We consider the auxiliary function
This function is also continuous and differentiable in ] a , b [ {\displaystyle {}]a,b[} . Moreover, we have g ( a ) = f ( a ) {\displaystyle {}g(a)=f(a)} and
Hence, g {\displaystyle {}g} fulfills the conditions of fact, and therefore there exists some c ∈ ] a , b [ {\displaystyle {}c\in {]a,b[}} , such that g ′ ( c ) = 0 {\displaystyle {}g'(c)=0} . Because of the rules for derivatives, we obtain
![{\displaystyle g\colon [a,b]\longrightarrow \mathbb {R} ,x\longmapsto g(x):=f(x)-{\frac {f(b)-f(a)}{b-a}}(x-a).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a2a6d3dd50759e2a4a3918e9fe204b862f6b08c)
![{\displaystyle {}]a,b[}](https://wikimedia.org/api/rest_v1/media/math/render/svg/270d5e0116fb92c1667fca123c454c114a042fd9)
. Moreover, we have
and
fulfills the conditions of
fact,
and therefore there exists some
![{\displaystyle {}c\in {]a,b[}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/893bc6fb00faaeb9f8bf828b68e80a64823b1d6d)
,
such that
.
Because of the rules for derivatives, we obtain
