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$\large \lim_{t \to a} \frac {\int_a^t f(x) \ dx - \frac {t-a}{2} \left (f(t) + f(a) \right ) }{(t-a)^3} = 0$

If $f(x)$ is a polynomial that satisfy the limit above for all $a$, then the degree of $f(x)$ can at most be?

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