Integral related to asymmetric smooth-step functions – mathoverflow.net

When trying to design asymmetric smooth-step functions $f_{\lbrack 0,1\rbrack}(x)$ that for 0<a<1 satisfy: $f(0)=0,\ f'(0)=0,\kappa\lbrack f\rbrack(0)=0$ $\kappa\lbrack f\rbrack(a)=0$ ...