I am trying to solve a statistical analysis problem where I wish to optimise the sensitivity of an experiment.
For this I need to maximise $$y = \frac{1 - e^{-x/a}}{\sqrt{x}}$$.
Having plotted this function on Desmos I can see that a maximum should exist, but differentiating and setting $\frac{dy}{dx}= 0$ gives me $$ae^{x/a}=(x+a)$$ which I don't know how to solve.
Any advice on how to approach this would be appreciated.
We have that $$y=(1-e^{-x/a})x^{-1/2}\quad (x,a\gt0)$$ $$y'=\frac1ae^{-x/a}x^{-1/2}-\frac12(1-e^{-x/a})x^{-3/2}=0$$ $$\iff\left(\frac{x}ae^{-x/a}-\frac12(1-e^{-x/a})=0\right)\land(x\ne0)$$ $$\iff\left(\left(\frac{x}a+\frac12\right)e^{-x/a}=\frac12\right)\land(x\ne0)$$ $$\iff\left(\left(-\frac{x}a-\frac12\right)e^{-x/a}=-\frac12\right)\land(x\ne0)$$ $$\iff\left(\left(-\frac{x}a-\frac12\right)e^{-x/a-1/2}=-\frac12e^{-1/2}\right)\land(x\ne0)$$ $$\iff\left(-\frac{x}a-\frac12=W_k\left(-\frac12e^{-1/2}\right)\quad\forall k\in\mathbb{Z}\right)\land(x\ne0)$$ $$\iff\left(x=-a\left(\frac12+W_k\left(-\frac12e^{-1/2}\right)\right)\quad\forall k\in\mathbb{Z}\right)\land(x\ne0)$$ $$\iff x=-a\left(\frac12+W_k\left(-\frac12e^{-1/2}\right)\right)\quad\forall k\in\mathbb{Z}\setminus\{0\}$$ Where $W_k(z)$ denotes the $k$th branch of the Lambert W function. For $a\gt0$ the only real solution is $$x=-a\left(\frac12+W_{-1}\left(-\frac12e^{-1/2}\right)\right)$$ For example, when $a=1$ we get a local maxima when $$x=-\frac12-W_{-1}\left(-\frac12e^{-1/2}\right)\approx1.25643120862616\dots$$ (using Wolfram for example).