Best approximations in $L_[0,infty)$ of the differentiation operator

A solution of Stechkin's problem concerning the approximation in $L_[0,infty)$ of the first-order differentiation operator in the class of functions of arbitrary bounded variation; the exact constant in the inequality $\|f'\|\leqslant K(\|f\|\bigvee\limits_0^\infty f')^{1/2}$ is found.