$\int_a^b xf(x)\mathrm{ d} x\leq 0$?
设 $\displaystyle f$ 在 $\displaystyle $ 上可积, 且有
$$\begin{aligned} \int_a^x f(t)\mathrm{ d} t\geq 0,\quad \int_a^b f(x)\mathrm{ d} x=0. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$
证明:$\displaystyle \int_a^b xf(x)\mathrm{ d} x\leq 0$. 下述证明有瑕疵. 因为没说 $\displaystyle f$ 连续 (才能保证 $\displaystyle F'(x)=f(x)$).
$$\begin{aligned} \int_a^b xf(x)\mathrm{ d} x=&\int_a^b x\mathrm{ d} F(x)\left(F(x)\equiv \int_a^x f(t)\mathrm{ d} t\geq 0\right)\\ =&\left.xF(x)\right|_a^b -\int_a^bF(x)\mathrm{ d} x\\ =&-\int_a^b F(x)\mathrm{ d} x\leq 0. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$
怎么办呢?
[纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/(https://mp.weixin.qq.com/s/Pt6_h5MqtomrUDYiPEwkxg)/(https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / 设 $\displaystyle F(x)=\int_a^x f(t)\mathrm{ d} t$, 则 $\displaystyle F(x)\geq 0, F(a)=F(b)=0$, 而
$$\begin{aligned} &\int_a^b xf(x)\mathrm{ d} x=\lim_{n\to\infty}\sum_{k=1}^n \xi_k f(\xi_k)\Delta x_k\\ \xlongequal[\tiny\mbox{中值}]{\tiny\mbox{Lagrange}}& \lim_{n\to\infty}\sum_{k=1}^n \xi_k \left\\ =&\lim_{n\to\infty}\left\{\sum_{k=1}^n \xi_k F(x_k)-\sum_{k=0}^{n-1} \xi_{k+1} F(x_k)\right\}\\ \stackrel{F(a)=F(b)=0}{=}&\lim_{n\to\infty}\sum_{k=1}^{n-1}(\xi_k-\xi_{k+1})F(\xi_k)\leq 0. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$
跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/
第二积分中值定理，左右两边积分异号相等
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