微积分(A)随缘一题[29]

设 \(f(x) \in C[0,\pi]\)，且 \(\int_0^\pi f(x)dx=0,\int_0^\pi f(x)\cos xdx=0\)

求证：\(\exists \zeta_1,\zeta_2 \in (0,\pi),\zeta_1 \ne \zeta_2,s.t.f(\zeta_1)=f(\zeta_2)=0\)

设 \(F(x)=\int_{0}^xf(t)dt\)，则 \(F(0)=F(\pi)=0\)

考虑到：\(0=\int_{0}^{\pi}f(x)\cos xdx=\int_0^\pi \cos xdF(x)=F(x)\cos x \bigg|_0^\pi+\int_0^\pi F(x)\sin xdx=\int_0^\pi F(x)\sin xdx\)

所以 \(\int_0^\pi F(x)\sin xdx=0 \Rightarrow \exists \zeta \in (0,\pi),s.t.F(\zeta)\sin \zeta=0 \Rightarrow F(\zeta)=0\)

所以 \(F(\zeta)-F(0)=F(\pi)-F(\zeta)=0\)

所以 \(\exists \zeta_1 \in (0,\zeta),\zeta_2 \in (\zeta,\pi),s.t. f(\zeta_1)=f(\zeta_2)=0\)