monotonicity of a univariate function

Question

I am considering the function $$ f(x) = (a + e^x)^{-b} + (a + e^{-x})^{-b} $$ where $a, b \in (0, \infty)$. What are the conditions on $a$ and $b$ that make $f(x)$ monotonic? Straightforward differentiation reduces it to showing that the transcendental equation $$ {a y + y^2 \over a y + 1} = y^{2 \over 1 + b} $$ has no solution on $(1, \infty)$. Any ideas? Thanks!

Answer 1

It's not necessarily monotone for small $a$. The only reason it appeared to be monotone is because when $a$ is sufficiently large, the $(a+e^{−x})^{−b}$ term is effectively constant.

It would be interesting to see if the exact value of $a$ for which $f$ on $(0,\infty)$ is monotone can be found.