Relationship between the extremes of a functional and its variation

Methods to solve variational problems, the tasks for the study for maximum and minimum of functionals are very similar methods to study the maximum and minimum functions. Therefore, appropriate to outline briefly the theory of maximum and minimum functions and in parallel we will introduce similar concepts and prove similar theorems for functional. The variation of functional is primary and linear with respect to a part of the increase of functional. In the study of functional, variation plays the same role played by the differential in the study of functions. We will show that if there is a variation in the basic sense of the linear increase of functional, then there is a variation in the derivative sense of the parameter with initial value, and that both definitions are equivalent and we will give another definition of functional variation. The above we will show in several examples and we will prove a theorem which is a link between the extremes of a functional and its variation