Which answer best explains how the modern periodic table was derived? the sole work of mendeleev, the father of the modern periodic table a select group of scientific investigations that all had similar outcomes through federal funding so that the modern periodic table was quickly completed much scientific examination, investigation, and argumentation of data by many scientists?

I think the correct answer would be the last option. The statement that would best explain the derivation of the  modern periodic table would be that it was made from much scientific examination, investigation, and argumentation of data by many scientists. It was indeed finalized by Mendeleev however his works are based on the works of previous scientists who focused on the elements in the periodic table. So, it was not solely a product of effort of only one scientist but rather a number of scientists. Without the the previous researches, Mendeleev would not be able to come to the modern periodic table.