On Sharp Constants in Inequalities for the Modulus of a Derivative

For every $1\le r\le\infty$, we solve a Kolmogorov-type problem of describing all triples of numbers $\mu _0,\mu _1,\mu _2\ge 0$ for which there exists a function $f$ with an absolutely continuous derivative on the interval $[0,1]$ such that $\|f\|_{L_\infty (0,1)}=\mu _0$, $|f'(x)|=\mu _1$, and $\|f''\|_{L_r(0,1)}=\mu _2$, where $x$ is a fixed point in the interval $[0,1]$.