A maximal variance problem

In this work, we provide a direct proof concerning a special type of concave density function on a bounded closed interval with minimal variance. This proof involves elementary methods, without using any advanced theories such as Weierstrass’s Approximation Theorem, from which the technical core result of the paper [C. Carlsson, R. Fullér, P. Majlender, On possibilistic correlation, Fuzzy Sets and Systems 155 (2005) 425–445] comes.