Calculus of Variations

Often in many problems which are encountered in practice one needs to find a minimum (or a maximum) of a function f(x) with respect to its variable x, i.e. the point \(x_{0}\) where the value of the function is the smallest (largest) within a certain interval \(a\le x\le b\) . The necessary condition for this to happen is for the point \(x_{0}\) to satisfy the equation \(f^{\prime }\left( x_{0}\right) =0\) . However, sometimes one may need to solve a much more difficult problem of finding the whole function, e.g. f(x), which results in a minimum (or a maximum) of some scalar quantity L which directly depends on it.