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

$\displaystyle J(y) = \int_{0}^{1} ((y')^2 -y)dx$ with $\displaystyle y(0)=0, y(1)=1$

I found the extremal to be: $\displaystyle y(x) = \frac{-1}{4}x^2 +\frac{5}{4}x$ 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:

$\displaystyle J(y+f) = J(y) + \int_{0}^{1}(f')^2dx \geq J(y)$ where f is continuously differentiable on the interval 0,1 with $\displaystyle y(0)=y(1)=0$

Thanks in advance!