- #1

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## Homework Statement

With

[tex]f(x)=-\lgroup\frac{a}{2}-\frac{x}{2b}\rgroup\ln\lgroup\frac{a}{2}-\frac{x}{2b}\rgroup-\lgroup\frac{a}{2}+\frac{x}{2b}\rgroup\ln\lgroup \frac{a}{2}+\frac{x}{2b}\rgroup[/tex]

and

[tex]z=\frac{df}{dx}=\frac{1}{c}[/tex]

find an expression of x in terms of c.

## Homework Equations

Well, relevant should be that the answer is supposed to be

[tex]x=ab \tanh\lgroup\frac{b}{c}\rgroup[/tex]

## The Attempt at a Solution

Differentiating f wrt x, I get

[tex]\frac{df}{dx}=-\frac{1}{2b}\lgroup \ln\lgroup\frac{a}{2}-\frac{x}{2b}\rgroup+\ln \lgroup\frac{a}{2}+\frac{x}{2b}\rgroup+2\rgroup[/tex]

But this is no good, because when I exponentiate on both sides of the equation (in order to solve for x)

[tex]-\frac{2b}{c}-2=-2b\frac{df}{dx}-2[/tex]

I end up with a something like

[tex]x=2b\sqrt\lgroup -\exp(-2\frac{b}{c}-2)-(\frac{a}{2})^2\rgroup,[/tex]

which is obviously incorrect.

I'd be happy if someone would help me.