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What is the maximum natural number n, so that all polynomials of degree n are integrated exactly with the formula $$ \int_a^b f(t) \approx \frac {b-a} {4} (f(a) +3f(\frac {a+2b} {3})) $$

user123763
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    What have you tried? Did you check the fomula for $t, t^2, t^3$? Note, that by linearity if the formula is true for $1, t, ..., t^n$ then it is true for any polynomial of degree $n$ – user68061 Jan 25 '14 at 10:26

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First prove that translation and scaling does not affect the formula. Thus without loss of generality put $(a,b)=(0,1)$ and $f(0)=0$. This is not necessary but makes the rest easy. Next, to find an upper bound on $n$, try monomials $x$,$x^2$,$x^3$,... until one of them fails. Finally, you can substitute in the highest degree polynomial that is possible and check, or as user68061 said check that the formula is linear; if it works for $f$,$g$ it also works for $f+g$.

user21820
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