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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.

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    The Legendre condition you found is a sufficient condition for a strong minimum. Additionally, you may notice that the functional $J[y]$ is defined on a convex set and is strictly convex. So, it's not only a strong minimum but also a unique global minimum. – 2017-02-14

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