附: 几道高微习题
题1 已知\(f(x)''>0\,,\forall x\in[0\,,1]\). 求证：
\[
\forall \lambda>0\,,\int_0^1f(x^\lambda)dx\ge f(\frac{1}{\lambda+1})\,.
\]
题2 已知\(f(x)\)在\([0,1]\) 商可微，\(0<f'(x)<1\) , \(f(0)=0\). 求证：
\[
\left(\int_0^1f(x)dx\right)^2>\int_0^1f^3(x)dx\,.
\]
题3 \(f(x)\) 在\([0,1]\) 上二阶可导，\(f(0)=f(1)=0\), 且\(f(x)\ne 0\). 求证
\[
\int_0^1\left|\frac{f''(x)}{f(x)}\right|dx\ge 4\,.
\]
题4 已知\(f(x)\in C^{(1)}[0\,,1]\), \(f(x)\ge 0\,,f'(x)\le 0\,, F(x)=\int_0^xf(t)dt\,.\)求证
\[
xF(1)\le F(x)\le2\int_0^1F(t)dt\,.
\]
题5 \(f(x)\) 有周期\(T\), 且满足\(|f(x)-f(y)|\le L|x-y|\), \(\displaystyle{\int_0^Tf(x)dx= 0}\).求证
\[
|f(x)|\le\frac{1}{2}LT\,.
\]