Differentiation of Real Functions

Mean Value Theorems

Lagrange Mean Value Theorem

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

Cauchy Mean Value Theorem

If $f,g:[a,b]\to \mathbb{R}$ are continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c\in (a,b)$ such that $(f(b)-f(a))g^{\prime }(c)=(g(b)-g(a))f^{\prime }(c).$