The following occurred to me while working on some research project, although it was not required, it seems interesting enough to post it for this audience. Anyhow, here it is. Let $$f(x)=\left(\frac{x}{e^x-1}\right)^2 + \left(\frac{x+1}{e^{x+1}+1}\right)^2.$$ Prove that $f(x)$ is a strictly decreasing function of $x$ over $\mathbb{R}$.

In fact, the statement holds true if $e$ is replaced by any real number $t>1$. Can one give a short and elegant proof of these statements?

The following occurred while working on some research project. Since the methods of proof I used were lengthy, I wish to see a skillful or insightful approach (perhaps even conceptual). Anyhow, here it is. Let $$f(x)=\left(\frac{x}{e^x-1}\right)^2 + \left(\frac{x+1}{e^{x+1}+1}\right)^2.$$ Can one give a short and elegant proof of these statements?

(1) $f(x)$ is a strictly decreasing function of $x$ over $\mathbb{R}$.

(2) In fact, the statement holds true if $e$ is replaced by any real number $t>1$.

The following occurred to me while working on some research project, although it was not required, it seems interesting enough to post it for this audience. Anyhow, here it is. Let $$f(x)=\left(\frac{x}{e^x-1}\right)^2 + \left(\frac{x+1}{e^{x+1}+1}\right)^2.$$ Prove that $f(x)$ is a strictly decreasing function of $x$ over $\mathbb{R}$.

In fact, the statement holds true if $e$ is replaced by any real number $t>1$.

The following occurred to me while working on some research project, although it was not required, it seems interesting enough to post it for this audience. Anyhow, here it is. Let $$f(x)=\left(\frac{x}{e^x-1}\right)^2 + \left(\frac{x+1}{e^{x+1}+1}\right)^2.$$ Prove that $f(x)$ is a strictly decreasing function of $x$ over $\mathbb{R}$.

In fact, the statement holds true if $e$ is replaced by any real number $t>1$. Can one give a short and elegant proof of these statements?