Find what expression is the greatest between $R\rm{e}^{-S\rm{e}^{-Tx}}$ and $x$

Question

I need to know what expression is the greatest between the following twos :

$R\textrm{e}^{-S\textrm{e}^{-Tx}}$ and $x$

$R$, $S$ and $T$ are strictly positive.

How can I determine it ? Thanks.

Answer 1

Because the exponential function is monotonically increasing and $-x$ is monotoniclly decreasing, $Re^{-Se^{-Tx}}$ is monotonically increasing. When $x=0, Re^{-Se^{-Tx}}=Re^{-S}.$ When $x \to \infty, Re^{-Se^{-Tx}}\to R$. Of course, $x$ in increasing as well and generally much more quickly. When $x=0$ it will be less than $Re^{-Se^{-Tx}}$ and by the time $x=R$ it will be greater. Given values for $R,S,T$ you could solve $x=Re^{-Se^{-Tx}}$ numerically to find the crossover point. An example with $R=4,S=3,T=2$ is shown below. The intersection is barely below $x=4$