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E_2(f) is the error for integrating the lagrange interpolation polynomial for f

What i have problem understanding is in the paragraph below. "Unfortunately, (7.7) gives a considerable overestimate of the error in Simpson’s rule; in particular it does not bring out the fact that E2(f) = 0 whenever f is a polynomial of degree 3." If E_2(f) is defined for polynomial of degree 2 how does this make sense. Can somebody explain what the author meant by this. It is from the book "Endre S¨uli and David Mayers, An introduction to numerical analysis.".

Blue Apricot
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    I don't understand what your difficulty is with this. $E_2(f)$ is the error in using Simpson's rule for integrating $f$ on the interval $[a,b]$. $f$ doesn't have to be a polynomial. If $f$ does happen to be a polynomial of degree $3$ (or less), then $E_2(f)$ happens to be $0$. So in such a case the bound $(b-a)^4 M_3/196$ in (7.7) is an overestimate. What doesn't make sense to you? – Robert Israel Dec 06 '23 at 20:03

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Is your difficulty that Simpson's rule fits $f$ with a polynomial of degree $2$, while the error is $0$ for degree $3$ polynomials? The easiest way to see this is by taking the interval to be symmetric around $0$, say $[-1,1]$. Notice that the odd-degree terms cancel in both the actual integral and Simpson's rule. So the error for $c_3 x^3 + c_2 x^2 + c_1 x + c_0$ is the same as that for $c_2 x^2 + c_0$, and that error is $0$.

Robert Israel
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