I'm trying to draw the graph of the function $f(x)=\frac{\ln(\sqrt{x^{2}+1})-1}{x}$ (from Demidovich's book problems in mathematical analysis). When I do the derivative, in order to find the extrema, I need to solve the equation $$\ln(\sqrt{x^{2}+1})-1=\frac{x^2}{x^2+1}$$ Using Geogebra i see that the curves intersect each other in some point $t\in (7,7.5)$ but I do not know how to get it.
I would be happy with any idea to solve it.
You can transform to $$ \frac{1+x^2}{e^2}=e^{-\frac2{1+x^2}} $$ Now bring the right side to the form $ue^u$ with $u=-\frac2{1+x^2}$ $$ -2e^{-2}=-\frac{2}{1+x^2}e^{-\frac2{1+x^2}} $$ By definition of the Lambert-W function, $u=W(v)$ solves $v=ue^u$. For $-e^{-1}<v<0$ there are two solutions, one in $(-\infty,-1)$ and one in $(-1,0)$. Obviously the first root is here $-2=u=-\frac{2}{1+x^2}$, which solves to $x=0$. The other solution is $u=W_0(-2/e^2)=-0.40637573995995996..$ and this can now be solved for $x$.