Given this function: $$ S(\tau) = (\tau^\eta − 2η\tau^{\eta−1} + \eta(\eta − 1)\tau^{\eta−2})e^{−\tau} $$ How can I find out what the maximum value of $S(\tau)$ is?
I need this value in order to get a normalized form of this wave shape. I could use root finding, but I'd prefer an analytical solution if possible.
PS: My math skills are weak, hopefully the question makes sense.
First we can begin the differentiation process by using the product rule: $$\frac{d(uv)}{dx} = u\frac{dv}{dx}+v\frac{du}{dx}$$
where $u = \tau^\eta − 2η\tau^{\eta−1} + \eta(\eta − 1)\tau^{\eta−2}$, and $v = e^{−\tau}$
Completing the differentiation and simplifying gets us to:
$$S'(\tau) = -{\tau}^{{\eta}-3}\left({\tau}^3-3{\eta}{\tau}^2+\left(3{\eta}^2-3{\eta}\right){\tau}-{\eta}^3+3{\eta}^2-2{\eta}\right)\mathrm{e}^{-{\tau}}$$
This isn't necessarily difficult, but it's $\textbf{very}$ tedious. If you want to see a step by step solution you can put your equation into this website: https://www.derivative-calculator.net, and then click on step-by-step solutions.
We can then solve $S'(\tau) = 0$ using the above solution by splitting it into $3$ separate equations:
$$e^{-\tau} = 0, \quad \tau^{\eta - 3} = 0, \quad {\tau}^3-3{\eta}{\tau}^2+\left(3{\eta}^2-3{\eta}\right){\tau}-{\eta}^3+3{\eta}^2-2{\eta}$$
We can now observe that the first equation will have no solution and the second equation has a unique solution at $\tau = 0$, if $\eta \geq 3$.
The third equation will take a little more effort. First we can collect like terms in $\tau$, giving us:
$$\tau^3 - 3\eta\tau^2 + (3\eta^2-3\eta)\tau - (\eta^3 - 3\eta^2 + 2\eta) = 0$$
Which can be solved using techniques that you would use to solve other cubic polynomials. Alternatively, there is a trick that can be applied here: By substituting $x =\tau - \eta$ we can eliminate the quadratic term and simplify, reducing the equation to:
$$x^3-3x\eta-2\eta = 0$$
This is still not trivial to solve but a trick I found that seems to work here is explained well with an example here: https://amsi.org.au/ESA_Senior_Years/SeniorTopic2/2e/2e_3links_3.html
Plugging all of this into an online calculator spits out a pretty ugly answer (which, albeit, does seem to have a nice solution at $\eta = 1, \tau = 3$).
If you follow this line of reasoning you will eventually reach an analytical solution but its extremely tedious and the answer you end up with is not pleasant. I suspect this question is intended to be answered using approximation techniques or with software.