I am having a hard time getting my head around Functionals and Calculus of Variations,

My question is: Given a functional and using the Euler-Lagrange equation to find an extremal,how do we show thatthe extremal provides a min/max (if it does)

The question I am working on is

with

I found the extremal to be: which I am told is a minimum to the functional problem.

However I am unsure on what is sufficient to show this, in the notes I have it is shown that:

where f is continuously differentiable on the interval 0,1 with

Thanks in advance!