The function $g(x)= (\sqrt{x^2 + k^2} - k)/2$ is a smooth approximation of $f(x)= \dfrac{|x|}{2}$. After a rotation of the plane of an angle $\alpha = \arctan(\tau) - \arctan(1/2)$, $f$ becomes $f_{\alpha}(x) = \tau x 1_{x\geq0} + (\tau-1) x 1_{x<0}$.
Do you know any smooth ($C^2$) approximation of $f_{\alpha}$ with $g'(0) = 0$ ? Thank you
I finally found the simple solution $g(x)= \sqrt{a^2 x^2 + b x + c} + dx$.