Polynomial degrees are back again

Calculus Level 5

limtaatf(x) dxta2(f(t)+f(a))(ta)3=0 \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)f(x) is a polynomial that satisfy the limit above for all aa, then the degree of f(x)f(x) can at most be?

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