Summary of the mean value theorem of higher numbers

The mean value theorem

Generally used for proof questions , Analysis steps ： Determine the interval 、 Determine auxiliary function 、 Determine the theorem used 、 Key point analysis .
f(x) stay [a,b] Continuous on ,

1. Bounded and maximum theorem ： $m\le f(x)\le M$
2. Intermediate value theorem ：$m\le \mu \le M,\exists ϵ\in [a,b],f(ϵ)=\mu$
3. Mean value theorem ：$a<x_1<x_2<\cdots <x_n<b,\exists ϵ\in [x_1,x_n],f(ϵ)=\frac{f(x_1)+f(x_2)+\cdots +f(x_n)}{n}$
4. Zero point theorem ：$f(a)\cdot f(b)<0,\exists ϵ\in [a,b],f(ϵ)=0$
5. Fermat's theorem ：$x_{0}It\text{'}s\; aboutcanguideAndbyextremelyvalue,f^{\prime }(x_0)=0$
6. Rolle's theorem ：$[a,b)canguide,f(a)=f(b),\exists ϵ\in [a,b],f^{\prime }(ϵ)=0$
7. Lagrange mean value theorem ：$(a,b)canguide,\exists ϵ\in [a,b],f(b)-f(a)=f^{\prime }(ϵ)(b-a)$
8. Cauchy mean value theorem ：$(a,b)canguide,g^{\prime }(x)\ne 0,\exists ϵ\in [a,b],\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f^{\prime }(ϵ)}{g^{\prime }(ϵ)}$
9. Taylor formula ( lagrange remainder )：$f(x)=f(x_0)+f^{\prime }(x_0)(x-x_0)+\cdots +\frac{f^{(n)}(x_0)}{n!}(x-x_0{)}^n+\frac{f^{(n+1)}(ϵ)}{(n+1)!}(x-x_0{)}^{n+1}$
10. Taylor formula ( Payano's remainder )：$f(x)=f(x_0)+f^{\prime }(x_0)(x-x_0)+\cdots +\frac{f^{(n)}(x_0)}{n!}(x-x_0{)}^n+O((x-x_0{)}^n)$
11. Mean value theorem of integral ：$\exists ϵ\in [a,b],\frac{{\int }_a^{b}f(x)dx}{b-a}=f(ϵ)$

There is a memory skill , The most commonly used are seven ,“ Zero dielectric ferroratesi ”, If you see the proof question, you can try it one by one ,