Polynomial degrees are back again

Calculus Level 5

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

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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