Reciprocal Recedes, Power Proceeds

We first have to maximize the function, so we differentiate and equate the resulting expression to zero, because the function approaches a maximum or minimum value it's slope tends to zero. Differentiating the function twice we can get to know whether the function attains a maximum or minimum value.

$\begin{aligned} f'(x) = f(x) \times \ln\left(\frac 1x - 1\right)&=0\\ \ln\left(\frac 1x \right)&=1\\ x&=\frac 1e\\ \end{aligned}$

Hence the function attains a maximum value at this point. Therefore $\ln(\ln(e^{1/e} ))=-1$ .

$x$ = $\cfrac {1}{e}$