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Let $f$ be a function such that $f'''$ is continuous and $f(0)=f'(0)=f'(1)=0$. Prove that there exists $c \in (0,1)$ such that

$$f(1)=-\frac{1}{12} f'''(c)$$

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    What is $F$? Or you mean $f$?2012-10-30
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    He means $F(x) = \int_0^x f(t)dt$.2012-10-30
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    @nullUser: I think it is more likely he just means $f$ instead of $F$, but maybe he'll clarify himself.2012-10-30
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    In this form, it cannot be true, since if you add $3x^2-2x^3$ to $f$, the initial conditions are preserved, the $LHS$ increases by $1/2$ and the $RHS$ increases by $1$.2012-10-30
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    it is $f$, I have edited2012-10-31

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