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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a2a6d3dd50759e2a4a3918e9fe204b862f6b08c)
This function is also
continuous
and
differentiable
in ![{\displaystyle {}]a,b[}](https://wikimedia.org/api/rest_v1/media/math/render/svg/270d5e0116fb92c1667fca123c454c114a042fd9)
. Moreover, we have
and
-
Hence, 
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
-
