Differentiable functions/Mean value theorem/Fact/Proof

We consider the auxiliary function

${\displaystyle g\colon [a,b]\longrightarrow \mathbb {R} ,x\longmapsto g(x):=f(x)-{\frac {f(b)-f(a)}{b-a}}(x-a).}$

This function is also continuous and differentiable in ${\displaystyle {}]a,b[}$. Moreover, we have ${\displaystyle {}g(a)=f(a)}$ and

${\displaystyle {}g(b)=f(b)-(f(b)-f(a))=f(a)\,.}$

Hence, ${\displaystyle {}g}$ fulfills the conditions of fact, and therefore there exists some ${\displaystyle {}c\in {]a,b[}}$, such that ${\displaystyle {}g'(c)=0}$. Because of the rules for derivatives, we obtain

${\displaystyle {}f'(c)={\frac {f(b)-f(a)}{b-a}}\,.}$