Variational methods are mathematical techniques used to find the extrema (minimum or maximum values) of functionals, which are mappings from a set of functions to real numbers. These methods are widely applied in fields such as calculus of variations, physics, and engineering to solve problems involving optimization and differential equations. In variational methods, one typically seeks a function that minimizes or maximizes a given functional. This often involves using tools like Euler-Lagrange equations to derive necessary conditions for optimality. The approach is essential in areas like quantum mechanics and image processing, where it helps in formulating and solving complex problems.