so I have such functional: $$\phi(y)=\int\limits_0^1 (y^2+2y'^2+y''^2)dx, \ \ y(0)=y(1)=0, \ y'(0)=1, \ y'(1) = -\sinh1.$$
By using Euler-Lagrange formula, I get $$y^{IV} - 2y'' + y = 0$$ After solving I get: $$y = \sinh(x)(1-x).$$
And now, how to determine whether it's minimum or maximum, maybe local or not? I realize that I should check the second variation, but I saw there are some theorems. Can you give me simple steps how to check? Thank you.