Find the constants $c_0, c_1,$ and $x_1$ so that the quadrature formula $$\int_0^1 f(x) dx = c_0 f(0) + c_1 f(x_1)$$ has the highest possible degree of precision.
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$x_1=1$, $c_0=c_1=1/2$. You do better! – Ted Shifrin Dec 15 '13 at 00:21
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@Amzoti yeah! edited – eChung00 Dec 15 '13 at 00:21
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By the mean value theorem for integrals, there exists some $\xi\in(0,1)$ such that $$f(\xi)=\int_0^1f(x)\,\mathrm d x.$$ Hence, choose $c_0=0$, $c_1=1$, and $x_1=\xi$ to obtain exact precision.
Of course, this doesn't really help much if you're supposed to find the value of $\xi$ because you don't know it in the first place.
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Presumably the answer is supposed to be independent of $f$ in a certain class...and computable. – Ted Shifrin Dec 15 '13 at 01:56