Cho $f,g: [a,b] \to R$ liên tục:
a.Nếu $f,g$  đều là hàm tăng.Chứng minh rằng:
$\frac{1}{b-a}\int\limits_{a}^{b}f(x)g(x)dx \geq  \frac{1}{b-a}\int\limits_{a}^{b}f(x) dx. \frac{1}{b-a}\int\limits_{a}^{b}g(x)dx $ 
b.Nếu $f$ hàm  tăng, $g$ hàm giảm. Chứng minh rằng:
$ \frac{1}{b-a}\int\limits_{a}^{b}f(x)g(x)dx \le \frac{1}{b-a}\int\limits_{a}^{b}f(x) dx. \frac{1}{b-a}\int\limits_{a}^{b}g(x)dx $ 
a.Với $\forall x\in [a,b]\Rightarrow f(a)\leq f(x) \leq f(b)$
$\Rightarrow \int\limits_{a}^{b}f(a) dx \leq \int\limits_{a}^{b}f(x) dx \leq \int\limits_{a}^{b}f(b) dx$
$\Rightarrow (b-a)f(a)\leq \int\limits_{a}^{b}f(x) dx \leq (b-a)f(b)$
$\Rightarrow f(a) \leq \frac{1}{b-a}\int\limits_{a}^{b}f(x) dx\leq f(b)$
$\Rightarrow $ Theo định lý về giá trị trung gian của hàm số liên tục:
$\exists  x_{0}\in [a,b]$ sao cho:
$f(  x_{0})=\frac{1}{b-a}\int\limits_{a}^{b}f(x) dx$
Hơn nữa: $f,g$ đồng biến trên $[a,b]$,suy ra:
$[f(x)-f(x_{0})][g(x)-g(x_{0})]\geq 0,\forall x \in [a,b]$
$\Rightarrow f(x)g(x)-f(x_{0})g(x)-f(x)g(x_{0})+f(x_{0})g(x_{0})\geq 0$
$\int\limits_{a}^{b}f(x)g(x)dx-f(x_{0})\int\limits_{a}^{b}g(x)dx-g(x_{0})\int\limits_{a}^{b}f(x)dx+(b-a)f(x_{0})g(x_{0})\geq 0$
$\Rightarrow \int\limits_{a}^{b}f(x)g(x)dx \geq f(x_{0})\int\limits_{a}^{b}g(x)dx+g(x_{0})\int\limits_{a}^{b}f(x)dx-(b-a)f(x_{0})g(x_{0})$
$\geq \frac{1}{b-a}\int\limits_{a}^{b}f(x)dx.\int\limits_{a}^{b}g(x)dx+g(x_{0})(b-a)f(x_{0})-(b-a)f(x_{0})g(x_{0})$
$\geq \frac{1}{b-a}\int\limits_{a}^{b}f(x)dx.\int\limits_{a}^{b}g(x)dx$
$\Rightarrow \frac{1}{b-a}\int\limits_{a}^{b}f(x)g(x)dx \geq  \frac{1}{b-a}\int\limits_{a}^{b}f(x) dx. \frac{1}{b-a}\int\limits_{a}^{b}g(x)dx $
$\Rightarrow$ (ĐPCM)
b.Giả thiết suy ra: $f,(-g)$ đều là hàm tăng,nên theo câu a:
$\frac{1}{b-a}\int\limits_{a}^{b}f(x)[-g(x)]dx \geq  \frac{1}{b-a}\int\limits_{a}^{b}f(x) dx. \frac{1}{b-a}\int\limits_{a}^{b}[-g(x)]dx $ 
$\Rightarrow \frac{1}{b-a}\int\limits_{a}^{b}f(x)g(x)dx \le \frac{1}{b-a}\int\limits_{a}^{b}f(x) dx. \frac{1}{b-a}\int\limits_{a}^{b}g(x)dx $ 
$\Rightarrow$ (ĐPCM)

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