The mean value theorem states the following:
If f ( x ) {\displaystyle \displaystyle f(x)} is continuous on [ a , b ] {\displaystyle \displaystyle [a,b]} and differentiable on ( a , b ) {\displaystyle \displaystyle (a,b)} , then there exists a point c ∈ [ a , b ] {\displaystyle \displaystyle c\in [a,b]} such that
f ′ ( c ) = f ( b ) − f ( a ) b − a {\displaystyle \displaystyle f'(c)={\frac {f(b)-f(a)}{b-a}}} .