Find all monotonic solutions on $\mathbb{R}_{+}^{\times}$ that do not vanish, of the functional equation

$$
f(x+y)=\frac{f(x) f(y)}{f(x)+f(y)}
$$
Suppose that $f$ is a solution, then we verify that $g=1 / f$ is a solution to the Cauchy equation. Now, $f$ is monotone if and only if $g$ is monotone (since $x \mapsto$ $1 / x$ is strictly decreasing on the set of strictly positive real numbers). Finally, the only possible solutions are

$$
f(x)=c / x
$$

where $c$ is a real constant.