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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: $$y'=ae^{-bx}(1-b)$$

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$y'=ae^{-bx}-baxe^{-bx} = ae^{-bx}(1-bx)$. –  David Mitra Nov 17 '11 at 8:27
    
Oops, yes I see that now, I had not factored it properly. So $(1-bx)=0$, which leads to $x=1/b$. –  NehriMattise Nov 17 '11 at 8:57
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1 Answer 1

From David Mitra:

$$y^\prime =ae^{−bx}−baxe^{−bx}=ae^{−bx}(1−bx)$$

From NehriMattisse:

So $(1−bx)=0$, which leads to $x=1/b$.

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