Mean Value Theorem

Definition

Mean Value Theorem

If $f:[a,b]\to \mathbb{R}$ is continuous on the $[a,b]$, and differentiable on the $(a,b)$, then $\exists c\in (a,b),f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$

Cauchy’s Mean Value Theorem

If $f,g:[a,b]\to \mathbb{R}$ is continuous on the $[a,b]$, and differentiable on the $(a,b)$, then $\exists c\in (a,b),(f(b)-f(a))g^{\prime }(c)=(g(b)-g(a))f^{\prime }(c)$, or $\exists c\in (a,b),\frac{g^{\prime }(c)}{f^{\prime }(c)}=\frac{g(b)-g(a)}{f(b)-f(a)}$ if $f(a)\ne f(b)\wedge f^{\prime }(c)\ne 0$

Mean Value Theorem for Definite Integrals

If $f:[a,b]\to \mathbb{R}$ is continuous on the $[a,b]$, then $\exists c\in (a,b),{\int }_a^{b}f(x)dx=f(c)(b-a)$