Simple Derivative plus tricky algebraic expression to simplify? I need to find the maximum of:
$$\frac{(1-e^{-\lambda \tau})}{\lambda \tau}-e^{-\lambda \tau}$$
apply quotient rule to the fraction term
$$\frac{(e^{-\lambda \tau}\lambda \tau-\tau(1-e^{-\lambda \tau}))}{(\lambda \tau)^2}+e^{-\lambda \tau}\tau$$
give common denominator
$$\frac{(e^{-\lambda \tau}\lambda \tau-\tau(1-e^{-\lambda \tau}))+e^{-\lambda \tau}\tau^3\lambda^2}{(\lambda \tau)^2}$$
Is this derivative correct?
I now need to set this equal to zero and solve for lambda!
$$(e^{-\lambda \tau}\lambda \tau-\tau(1-e^{-\lambda \tau}))+e^{-\lambda \tau}\tau^3\lambda^2=0$$
Can this be solved analytically?
Is so could I please have a hint as to how to approach the problem?
Baz
 A: $$\frac{(1-e^{-\lambda \tau})}{\lambda \tau}-e^{-\lambda \tau}$$
We see, the function depends on $\lambda\tau$. Set $\lambda\tau=x$ therefore.
$$\frac{(1-e^{-x})}{x}-e^{-x}$$
$$\curvearrowright x\neq0$$
Calculating the extrema:
$$\frac{d}{dx}\frac{(1-e^{-x})}{x}-e^{-x}\stackrel{!}{=}0$$
$$\frac{x^2e^{-x}+xe^{-x}+e^{-x}-1}{x^2}\stackrel{!}{=}0$$
$$x^2e^{-x}+xe^{-x}+e^{-x}-1\stackrel{!}{=}0$$
We see, the function on the left-hand side of the equation is an algebraic equation that depends on $x$ and $e^{-x}$. Both are algebraically independent of each other for all $x$ except $0$. Unfortunately, the equation cannot have therefore a solution that is an elementary function (see Wikipedia: Elementary function for the definition of the elementary functions).
A known special function for inverting functions of $x$ and $e^x$ is the Lambert W function. It is the inverse of $xe^x$. But the function in the equation cannot be represented in the form of equation (1) of my answer at Algebraic solution to natural logarithm equations like 1−x+xln(−x)=0. Unfortunately, the equation cannot be solved with help of Lambert W therefore.
If a function in dependence of a variable in polynomials and in exponential functions cannot be solved with Lambert W, some generalizations of Lambert W function are available. See the references in Wikipedia: Lambert W function - Generalizations as an entry. Ask if you need further literature references.
The numerical solution for the maximum is
$$\lambda\tau=1.793282132900...,\ f(\lambda\tau)=0.2984256075256...$$
$$\lambda=\tau=1.339134844928...,\ f(\lambda\tau)=0.2984256075256...$$
