Is there any way to solve df(t)/dt = 0 for t and f(tsolved) value if we know F(s), the Laplace transform of f?

Question

I am trying to know max(f(t)) value but I have only F(s) equation of it and I thought that by solving df(t)/dt == 0 by Laplace using F(s) which is s*F(s) == 0, I can easily solve what is t for max(f(t)) and using solved t I will find f(tsolved) value which will give me global max or min value.

Two questions: how can I solve df(t)/dt == 0 diff equation by F(s) and how can I get the value of f(tsolved) again from F(s)

Answer 1

There are at least two very different things one might mean by "Solve $f'(t)=0$"; it's not at all clear from the OP which one is intended:

Problem 1. "Given" $f$, find $t$ such that $f'(t)=0$.

Problem 2. Find $f$ such that $f'(t)=0$ for all $t$.

Indeed, it seems possible that the OP doesn't appreciate the distinction. The two tags ordinary-differential-equations and global-optimization would appear to indicate this, since the ode tag is relevant to Problem 2 but not to Problem 1, and vice-versa for the optimization tag.

Anyway, it seems very unlikely that the Laplace transform can be used for Problem 1. Otoh one could use the LT on Problem 2, but that would be a very curious thing to do, since the solution to Problem 2 is so obvious without the LT.

By the way, you have a detail wrong about how the Laplace transform plays with differential equations: Saying $f'==0$ does not say $sF(s)=0$, rather it says $sF(s)-f(0)=0$.