Check whether the extremal has weak minima or weak maxima

Question

The functional $$\int_0^1(y'^2 + x^3)dx,$$ given $y(1)=1,$ achieves its

1. weak maximum on all its extremals

2. weak minimum on all its extremals

3. weak maximum on some, but not on all of its extremals

4. weak minimum on some, but not all of its extremals

To check weak minima or weak maxima we have to check whether $F_{y'y'}> 0$ or $<0$. Here , $F=y'^2+x^3\implies F_{y'}=2y'\implies F_{y'y'}=2>0$ for all $y'$. So the functional achieves strong maximum But this mismatch the given options. I can't understand my fault.

Please explain how can I check it?

Is there any other method to find it properly ?

Please help....

Answer 1

Since $F_{y'y'}=2>0$ for all $y'$ so , the functional attains strong minimum . So I think all the options are incorrect..