Hello, was wondering if you could help me with a question im stuck on...

The functional I is defined by

I[y] = integral from 1 to 0 of (1/2 y''^2) dx + (y(1))^2 + y'(1),

where y(0) = y'(0) = 0 and y belongs to C^4[0,1].

Show from first principles that a extremum y0(x) of I satisfies the euler equation y0'''' = 0

together with the transversality conditions y0''(1) = -1, y0'''(1) = 2y0(1).

show that y0= -1/10x^2(2+x).

Thanks : ) x