I am working on the question below and I am getting stuck.
Consider the surge function $y=axe^{-bx}$ with $a$ and $b$ positive constants.
(a) Find the local maxima, local minima, and inflection points.
(b) How does varying $a$ and $b$ affect the shape of the graph?
(c) On one set of axes, sketch the graph of this function for a few values of $a$ and $b$.
I've played around with the function on Wolfram Alpha in order to get a feel for how different values of $a$ and $b$ affect the graph. I also see that the local maximum seems always to be at $1/b$. I cant seem to figure out how to show this result using first and second derivatives, etc.
I've found the first derivative to be: $$y'=ae^{-bx}(1-b)$$
Best Answer
From David Mitra:
$$y^\prime =ae^{−bx}−baxe^{−bx}=ae^{−bx}(1−bx)$$
From NehriMattisse:
So $(1−bx)=0$, which leads to $x=1/b$.