It is well known that the problem to find a sharp constant in a Kolmogorov-type inequality for functions defined on the real axis, is equivalent to the extremal Kolmogorov problem to find the exact upper bound of the norm of an intermediate derivative of a function on the class of functions with restrictions on the norms of the function and its higher derivative. Despite a large number of works devoted to Kolmogorov-type inequalities, sharp constants for derivatives of arbitrary order are known only in a few cases. Therefore, the modification of the Kolmogorov problem considered by Boyanov and Naidyonov is interesting. In this modification, the norm of the intermediate derivative on the entire line is substituted by its norm on an arbitrary finite segment. In this chapter, the Boyanov-Naidyonov problem is solved on classes of functions with a given comparison function for norms of the positive and negative parts of the intermediate derivative of the function. In particular, this problem is solved on the Sobolev classes and on the spaces of trigonometric polynomials and polynomial splines. In addition, a solution to an analogue of the Erdös problem is obtained; we characterize a polynomial (spline) with a given uniform norm that has maximal possible total length of the arcs of the graph of its positive (negative) part on a given segment.