Determine whether the functional has weak or strong extrema?

Question

The functional $$J[y]=\int_{0}^{1}((y')^2+x^2)dx$$ where $y(0)=-1$ and $y(1)=1$ on $y=2x-1$, has

1. weak minimum
2. weak maximum
3. strong minimum
4. strong maximum

I have searched similar questions on this site, so I found that if $F_{y'y'}>0$ or $<0$ determines weak minimum or weak maxima. So in my question $F_{y'y'}=2>0$, so I have weak minimum, so option 1 is correct. Is my solution correct? If not, how to check the weak or strong extrema? Thank you.