# maximisation

• October 21st 2010, 07:22 AM
trout
maximisation
I have a question which I thought was simple enough, but have somehow got lost for quite a while on it. I need to maximise the following equation with respect to the variable x. a and b are constants.

$max \ f(x) = \frac{a}{x}\left[e^{-b}\left(1+\frac{1}{b}\right)-e^{-b x}\left(x+\frac{1}{b}\right)\right]$

Making use of the quotient rule, I got the partial derivative with respect to x to be:

$
\frac{\delta f(x)}{\delta x}
= -\frac{a e^{-b}}{x^2}\left(1+\frac{1}{b}\right)+\left(\frac{x b^2(a e^{-b x})+ba e^{-bx}}{{x^2 b^2}}\right)+b e^{-b x} =0
$

Now solving for x, I thought, would be the easy part. But I've spent quite a while now just lost in the algebra. Is there any tips or hints with regards to getting the answer out?
• October 21st 2010, 08:40 AM
BobP
Do you have any likely values for a and b ?
I chose arbitrarily a=1, b=2 and plotted the graph of f(x).
The graph has a vertical asymptote at x=0 and goes to zero (from below) as x goes to infinity.
You could see what happens for other values (of a and b).
• October 21st 2010, 08:51 AM
trout
Thanks. Sorry, I should have stated likely parameter values in my post. Plausible values are a=10000, and b=0.05. So for that example, the max is at x=36.