Maximizing a Function with Constants a=10000 and b=0.05 | Homework Question

[trout_]

Homework Statement

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. Plausible values for the constants are a=10000, b=0.05.

Homework Equations

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

The Attempt at a Solution

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

\begin{equation}
\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
\end{equation}

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?As a side note, is latex code preferred when posting on these forums? Or is typing it out as you would in a text document preferred?

 

[LCKurtz]

Homework Statement

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. Plausible values for the constants are a=10000, b=0.05.

Homework Equations

$$\begin{equation}<br /> f(x) = \frac{a}{x}\left[e^{-b}\left(1+\frac{1}{b}\right)-e^{-b x}\left(x+\frac{1}{b}\right)\right]<br /> \end{equation}$$

The Attempt at a Solution

Making use of the quotient rule, I got the partial derivative with respect to x to be:
$$\begin{equation}<br /> \frac{\delta f(x)}{\delta x}<br /> = -\frac{a e^{-b}}{x^2}\left(1+\frac{1}{b}\right)+\left(\frac{x b^2(a e^{-b x})+ba e^{-bx}}{{x^2 b^2}}\right)+b e^{-b x} =0<br /> \end{equation}$$

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?As a side note, is latex code preferred when posting on these forums? Or is typing it out as you would in a text document preferred?

You just need tex tags as I have inserted for you here. (Click on the expressions to see the tex). You aren't going to be able to solve explicitly for x. If I get time I will plug it into Maple for you and see what I get.

[Edit - added] Your function has a vertical asymptote at x = 0 and appears to decrease to zero as x increases. No max for your values of a and b.