Question

State Lagrange's mean value theorem.

Answer 1

Lagrange's Mean Value Theorem:

Let $f\left(x\right)$ be a function defined on $\left[a,b\right]$ such that
(i) it is continuous on ​$\left[a,b\right]$ and
(ii) it is differentiable on $\left(a,b\right)$.

Then, there exists a real number $c\in \left(a,b\right)$ such that $f\text{'}\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$.