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## Homework Statement

J(f)=[itex]\int[/itex] 2xf−f′

^{2}+3f

^{2}f′dx

f(0)=0,f(1)=−1.

## Homework Equations

F

_{f}-[itex]\frac{d}{dx}[/itex]F

_{f'}=0

## The Attempt at a Solution

F

_{f}=2x+6f f''

F

_{f'}=-2f' + 6f

^{2}

Plugging in, I get:

2x+6f f''- [itex]\frac{d}{dx} (-2f' + 6f

^{2})

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.