For example, consider the functional
Euler-Lagrange Equation is given by
Solving by integrating twice, we get , applying boundary conditions f(0)=0, f(1)=1, we get extrema as .
That is, if you put and hence, the functioal becomes
That is, if you take any other that satisfies the boundary condition, the functional value for that function will be definitely greater than . For example, suppose we choose , then , if , then , which is greater than 4.2. Similarly,
That is, functional takes minimum value for the function