But.. What does all these mean? Can we visualise it?
Note that the range of the functional here is – hence, we can easily visualise it with GeoGebra!
First we need to find a family of functions which satisfies the boundary condition – is ideal (and contains the extrema of our interest for ).
Now, we can find F(x,f(x),f'(x)), putting – geogebra will do the needful. If needed, we can restrict the function F(x) to the region of integration by giving .
Now, the functional is – which can be given in GeoGebra as
Open the above GeoGebra link to see a sample. There are two controls – one drop down (there are two family of functions which satisfies boundary conditions – and – and a slider titled . Try varying the slider and function – see when functional has the least value!