Extremum Function of a Functional

Shantih

1. The problem statement, all variables and given/known data

J(f)=[itex]\int[/itex] 2xf−f′²+3f²f′dx
f(0)=0,f(1)=−1.

2. Relevant equations

F_f-[itex]\frac{d}{dx}[/itex]F_f'=0

3. The attempt at a solution

F_f=2x+6f f''
F_f'=-2f' + 6f²

Plugging in, I get:
2x+6f f''- [itex]\frac{d}{dx} (-2f' + 6f²)

2x+6f f''-12f f'-2f''=0

Which doesn't look correct to me.
I'm guessing my mistake was in the differentiation, but I don't see it.

Shantih

I figured it out, algebra errors.