Today's calculation of Integral 270
Source: 1994 Tohoku University entrance exam
January 18, 2008
calculusintegrationfunctiongeometryalgebrabinomial theoremcalculus computations
Problem Statement
Let be the continuous function at such that \int_0^1 x^kf(x)\ dx\equal{}0 for integers k\equal{}0,\ 1,\ \cdots ,\ n\minus{}1\ (n\geq 1).
(1) For all real numbers find the minimum value of g(t)\equal{}\int_0^1 |x\minus{}t|^n\ dx.
(2) Show the following equation for all real real numbers
\int_0^1 (x\minus{}t)^nf(x)\ dx\equal{}\int_0^1 x^nf(x)\ dx
(3) Let be the maximum value of the function for
Show that \left|\int_0^1 x^nf(x)\ dx\right|\leq \frac{M}{2^n(n\plus{}1)}