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169

Part of 2007 Today's Calculation Of Integral

Problems(1)

Today's calculation of Integral 169

Source: Osaka women university entrance exam 1985

1/15/2007
(1) Let f(x)f(x) be the differentiable and increasing function such that f(0)=0.f(0)=0.Prove that ∫01f(x)f′(x)dx≥12(∫01f(x)dx)2.\int_{0}^{1}f(x)f'(x)dx\geq \frac{1}{2}\left(\int_{0}^{1}f(x)dx\right)^{2}. (2) gn(x)=x2n+1+anx+bn (n=1, 2, 3, ⋯ )g_{n}(x)=x^{2n+1}+a_{n}x+b_{n}\ (n=1,\ 2,\ 3,\ \cdots) satisfies ∫−11(px+q)gn(x)dx=0\int_{-1}^{1}(px+q)g_{n}(x)dx=0 for all linear equations px+q.px+q. Find an, bn.a_{n},\ b_{n}.
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