\right)^{p} \ldots$ is greatest.
(For complete article an image is attached)
Now, here I am not able to understand why $a^mb^nc^p….$ depends on $\left(\frac{a}{m}\right)^{m}\left(\frac{b}{n}\right)^{n}\left(\frac{c}{p}\right)^{p} \ldots$ for being the greatest?
Also, can anyone please explain how the author got $m^{n} n^{n} p^{p} \ldots\left(\frac{a+b+c+\ldots}{m+n+p+\ldots}\right)^{m+n+p+\ldots}$ as the greatest value?
Any help would be appreciated.
Best Answer
Because $ m^m n^n \ldots $ is a constant, so you can multiply/divide/add/subtract it and still retain the relative sizing.
AM-GM gives the equality condition easily. Did they talk about it before?