$$\frac{d}{dx}\int\_{0}^{1}|log\_2(1+t)-(t+x)|\,dt=0$$
Is this solvable at all?



How to solve this equation for x?
The points where $\log\_2(1+t)-(t+x)$ changes sign can be expressed in terms of the Lambert W function, and then the integral can be evaluated explicitly
and differentiated. According to Maple, you want to solve
$$ \left( -2\,{\rm W}\_{-1} \left(-{2}^{x-1}\ln \left( 2 \right) \right)-2
 \right) {\rm W} \left(-{2}^{x-1}\ln \left( 2 \right) \right)+
 \left( 2\,x\ln \left( 2 \right) -\ln \left( 2 \right) -2\,\ln 
 \left( -{\rm W}\_{-1} \left(-{2}^{x-1}\ln \left( 2 \right) \right)
 \right) +2\,\ln \left( \ln \left( 2 \right) \right) +2 \right) 
{\rm W}\_{-1} \left(-{2}^{x-1}\ln \left( 2 \right) \right)+\ln \left( 2
 \right) =0
$$
There is unlikely to be a "closed-form" solution, but numerically a solution
is approximately $x = 0.06444652198$.


